Giáo trình giải tích lồi (tài liệu tham khảo)

Such applications are: the study of well-conditioned convex functions, uniformly convex and uniformly smooth convex functions, best approximation prob- lems, characterizations of conve[r]

C o n v e x A n a l y s i s

G e n e r a l V e c t o r

S p a c e s C Zalinescu

A n a l y s i s i n G e n e r a l V e c t o r

C o n v e x A n a l y s i s

1 n

G e n e r a l V e c t o r

S p a c e s

C Zalinescu Faculty of Mathematics University "Al I Cuza" lasi, Romania

B% World Scientific

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Zalinescu, C ,

1952-Convex analysis in general vector spaces / C Zalinescu p cm

Includes bibliographical references and index ISBN 9812380671 (alk paper)

1 Convex functions Convex sets Functional analysis Vector spaces I Title

QA331.5.Z34 2002 2002069000 515'.8-dc21

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Preface

The text of this book has its origin in a course we delivered to students for Master Degree at the Faculty of Mathematics of the University "Al I Cuza" Ia§i, Romania

One can ask if another book on Convex Analysis is needed when there are many excellent books dedicated to this discipline like those written by R.T Rockafellar (1970), J Stoer and C Witzgall (1970), J.-B Hiriart-Urruty and C Lemarechal (1993), J.M Borwein and A Lewis (2000) for finite dimensional spaces and by P.-J Laurent (1972), I Ekeland and R Temam (1974), R.T Rockafellar (1974), A.D Ioffe and V.M Tikhomirov (1974), V Barbu and Th Precupanu (1978, 1986), J.R Giles (1982), R.R Phelps (1989, 1993), D Aze (1997) for infinite dimensional spaces

We think that such a book is necessary for taking into consideration new results concerning the validity of the formulas for conjugates and sub-differentials of convex functions constructed from other convex functions by operations which preserve convexity, results obtained in the last 10-15 years Also, there are classes of convex functions like uniformly convex, uniformly smooth, well behaving, well conditioned functions that are not studied in other books Characterizations of convex functions using other types of derivatives or subdifferentials than usual directional derivatives or Fenchel subdifferential are quite recent and deserve being included in a book All these themes are treated in this book

We have chosen for studying convex functions the framework of locally convex spaces and the most general conditions met in the literature; even when restricted to normed vector spaces many results are stated in more general conditions than the corresponding ones in other books To make

this possible, in the first chapter we introduce several interiority and closed-ness conditions and state two strong open mapping theorems

In the second chapter, besides the usual characterizations and properties of convex functions we study new classes of such functions: convex, cs-closed, cs-complete, lcs-cs-closed, ideally convex, bcs-complete and li-convex functions, respectively; note that the classes of li-convex and lcs-closed functions have very good stability properties This will give the possibility to have a rich calculus for the conjugate and the subdifferential of convex functions under mild conditions In obtaining these results we use the method of perturbation functions introduced by R.T Rockafellar The main tool is the fundamental duality formula which is stated under very general conditions by using open mapping theorems

The framework of the third chapter is that of infinite dimensional normed vector spaces Besides some classical results in convex analysis we give characterizations of convex functions using abstract subdifferentials and study differentiability of convex functions Also, we introduce and study well-conditioned convex functions, uniformly convex and uniformly smooth convex functions and their applications to the study of the geometry of Banach spaces In connection with well-conditioned functions we study the sets of weak sharp minima, well-behaved convex functions and global error bounds for convex inequality systems The chapter ends with the study of monotone operators by using convex functions

Every chapter ends with exercises and bibliographical notes; there are more than 80 exercises The statements of the exercises are generally ex-tracted from auxiliary results in recent articles, but some of them are known results that deserve being included in a textbook, but which not fit very well our aims The complete solutions of all exercises are given The book ends with an index of terms and a list of symbols and notations

Even if all the results with the exception of those in the first section are given with their complete proofs, for a successful reading of the book a good knowledge of topology and topological vector spaces is recommended Finally I would like to thank Prof J.-P Penot and Prof A Gopfert for reading the manuscript, for their remarks and encouragements

Contents

Preface vii Introduction xi

Chapter Preliminary Results on Functional Analysis

1.1 Preliminary notions and results 1.2 Closedness and interiority notions

1.3 Open mapping theorems 19 1.4 Variational principles 29

1.5 Exercises 34 1.6 Bibliographical notes 36

Chapter Convex Analysis in Locally Convex Spaces 39

2.1 Convex functions 39 2.2 Semi-continuity of convex functions 60

2.3 Conjugate functions 75 2.4 The subdifferential of a convex function 79

2.5 The general problem of convex programming 99

2.6 Perturbed problems 106 2.7 The fundamental duality formula 113

2.8 Formulas for conjugates and e-subdifferentials, duality relations

and optimality conditions 121 2.9 Convex optimization with constraints 136

2.10 A minimax theorem 143

2.11 Exercises 146 2.12 Bibliographical notes 155

Chapter Some Results and Applications of Convex

Analy-sis in N o r m e d Spaces 159

3.1 Further fundamental results in convex analysis 159 3.2 Convexity and monotonicity of subdifferentials 169 3.3 Some classes of functions of a real variable and differentiability

of convex functions 188 3.4 Well conditioned functions 195 3.5 Uniformly convex and uniformly smooth convex functions 203

3.6 Uniformly convex and uniformly smooth convex functions on

bounded sets 221 3.7 Applications to the geometry of normed spaces 226

3.8 Applications to the best approximation problem 237 3.9 Characterizations of convexity in terms of smoothness 243 3.10 Weak sharp minima, well-behaved functions and global error

bounds for convex inequalities 248 3.11 Monotone multifunctions 269

3.12 Exercises 288 3.13 Bibliographical notes 292

Exercises - Solutions 297

Bibliography 349 Index 359

Introduction

The primary aim of this book is to present the conjugate and subdifferential calculus using the method of perturbation functions in order to obtain the most general results in this field The secondary aim is to give important applications of this calculus and of the properties of convex functions Such applications are: the study of well-conditioned convex functions, uniformly convex and uniformly smooth convex functions, best approximation prob-lems, characterizations of convexity, the study of the sets of weak sharp minima, well-behaved functions and the existence of global error bounds for convex inequalities, as well as the study of monotone multifunctions by using convex functions

The method of perturbation functions is based on the "fundamental duality theorem" which says that under certain conditions one has

inf $ ( i , 0) = max_ ( - $*(0,y*)) (FDF)

For many problems in convex optimization one can associate a useful perturbation function We give here four examples; see [Rockafellar (1974)] for other interesting ones

Example (Convex programming; see Section 2.9) Let f,9i, -,9n '•

X ->• E be proper convex functions with d o m / n C\7=i domgi ^ The

problem of minimizing f(x) over the set of those x S X satisfying gi{x) < for a l i i = , , n is equivalent to the minimization of $(x, 0) for x X, where

- {

and Y := En; the element y* obtained from the right-hand side of (FDF)

will furnish the Lagrange multipliers

Example (Control problems) Let F : X xY -±Rbe a, proper convex

function and A : X —> Y a linear operator A control problem (in its abstract form) is to minimize F(x,y) for x € X and y = Ax + yo- The perturbation function to be considered is $ : X x Y —> M defined by $(x,y) := F{x,Ax + y0+y)

Example (Semi-infinite programming) We are as in Example but

{ , , n } is replaced by a general nonempty set I\ In this case Y = W and $(x,y) := f(x) if gt(x) < yi for all i £ / , $(x,y) := co otherwise

Formula (FDF), or more precisely the Fenchel-Rockafellar duality for-mula, can also be used for deriving results similar to that in the next ex-ample

Example ([Simons (1998b)]; see Exercise 2.37) Let X be a linear space,

(Y, ||-||) be a normed linear space, A : X —> Y be a linear operator, y0 £ Y

and / : X —> M be a proper convex function Then f(x) + \\Ax + yo\\ > 0 for all x € X if and only if there exists y* € Y* such that f(x) — (Tz + ?/o,2/*) - | | j / * | |2 > for a l l \ £ X

It is worth mentioning that adequate perturbation functions can be used for deriving formulas for the conjugate and e-subdifferential for many types of convex functions; this method is used by Rockafellar (1974) for foA and / l + • • • + fn, but we use it for almost all the operations which preserve convexity (see Section 2.8)

The formula (FDF) is automatically valid when infx €x $(a;,0) = - c o

and is equivalent to the subdifferentiability at £ Y of the marginal func-tion h : Y -> M, h(y) := mfxeX$(x,y), when i n fx ex $(#,0) £ E A

sufficient condition for this is the continuity of the restriction of h to the affine hull of its domain at 0; note that is in the relative algebraic interior of the domain of h in this case (without this condition one can give simple examples in which the subdifferential of h at is empty)

Considering the multifunction R : I x l = j whose graph is the set

grTl = {(x,t,y) | (x,y,t) £ e p i $ } , the continuity of / ^ ( d o m ^ ) at is

ensured if It is relatively open at some (a;0,io) with (a;o,io,0) € gr7? This

Robinson-Ursescu theorem The preceding examples show that the consideration of more general spaces is natural: In Example Y is a locally convex space while in Example X can be endowed with the topology a(X, X') The original result of Ursescu (1975) is stated in very general topological vec-tor spaces The inconvenient of Ursescu's theorem is that one asks the multifunction to be closed, condition which is quite strong in certain situ-ations For example, when calculating the conjugate or subdifferential of max(/, g) with / , g proper lower semicontinuous convex functions one has to evaluate conjugate or the subdifferential of • / + • g which is not lower semicontinuous convex Fortunately we dispose of another open mapping theorem in which the closedness condition is replaced by a weaker one, but one must pay for this by asking (slightly) more on the spaces involved

As said above, the second aim of the book is to give some interesting applications of conjugate and subdifferential calculus, less treated in other books

In many algorithms for the minimization problem (P) f(x), s.t x G

X, one obtains a sequence (xn) which is minimizing (i.e (f(xn)) -» inf / )

or stationary (i.e (da/(z„)(0)) —> 0) It is important to know if such a sequence converges to a solution of (P) Assuming that S := a r g m i n / :=

{x | f(x) = i n f / } 7^ 0, one says that / is well-conditioned if (ds(xn)) -»

whenever (xn) is a minimizing sequence, and / is well-behaved

(asymptot-ically) if (xn) is minimizing whenever (xn) is a stationary sequence; when

5 is a singleton conditioning reduces to the known notion of well-posedness in the sense of Tikhonov If / is well-conditioned with linear rate the set argmin / is a set of weak sharp minima When / is convex, we es-tablish several characterizations of well-conditioning using the conjugate or the subdifferential of / When is a singleton one of the characterizations is close to uniform convexity of / at a point

One says that the proper function / : (X, ||-||) ->• K is strongly convex if

f(\x + (1 - X)y) < Xf(x) + (1 - X)f(y) - f A(l - A) ||z - yf

for some c > and for all x, y € d o m / , A G [0,1] This notion is not very adequate for non-Hilbert spaces; for general normed spaces, one says that / is uniformly convex if there exists p : 1+ -+ E+ with p(0) = such that

f(Xx + (1 - X)y) < Xf(x) + (1 - X)f(y) - A(l - X)p (\\x - y\\)

the class of uniformly convex functions is important because on a Banach space every uniformly convex and lower semicontinuous proper function has a unique minimum point and the corresponding minimization prob-lem is well-conditioned It turns out that uniformly convex functions have very nice characterizations using their conjugates and subdifferentials The dual notion for uniformly convex function is that of uniformly smooth con-vex function An important fact is that / is uniformly concon-vex (uniformly smooth) if and only if /* is uniformly smooth (uniformly convex)

Another interesting application of convex analysis is in the study of monotone operators This became possible by using a convex function associated to a multifunction introduced by M Coodey and S Simons So, one obtains quite easily characterizations of maximal monotone operators, local boundedness of monotone operators and maximal monotonicity of the sum of two maximal monotone operators using continuity properties of convex functions, the formula for the subdifFerential of a sum of convex functions and a minimax theorem (whose proof is also included)

A more detailed presentation of the book follows

The book is divided into three chapters, every chapter ending with ex-ercises and bibliographical notes; there are more than 80 exex-ercises It also includes the complete solutions of the exercises, the bibliography, the list of notations and the index of terms

No prior knowledge of convex analysis is assumed, but basic knowledge of topology, linear spaces, topological (locally convex) linear spaces and normed spaces is needed

In Chapter 1, as a preliminary, we introduce the notions and results of functional analysis we need in the rest of the book For easy reference, in Section 1.1 we recall several notions, notations and results (without proofs) which can be found in almost all books on functional analysis; let us men-tion four separamen-tion theorems for convex sets, the Dieudonne and Alaoglu-Bourbaki theorems, as well as the bipolar theorem

in-terior %A of a subset A of a linear space X, we introduce, when X is a

topological vector space, the sets lcA and tbA, which reduce to lA when the

affine hull aff A of A is closed or barreled, respectively, and are the empty set otherwise The quasi interior of a set and united sets are also studied

In Section 1.3 we state and prove the famous Ursescu's theorem as well as a slight amelioration of Simons' open mapping theorem As application of these results one reobtain the Banach-Steinhaus theorem and the closed graph theorem as well as two results of Carja which are useful in control-lability problems Because the notions (with the exception of cs-closed and ideally convex sets) and results from Sections 1.2 and 1.3 are not treated in many books (to our knowledge only [Kusraev and Kutateladze (1995)] contains some similar material), we give complete proofs of the results

The chapter ends with Section 1.4 in which we state and prove the Ekeland's variational principle, the smooth variational principle of Borwein and Preiss, as well as two (dual) results of Ursescu which generalize Baire's theorem

Chapter is dedicated, mainly, to conjugate and e-subdifferential calcu-lus Because no prior knowledge of convex analysis is assumed, we introduce in Section 2.1 convex functions, give several characterizations using the epi-graph, or the gradients in case of differentiability, point out the operations which preserve convexity and study the important class of convex functions of one variable; the existence of the (e-)directional derivative and some of its properties are also studied We close this section with a characterization of convex functions using the upper Dini directional derivative

Section 2.2 is dedicated to the study of continuity properties of convex functions To the classes of sets introduced in Section 1.2 correspond cs-closed, cs-complete, lcs-cs-closed, ideally convex, bcs-complete and li-convex functions We mention the fact that almost all operations which preserve convexity also preserve the lcs-closedness and the li-convexity of functions as seen in Proposition 2.2.19 The most part of the results of this section are not present in other books; among them we mention the result on the convexity of a quasiconvex positively homogeneous function and the results on cs-closed, cs-complete, cs-convex, lcs-closed, ideally convex, bcs-complete and li-convex functions

for-mulas for the subdifferentials of Af and / i • • • • • / „ which are valid without additional hypothesis The classical theorem which states that the £-sub-differential of a proper convex function is nonempty and w*-compact at a continuity point of its domain, as well as the formula for the e-directional derivative as the support function of the e-subdifferential is also estab-lished The less classical result which states that the same formula holds for e > when the function is not necessarily continuous (but is lower semi-continuous) is established, too We mention also Theorem 2.4.14 related to the subdifferential of sublinear functions; some of its statements are not very spread Other interesting results are introduced for completeness or further use

In Section 2.5 we introduce the general problem of convex program-ming and establish sufficient conditions for the existence and uniqueness of solutions, respectively We mention especially Theorems 2.5.2 and 2.5.5; Theorem 2.5.2 ameliorates a result of Polyak (1966), which shows that the refiexivity of the space, needed in proving the existence of solutions, is al-most necessary, while Theorem 2.5.5 shows that the coercivity condition is essential for the existence of solutions

Section 2.6 is dedicated to perturbed functions One introduces primal and dual problems, the marginal function, and give some direct properties of them Then one obtains the formula for the e-subdifferential of the marginal function using the (e + ^-subdifferentials (with 77 > 0) of the perturbed function Applying this result one obtains formulas for the e-subdifferential of several types of convex functions

In the main result of Section 2.7 we provide nine (non-independent) sufficient conditions which ensure the validity of the fundamental duality formula (FDF) The most known of them is that (x0,0) € d o m $ and

$(xo, •) is continuous at for some x o £ l For the proof of the sufficiency of some conditions one uses the open mapping theorems established in Section 1.3 A related result involves also a convex multifunction; this will be useful for obtaining the formulas for the conjugate and the e-sub-differential of a function of the forms go A with A a densely defined and closed linear operator and of g o H with g being increasing and H convex Section 2.8 is dedicated entirely to conjugate and e-subdifferential cal-culus for convex functions The considered functions ip have the form:

ip(x) = F(x, A{x)) and — f + g o A with A a continuous linear

oper-ator, <p = / + g, ip(x) = ini{g(y) \ y £ C(x)} with a convex process,

func-tion, cp = m a x { / i , , / „ } and = A / - Besides classical conditions one points out very recent ones For the proof one constructs an adequate perturbation function and uses the fundamental duality theorem

In Section 2.9 we apply the fundamental duality theorem for obtain-ing necessary and sufficient optimality conditions in convex optimization problems with constraints These conditions involve the subdifferentials of the functions considered or the corresponding Lagrangian The results are well-known However we mention the formula for the normal cone to a level set stated in Corollary 2.9.5 for not necessarily finite-valued functions which is quite new

The minimax theorem presented in Section 2.10 will be used in the section dedicated to monotone multifunctions

Throughout Chapter the involved spaces are normed spaces In Sec-tion 3.1 besides the classical theorems of Borwein, Br0ndsted-Rockafellar, Bishop-Phelps and Rockafellar (on the maximal monotonicity of the sub-differential of a convex function) we present a recent theorem of Simons and use it for a very short proof of Rockafellar's theorem (mentioned be-fore) As a consequence of the Br0ndsted-Rockafellar theorem we obtain other three conditions for the validity of the formulas for the conjugate and subdifferential of the function F(-,A(-)) (and therefore for the functions f + go A and f + g)

The aim of Section 3.2 is to characterize the convex functions using other types of subdifferentials In fact we use an abstract subdifferential An example of such subdifferential is Clarke's one for which we establish sev-eral properties The main tool for such characterizations is the well-known Zagrodny's approximate mean value theorem; the version we present sub-sumes several results met in the literature We present also an integration theorem of Thibault and Zagrodny which yields the fact that two lower semicontinuous convex functions on a Banach space which has the same Fenchel subdifferential coincide up to an additive constant

In Section 3.3 we introduce the class A of functions : K+ —> R+ with

respect to arbitrary bornologies Using one of the characterizations and the Br0ndsted-Rockafellar theorem one obtains the following interesting result of Asplund and Rockafellar: Let X be a Banach space and / : I - > l a proper lower semicontinuous convex function; if / * is Frechet differentiable at x* e int(dom/*) then V/*(x*) € X

In Section 3.4 we introduce the well-conditioned convex functions and give several characterizations of this notion using the conjugate and the subdifferential of the function An important special case is that of well-conditioning with linear rate This situation is studied in Section 3.10

In Section 3.5 we study uniformly convex and uniformly smooth convex functions, respectively To any convex function / one associates the gages

pj and 07 of uniform convexity and uniform smoothness, respectively The

gage pf has an important property: the mapping < t i-» t~2pf(t) is

nondecreasing Because a/* = ( p / ) * and a/ = ( p / * )# f°r a n v proper lower

semicontinuous convex function / , the mapping < i t~2crf(t) is

non-increasing for such a function; moreover, one obtains that for such an / , / is uniformly convex if and only if / * is uniformly smooth and / is uniformly smooth if and only if / * is uniformly convex Then one establishes many characterizations of uniformly convex functions and of uniformly smooth convex functions In these characterizations appear functions (gages or moduli) belonging to different subclasses of A introduced in Section 3.3 These gages and moduli are sharp enough in order to obtain that / is c-strongly convex if and only if /* is Frechet differentiable on X* and V / * is c_1-Lipschitz Even if the results are established in general Banach

spaces the natural framework for uniformly convex and uniformly smooth convex function is that of reflexive Banach spaces This is due to the fact that when there exists a proper lower semicontinuous and uniformly convex function on a Banach space whose domain has nonempty interior, the space is necessarily reflexive

Section 3.6 is dedicated to the study of those convex functions which are uniformly convex on bounded sets and uniformly smooth on bounded sets, respectively Under strong coercivity of the function one shows that these notions are dual

In Section 3.7 we study the function /v : X ->• E, f^x) = / „ " tp(t) dt,

One obtains also characterizations of (local) uniform convexity and (local) uniform smoothness of X with the help of the properties of fv For example

one obtains: X is uniformly convex •£> X* is uniformly smooth •& fv is

uni-formly convex on bounded sets -O (fv)* is uniformly smooth on bounded

sets •& {ftp)* is Frechet differentiable and V ( /v) * is uniformly continuous

on bounded sets Note that a part of the results of this section can be found in the book [Cioranescu (1990)], but the proofs are different; note also that some notions are introduced differently in Cioranescu's book

Another application of convex analysis is emphasized in Section 3.8; here we apply the results on the existence, the uniqueness and the charac-terizations of optimal solutions of convex programs to the problem of the best approximation with elements of a convex subset of a normed space

In Section 3.9 it is shown that there exists a strong relationship between the well-posedness of the minimization problem min/(:r) s.t x £ X, and the differentiability at of the conjugate / * of / ; when / is convex these properties are equivalent Using this result we establish a very interesting characterization of Chebyshev sets in Hilbert spaces and show that the class of weakly closed Chebyshev sets coincides with the class of closed convex sets in Hilbert spaces

Section 3.10 deals with sets of weak sharp minima, well-behaved convex functions and the study of the existence of global error bounds for convex inequalities These notions were studied separately for a time, but they are intimately related As noted above, argmin / is a set of weak sharp minima for / exactly when / is well-conditioned with linear rate But the inequality f{x) < has a global error bound exactly when argmin[/]+

is a set of weak sharp minima for [/]+ := max(/, 0) We give several

characterizations of the fact that argmin / is a set of weak sharp minima for / , one of them being the fact that up to a constant, the conjugate / * is sublinear on a neighborhood of the origin Several numbers associated to a convex function are introduced which are related to the conditioning number from numerical analysis Although the most part of the results from this section are stated in the literature in finite dimensional spaces, we present them in infinite dimensions

X =} X*, the minimax theorem and a few results of convex analysis One

obtains: two characterizations of maximal monotone multifunctions; the fact that the condition int(domTi — domT2) is equivalent to other three conditions involving dom Tj and dom XT{ , and is sufficient for the maximal

monotonicity of T\ + T2; dom T and Im T are convex if X is reflexive and T

is maximal monotone; domT is convex if int(domT) ^ and T is maximal monotone; T is locally bounded at XQ € (co(domX'))1 if T is a monotone

multifunction; Rockafellar's theorem on the local boundedness of maximal monotone multifunctions The result stating that for a maximal monotone multifunction T on the Banach space X for which dom T is convex the local boundedness of T at x dom T implies that x € int (dom T) seems to be new When applied to the subdifferential of a proper lower semicontinuous convex function / on the Banach space X, this result gives (for example): / is continuous &• d o m / is open <& df is locally bounded at any x £ d o m /

Preliminary Results on Functional Analysis

1.1 Preliminary Notions and Results

In this section we introduce several notions and results on separation of sets as well as some properties of topological vector spaces and locally convex spaces which are frequently used throughout the book, for easy reference

Let X be a real linear (vector) space Throughout this work we shall use the following notation (x, y being elements of X): [x, y] := {(1 — A)x + Xy \ A e [0,1]}, [x,y[:= {(1 - X)x + \y | A € [0,1[}, ]x,y[:= {(1 - X)x + Xy | A 6]0,1[}, called closed, semi-closed and open segment, respectively Note that [x,x] =]x,x[= {x}\

If ^ A, B C X, the Minkowski sum of A and B is A + B := {a +

b | a £ A,b £ B) Moreover, if x £ X, X £ R and ^ T C R, then x + A := A + x := A + {x}, A • A = {70 | A € A, a £ A} and XA — {A} • A

We shall consider that A + = and A • = • A =

A nonempty set A C X is star-shaped at a (G A) if [a, x] C A for all 2: £ A; A is convex if [x, y] C A for all x,y £ A; Ais a, cone if 1+ • A C A (in particular € A when A is a cone), R+ is the set of nonnegative reals;

A is afflne if Ax + (1 - X)y A for all x,y e A and A e R; A is balanced if

Ax € ^4 for all x e A and A £ [-1,1]; A is symmetric if A = - A Hence ^4

is balanced if and only if A is symmetric and star-shaped at We consider that the empty set is convex and afflne It is easy to prove that

A is afflne o a l , 3XQ linear subspace of X : A = a + X0

<=>Va€^4(3a€^4) : A — a is a linear subspace

When A is affine and a £ A, the linear space XQ := A — a is called the

linear space parallel to A; we consider that the dimension of A is

dimXo-Note that if (Aj)j€/ is a family of affine, convex, balanced subsets or

cones of X then f]ieIAi has the same property (Exercise!); we use the

usual convention that HieO7^ = -^- Taking into account this remark, we

can introduce the notions of affine, convex and conic hull of a set So, the

affine, convex and conic hull of the subset A of X are:

aff A := [){V C X \ A C V, V affine}, co A := P){C C X | A C C, C convex}, cone A := f]{C CX\AcC, C cone},

respectively Of course, the linear hull of the subset A of X is the linear subspace spanned by A:

linA := P | { ^ o C X \ A C X0, X0 linear subspace of X}

It is easy to verify (Exercise!) that

aff A = { V " XiXi n G N, (Ai)i<i < n C R, V " X, = 1,

(Xi)l<i<n C A> ,

coA= {Yl^-^i n S N, (Aj) C K+, (xt) C A, Yl"-^ = 1) '

cone A - {Ax | A > 0, x £ A} = 1+ • A,

where N is the set of positive integers

Let us mention some properties of the affine and convex hulls Consider

Y another linear space, T : X -> Y a linear operator, A,B c X, C C Y

nonempty sets Then: (i) aff(AxC) = aff Ax aff C; (ii) aff T(A) = T(aff A) (iii) aft(A + B) = aff A + aff B; (iv) Vo € A : aff A = a + aff (A - A) (v) aff (A - A) = UA>oA(A - A) if A is convex; (vi) aff A = linA if e A

(vii) co(A x C) = co A x coC; (viii) co T(A) = T(coA); (ix) co(A + ) =

co A + coB; (x) co(cone A) = cone(co A) (Exercises!)

Let M C X be a linear subspace, and let A C X be nonempty; the

algebraic interior of A with respect to M is

a i n tM^ : = {a X | Vrr € M, 36 > 0, VA € [0,6] : a + Xxe A}

-We distinguish two important cases: (i) M = X; in this case we write

A% instead of aintM A; A1 is called the algebraic interior of A, (ii) M =

aS(A — A); in this case aintM A is denoted by M and is called the relative

algebraic interior of A Therefore a £ A1 ii and only if aff A = X and

a G 14 (Exercise!)

When the set A is convex we have (Exercise!) that:

V a G i : \m(A-a) = cone(A-A),

whence aff A = a + cone(A — A) for every a £ A (hence cone(A — A) is the linear subspace parallel to aff A),

a e i ' ^ V i e l , A > : a + Xx £ A& cone(A - a) = X,

and

a G M o V a ; G A, A > : (1 + X)a - Xx G A <$ cone(yl — a) = cone(A — A)

<=> cone(A — a) is a linear subspace

O M n(C — a) is a linear subspace; (1.1)

the dimension of the convex set A is dim A := dim(aff A) = dim (cone(A —

A))

Some properties of the algebraic interior are listed below Let ^

A,B CX, i s l a n d A G E \ {0}; then: (i) *(x + A) =x + iA; (ii) ^XA) =

X • % (iii) A + Bi C (A + B ) ' ; (iv) A + Bl = {A + BY if Bi = B;

(v) iA + iB c 1{A + B); (vi) *(A + B) = A + *B if A, B are convex, VI /

and *B T^ 0; (vii) M 7^ if A is convex and dim A < 00; (viii) if A is convex then [a, x[ C VI for all a G M and i £ A

In the sequel the results will be established for real topological vector spaces (tvs for short) or real locally convex spaces (lcs for short) When X is a tvs it is well-known that the class Nx of closed and balanced neigh-borhoods of G X is a base of neighneigh-borhoods of 0; when X is a lcs then the class J^cx of the closed, convex and balanced neighborhoods of G X is

also a base of neighborhoods of

If X, Y are real linear spaces, we denote by L(X, Y) the real linear space of linear operators from X into Y The space L(X,R) is denoted by X' and is called the algebraic dual of X; an element of X' is called a linear

the linear space of continuous linear operators from X into Y; the space £ ( X , E) is denoted by X* and is called the topological dual of X

Let now A be an absorbing subset of the linear space X, i.e £ A1;

the Minkowski gauge of A is defined by

pA : X -> M, ^ ( a : ) := inf{A > | a; £ \A)

It is obvious that PA = P[o,i].4- Moreover, if A, B C X and C C Y are absorbing and star-shaped sets, where Y is another linear space, one has (Exercise!):

{x X | pA(a) < 1} C A C {x € X | P A ( Z ) < 1},

VxGX : pAns(a;) = max {PA (a;), pB(a;)},

Va; X, 2/ € Y : PAxc(x,y) = max{pA

(x),pc(y)}-Other useful properties of the Minkowski gauge are mentioned in the next result Recall that p : X -» M is sublinear if p(0) = 0, p(x + y) <

p(x) +p(y) [with the convention (+oo) + (—oo) = +oo] and p(Xx) = Xp(x)

for all x, y £ X, X € P := ]0, oo[; p is a semi-norm if p is a finite, sublinear and even [i.e p(—x) = p(x) for every x € X] function

Proposition 1.1.1 Let A be a convex and absorbing subset of the linear

space X

(i) Then PA is finite, sublinear and A1 = {x E X \ PA{X) < 1};

furthermore, if A is symmetric then PA is a semi-norm, too

(ii) Assume, moreover, that X is a topological vector space and V is a

neighborhood of Q £ X Then pv is continuous and

intV = {xeX\pv{x) < 1}, clV = {xeX \pv{x) < 1}

The following result will be useful, too

Theorem 1.1.2 Let C be a convex subset of the topological vector space

X Then

(i) c l C is convex;

(ii) if a £ int C and x £ cl C, then [a, x[ C int C; (iii) i n t C is convex;

Using the Minkowski gauge one obtains the geometrical versions of the Hahn-Banach theorem, i.e separation theorems In the sequel we give several separation theorems for convex subsets of topological vector spaces or locally convex space

Theorem 1.1.3 (Eidelheit) Let A and B be two nonempty convex subsets

of the topological vector space X If int A ^ and B flint A = then there exist x* e X* \ {0} and a £ E such that

VxeA,\/yeB : (x,x*) < a < (y,x*), (1.2) or equivalently, supa;*(yl) < inf x*(B)

The separation condition (1.2) can be given in a different manner Let

x* e X* \ {0} and a £ t Consider the sets H^a:={x£X\(x,x*)<a},

H^,a~{xeX\ (x,x*) <a},

Hx*,a := {x e X \ (x,x*) = a } ;

similarly one defines H^ and H>.a All these sets are convex and

non-empty The set Hx*^a is called a closed hyperplane, H<. a and H>* a are

called o p e n half-spaces, while H-. a and H~.a are called closed

half-spaces Hx-<a, H-, a and H-, a are closed sets, while H<» a and H>, a are

open sets; moreover, c\H< a = H% a and {Hf,a)1 = int H~ >a = H< a

(Exercises!)

Theorem 1.1.3 states the existence of x* G X* \ {0} and a G K such that

A c H-, a and B C H-, a; in this situation we say that Hx*tC[ separates

A and B; the separation is proper when AUB <£ Hx*i<x and the separation

is strict when A n Hx*^a = or B n Hx*ia =

When x0 A and ffx*,a separates A and {xo} we say that ifx*,a is a

supporting hyperplane of A at xo; XQ is called a support point and

x* is called a support functional Therefore x* € X* \ {0} is a support

functional for A if and only if x* attains its supremum on A Generally,

Hx*<a, with x* ^ 0, is a supporting hyperplane for A if A C B.-,a (or

i c H ^i a) a n d A n i fs ,a^

Corollary 1.1.4 Let A be a convex subset of the topological vector space

X having nonempty interior and x € A \ int A Then x is a support point

In the case of locally convex spaces one has the following result for the separation of two sets

T h e o r e m 1.1.5 Let X be a locally convex space and A,B C X be two

nonempty convex sets If A is closed, B is compact and An B = 0, then there exist x* G X* \ {0} and ct\,a.2 G K such that

Vx G A, Vy e B : (x,x*) < ax < a2 < (y,x*),

or equivalently, sup x* (A) < inf x* (B)

The two preceding results can be stated in a more general setting T h e o r e m 1.1.6 Let A and B be two nonempty convex subsets of the

topological vector space X such that int(A — B) ^ Then

0 £ i n t ( A - £ ) < £ • 3x* G X * \ { } : supa;*(yl) < inf x*{B) T h e o r e m 1.1.7 Let A and B be two nonempty convex subsets of the

locally convex space X Then

Q$d{A-B)&3x* eX* : supx*{A) < inf x*(B)

The preceding theorem shows the usefulness of having criteria for the closedness of the difference (or sum) of two convex sets In order to give such a criterion, let A be a nonempty convex subset of the topological vector space X The recession cone of A is defined by

vecA := {u G X | Vo G A : a + uGA}

It is easy to show that rec A is a convex cone and A + rec A = A When A is a closed convex set we have that

vecA = f]t>ot(A-a) (1.3)

for every a A In this case it is obvious that rec A is a closed convex cone which is also denoted by Aoo It is easy to see that when X is a finite dimensional separated topological vector space and A is a closed convex nonempty subset of X, A^ = {0} if, and only if, A is bounded This is no longer true when d i m X ~ oo

Example 1.1.1 Let X := £p with p £ [l,oo] and A := {(x„)„>i G £p |

\xn\ < n Vn G N} It is obvious that A is a closed convex set which is

u = (un) G Aoo then tu G A for every t > 0; so \tun\ < n for every t > 0,

whence u„ = Hence u =

The following famous theorem was obtained in [Dieudonne (1966)]

Theorem 1.1.8 (Dieudonne) Let A, B be nonempty closed convex subsets

of the locally convex space X If A or B is locally compact and A^ n J3oo is a linear subspace, then A — B is closed

In convex analysis (as well as in functional analysis) one often uses the following sets associated to a nonempty subset A of the locally convex space

X:

A° A+ A^

{x* eX*\Vx€A

{x* ex* \Vxe A {x* ex* IVze A

(x,x*)>-l}, (x,x*)>0}, (x,x*) = 0},

called the polar, the dual cone and the orthogonal space of A, respec-tively One verifies easily that A° is a w*-closed convex set which contains 0, that A+ is a w*-closed convex cone, and, finally, that A1- is a u;*-closed

linear subspace of X*, where w* = a(X*,X) is the weak* topology on X* Similarly, for ^ B C X* we define the polar, the dual cone and the orthogonal space; for example, the polar of B is

B° := {x G X | Vx* G B : (x,x*) > - }

It is obvious that B° is a closed convex set containing 0, B+ is a closed

convex cone, and, finally, B1- is a closed linear subspace

One verifies easily that when A,BcX and A G P we have: (i) A° is convex and G A°; (ii) A U {0} C (A°)° =: A°°; (hi) AcB => A° D B°; (iv) (AUB)° = A°nB°; (v) if € AnB then (A + B)+ = (AUB)+ =A+n

B+; (vi) (\A)° = {A°; (vii) A° = A+ if A is a cone, and A° = A+ = A1- if

A is a linear subspace; (viii) (T(A))° = ( T * ) "1^0) , if T G &(X,Y), where

Y is another locally convex space

A very useful result is the bipolar's theorem Let X be a topological vector space and A C X; the set coA := cl(coA) is called the closed

convex hull of the set A; it is the smallest closed convex set containing A

Similarly, coneA :— cl(coneyl) is called the closed conic hull of A

con-vex space X Then

A00 = co(A U {0}), A+ + = cone(co A), A±JL = cl(lin A)

It follows that for the nonempty subset A of the lcs X one has: (a) A°° = A o A is closed, convex and € A; (b) A+ + = A <£> A is a closed convex

cone; (c) A-"-1- = A <=> A is a closed linear subspace

Another famous result is the following

Theorem 1.1.10 (Alaoglu-Bourbaki) Let X be a locally convex space

and U C X be a neighborhood of the origin Then U° is w*-compact

We finish this preliminary section with some notions and results con-cerning completeness and metrizability of topological vector spaces

The subset A of the topological vector space X is complete

(quasi-complete) if every (bounded) Cauchy net (xi)i^i C A is convergent to an

element x £ A Of course, any complete set is closed and any closed subset of a complete set is complete (Exercise!) Recall that the topological space

(X, T) is first countable if every element of X has a (at most) countable

base of neighborhoods Note that a subset A of a first countable tvs X is complete if and only if every Cauchy sequence of A is convergent to an element of A; in particular, a first countable tvs is complete if and only if it is quasi-complete (Exercise!)

We shall use several times the hypothesis that a certain topological vector space is first countable The next result refers to the first countability of locally convex spaces

Proposition 1.1.11 Let (X,T) be a locally convex space Then

(i) (X, T) is first countable O a (at most) countable family of

semi-norms on X such that r = rg> O r is semi-metrizable, i.e there exists a semi-metric d on X such that T = T^; the semi-metric d may be chosen to be invariant to translations (i.e d(x + z,y + z) = d(x,y) for allx,y,z € X)

(ii) (X, T) is separated and first countable if and only if T is metrizable,

i.e there exists a metric d on X such that r = T&

too One says that the topological vector space X is barreled if every absorbing, convex and closed subset of X is a neighborhood of £ X As application of the Baire theorem one obtains that every Frechet space is barreled

It is well-known that in a finite dimensional separated topological vector space any convex and absorbing set is a neighborhood of the origin

1.2 Closedness and Interiority Notions

Consider X a real topological vector space We say that the series Yln>i Xn

is convergent (resp Cauchy) if the sequence (£n)n£N is convergent (resp

Cauchy), where Sn := Y^k=i xk f °r every n £ N; of course, any convergent

series is Cauchy

Let A C X; by a convex series with elements of A we mean a series of the form Ylm>i ^mXm with (Am) C IR+, {xm) C A and Y,m>i ^™ = ^

if, furthermore, the sequence (xm) is bounded we speak about a b-convex

series We say that A is cs-closed if any convergent convex series with

elements of A has its sum in A;* A is cs-complete if any Cauchy convex series with elements of A is convergent and its sum is in A Similarly, the set

A is called ideally convex if any convergent b-convex series with elements

of A has its sum in A and A is bcs-complete if any Cauchy b-convex series with elements of A is convergent and its sum is in A It is obvious that any closed set is ideally convex, every ideally convex set is convex, every cs-complete set is cs-closed and every cs-complete convex set is cs-cs-complete; if X is complete, then A C X is complete (bcomplete) if and only if A is cs-closed (ideally convex) If Xo is a linear subspace of X and A is a cs-cs-closed (ideally convex) subset of X, then Xo C\ A is a cs-closed (ideally convex) subset of Xo (endowed with the induced topology) Moreover, if A C X and B C Y are nonempty, then A x B is cs-closed (cs-complete, ideally convex, bcs-complete) if and only if A and B are cs-closed (cs-complete, ideally convex, bcs-complete) If X is first countable, a linear subspace

XQ of X is cs-closed (cs-complete) if and only if it is closed (complete);

moreover, if X is a locally convex space, X0 is closed if and only if Xo

is ideally convex Note also that T(A) is cs-closed (cs-complete, ideally convex, bcs-complete) if A C X is cs-closed (cs-complete, ideally convex,

bcs-complete) and T : X -> Y is an isomorphism of topological vector spaces (Exercise!), Y being another tvs We consider that the empty set is convex, ideally convex, bcs-complete, cs-complete and cs-closed

It is worth to point out that when X is a locally convex space, every b-convex series with elements of X is Cauchy (Exercise!)

The class of cs-closed sets (and consequently that of ideally convex sets) is larger than the class of closed convex sets, as the next result shows

Proposition 1.2.1 Let A C X be a nonempty convex set

(i) / / A is closed or open then A is cs-closed

(ii) If X is separated and dim A < oo then A is cs-closed

Proof, (i) Let £n >! Anin be a convergent convex series with elements

of A; denote by x its sum

Suppose that A is closed and fix a £ A Then, for every n G N we have that X)fc=i ^kXk + (l — Yl'kLn+i ^*) a S A Taking the limit for n -> oo,

we obtain that x G cl A = A

Suppose now that A is open Assume that x $ A By Theorem 1.1.3, there exists x* G X* such that (a - x, x") > for every a G A In particular

(xn — x,x*) > for every n G N Multiplying by \n > and adding for

n G N we get (since A„ > for some n) the contradiction

0 < ^n > 1 A„ (xn - x, x*) = (X]„>i XnXn' x*)~ \52n>i A n) ^'x") = °'

Therefore x G A So in both cases A is cs-closed

(ii) We prove the statement by mathematical induction on n := dim A If n = A reduces to a point; it is obvious that A is cs-closed in this case Suppose that the statement is true if dim A < n G N U {0} and show it for dim A = n + Without any loss of generality we suppose that G A; then Xo := aff A is a linear subspace with d i m X0 = n +

Because on a finite dimensional linear space there exists a unique separated linear topology and in such spaces the interior and the algebraic interior coincide for convex sets, we have that iA — intx0 A ^ Let J2n>i ^n%n

be a convergent convex series with elements of A and sum x Assume that

x fi A Because A is convex, the set P :— {n G N | A„ > 0} is infinite,

and so we may assume that P — N Applying now Theorem 1.1.3 in XQ, there exists XQ G XQ \ {0} such that (x — x, XQ) > for every x G A But X)n>i An (xn —X,XQ) = Since {xn —X,XQ) > and Xn > for every

dimAo < dimHx*t\ = n, from the induction hypothesis we obtain the

contradiction x e A0 c A Therefore x € A The proof is complete •

Other properties of cs-closed and ideally convex sets are given in the following result

Proposition 1.2.2 (i) If Ai C X is cs-closed (resp ideally convex) for

every i £ I then P |i e / A{ is cs-closed (resp ideally convex)

(ii) / / Xi is a topological vector space and Ai C Xi is cs-closed (resp

ideally convex) for every i I, then Ylizi -^-i is cs-closed (resp ideally convex) in Yliei Xi (which is endowed with the product topology)

Proof The proof of (i) is immediate, while for (ii) one must take into account that a sequence (xn)n €N C X := YlieI Xi converges t o x G X (resp

is bounded) if and only if (xln) converges to xl in Xi (resp is bounded) for

every i £ I • We say that the subset C of Y is lower cs-closed (Ics-closed for short)

if there exist a Frechet space X and a cs-closed subset B of X x Y such that C = Pry (B) Similarly, the subset C of Y is lower ideally convex (li-convex for short) if there exist a Frechet space X and an ideally convex subset B of X xY such that C = Pry (B) It is obvious that any cs-closed (resp ideally convex) set is Ics-closed (resp li-convex), any Ics-closed set is li-convex and any li-convex set is convex, but the converse implications are not true, generally Note also that T(A) is Ics-closed (resp li-convex) if

A C X is Ics-closed (resp li-convex) and T : X -> Y is an isomorphism of

topological vector spaces (Exercise!) The classes of Ics-closed and li-convex sets have very good stability properties as the following results show We give only the proofs for the "li-convex" case, that for the "Ics-closed" case being similar

Proposition 1.2.3 Suppose that Y is a Frechet space and C C F x Z is

a li-convex (Ics-closed) set Then Prz(C) is a li-convex (Ics-closed) set

Proof By hypothesis, there exists a Frechet space X and an ideally convex subset B C X x (Y x Z) such that C = P r yxz ( B ) - Since I x i s

a Frechet space and Prz(C) = P r z ( B ) , we have that Prz(C) is a li-convex

subset of Z •

Proposition 1.2.4 Let I be an at most countable nonempty set

(i) If Ci C Y is li-convex (Ics-closed) for every i £ I then (~)ieI Ci is

(ii) / / Y{ is a topological vector space and Ci C Yi is li-convex

(Ics-closed) for every i I, then Yli&iCi 8S li-convex (Ics-closed) in W^jYi

Proof, (i) For each i £ I there exist X» a Frechet space and an ideally

convex set Bt C X» xY such that d = PrY(Bi) The space X := fliei -%i

is a Frechet space as the product of an at most countable family of Frechet spaces Let

Bi •= {{(xj)jei,y) e X x Y | (xi,y) £ Bi}

Then Bi is an ideally convex set by Proposition 1.2.2(h) It follows that

B := f]ieI Bi is ideally convex by Proposition 1.2.2(i) Since PrY(B) =

Hie/ Ci, ^ f °u o w s that flie/ Ci is li-convex

(ii) For each i £ I there exist Xj a Frechet space and an ideally convex set Bi cXiX Yi such that d = Pry; (Bi) By Proposition 1.2.2(h), ]JieI Bt

is an ideally convex subset of r i i e / ( ^ j x ^ ) - The space X := riig/ ^i 1S a

Frechet space; let Y := FJie/ ^i- Consider the set

B:= {((xi)ieI,(yi)ieI) £XxY\ (xi,yi) £ Bi Vt / }

Since T : I l i e / (X* x y4) - • X x y , T ( ( xi, yi)i 6/ ) := ((xi)ieI,(yi)iei) is

an isomorphism of topological vector spaces, B — T (Y\ieI Bi) is ideally

convex As C := riig/ C* = ^VY(B), C is li-convex D

Before stating other properties of li-convex and lcs-closed sets, let us define some notions and notations related to multifunctions

Let E,F be two nonempty sets; a mapping ft : E —• 2F is called a

multifunction, and it will be denoted by ft : E =X F The set domft :=

{x £ E \ 5l(x) -£ 0} is called the d o m a i n of the multifunction 51; the i m a g e

of 51 is Imft := UigB^(a ;)i the g r a p h of 51 is the set grft :— {(x,y) \ y £

5l(x)} C E x F; the inverse of the multifunction 51 is the multifunction

ft"1 : F =} E defined by ft_1(?/) := {x £ E | y £ %(x)} Therefore

d o m f t -1 = Imft, I m f t -1 = domft and g r f t -1 = {(y,x) | (x,y) £ grft}

Frequently we shall identify a multifunction with its graph For A C E and

B C F one defines 51(A) := \JxeA5l(x) and ft"1^) := Uj,eB#- 1(2/); i n

particular Imft = ft(.E) and domft = ft_1(F) If : F =4 G is another

multifunction, then the composition of the multifunctions S and 51 is the multifunction Soft :E=lG, (Soft) (a) := \Jy€Ci{x) S(y) If ft, S : E =} F and

F is a linear space, the sum of ft and S is the multifunction ft + S : E =t F,

Let now K : X r{ ; we say that 51 is convex (closed, ideally

con-vex, bcs-complete, cs-concon-vex, cs-complete, li-concon-vex, lcs-closed)

if its graph is a convex (closed, ideally convex, bcs-complete, cs-convex, cs-complete, li-convex, lcs-closed) subset of X x Y Note that 31 is convex if and only if

V i , a ' eX, V A e [0,1] : \3l(x) + (l-\)3l(x')c3l(\x + {l-\)x') Proposition 1.2.5 Let A, B C X, 31, S : X =4 Y and7:Y =} Z

(i) If X is a Frechet space and A, 31 are li-convex (resp lcs-closed),

then 01(A) is li-convex (resp lcs-closed)

(ii) If X is a Frechet space and A, B are li-convex (resp lcs-closed),

then A + B is li-convex (resp lcs-closed)

(hi) If Y is a Frechet space and 31, T are li-convex (resp lcs-closed),

then T o 31 is li-convex (resp lcs-closed)

(iv) IfY is a Frechet space and 31,$ are li-convex (resp lcs-closed), then

31 + § is li-convex (resp lcs-closed)

Proof, (i) We have that

%{A) =PrY((AxY)ngrJl)

Using successively Propositions 1.2.4(h), 1.2.4(i) and 1.2.3, it follows that

31(A) is li-convex

(ii) Let T : X xX —> X, T(x,y) := x + y Since T is a continuous linear operator, g r T is a closed linear subspace; in particular T is a li-convex multifunction Since Ax B is li-convex, by (i) A + B = T(A x B) is a li-convex set

(hi) We have that

gr(To3?) = P rXx ( g r : R x Z) n (X x grT))

The conclusion follows from Propositions 1.2.4(h), 1.2.4(i) and 1.2.3 (iv) The sets

T~{(x,z,y,y')\x€X, y,y' eY, z = y + y'} C (X x Y) x (Y x Y), A:= {(x,z,y,y') \ (x,y) G g r # , y',z € Y} and B := {(x,z,y,y') \ (x,y') G

grS, y,z € Y} are li-convex sets (the first being a closed linear subspace)

Let Y be another topological vector space and A C X xY; we introduce the conditions (Ha;) and (Hwa;) below, where x refers to the component

xeX:

(Ha:) If the sequences ((xn, yn)) C A and (A„)„>! C ffi+ are such that

E „ > i K = 1, En> i ^nVn^as sum y and X)„>i A«a;n is Cauchy,

then the series X^n>i ^nxn *s convergent and its sum x G X verifies

{x,y)EA

(Hwi) If the sequences ({xn,yn))n>1 C A and (An)n>i C K+ are such

that ((xn,yn)) is bounded, En>x An = 1, £ „ > i A„2/n has sum y

and ^r a > Anxn is Cauchy, then the series X^n > ^«x« ^s convergent

and its sum x € X verifies (x, y) A

Of course, when X is a locally convex space, deleting "^2n>1 Ana;n is

Cauchy" in condition (Hwa;) one obtains an equivalent statement

In the next result we mention the relationships among conditions (Ha;), (Hwi), ideal convexity, cs-closedness, cs-completeness and convexity The proof being very easy we omit it

Proposition 1.2.6 Let A C X xY and BcXxYxZbe nonempty

sets

(i) Assume that Y is complete Then B satisfies (H.(x,y)) if and only

if B satisfies (Ha;); B satisfies (Hwa;) if and only if B satisfies (Hw(x,j/))

(ii) Assume that X is complete Then A satisfies (Ha;) if and only if A

is cs-closed; A satisfies (Hwa;) if and only if A is ideally convex

(iii) Assume that Y is complete Then A satisfies (Ha;) if and only if A

is cs-complete; A satisfies (Hwa;) if and only if A is bcs-complete

(iv) If A satisfies (Ha;) then A satisfies (Hwx); if A satisfies (Hwa;)

then A is convex

(v) Assume that X is a locally convex space and Prx(A) is bounded If

A satisfies (Ha;) then Pry(^4) is cs-closed; if A satisfies (Hwa;) then Pry (A) is ideally convex

course, rint A C %A Consider also the sets c lA if aff A is a closed set,

1 otherwise,

r i A : = | rint A if aff A is a closed set, otherwise,

i6 | lA if XQ is a barreled linear subspace,

- { otherwise,

where X0 = \in(A — a) for some (every) a G A; XQ is the linear subspace,

parallel to aff A

In the sequel, in this section, A C X is a nonempty convex set Taking into account the characterization (1.1) of *A, we obtain that

x € tcA <£> cone(A — x) is a closed linear subspace of X

& M X(A — x) is a closed linear subspace of X,

and

x £ A -£> cone(^4 — a;) is a barreled linear subspace of X

<£> M n(A — x) is a barreled linear subspace of X

If X is a Frechet space and aff A is closed then %CA = lbA, but it is possible

to have ibA ^ and icA = (if aff A is not closed)

The quasi relative interior of A is the set

qri A := {x € A | cone(yl — z) is a linear subspace of X} Taking into account that in a finite dimensional separated topological vector space the closure of a convex cone C is a linear subspace if and only if C is a linear subspace (Exercise!), it follows that in this case qri A = %A =

ic A = ibA

Below we collect several properties of the quasi relative interior

Proposition 1.2.7 Let A C X be a nonempty convex set and T £

L{X,Y) Then:

a £ qri A •& a G A and cone(A — a) = cone(A - A)

O a E i and a - A C cone(yi - a), (1.4)

and T(qriA) C qriT(A) In particular qriA is a convex set Assume that qriA ^ 0; then qriA = A and

*(T(A)) C T(qriA) C qriT(A) C T(A) c T(qriA) (1.6)

Moreover, if Y is separated and finite dimensional then

T(qiiA)=i(T(A)) (1.7)

Proof The first equivalence in Eq (1.4) is immediate from the definition of qriA Of course, cone (A — a) = cone (A — A) implies that a — A C cone(A - a) Conversely, if a — A C cone(A - a) then -cone(A - a) = cone(a — A) C cone (A — a), which shows that cone (A — a) is a linear subspace Therefore Eq (1.4) holds

If a £ lA, from Eq (1.1) we have that a £ A and cone(A — a) is

a linear subspace, and so cone(A — a) is a linear subspace, too Hence

a € qriA The equality qriA = A n qriA follows immediately from the

relation coneC = cone C, valid for every nonempty subset C of X Let a € qri A, x e A, A € [0,1[ and a\ :— (1 — X)a + Xx Then A - A D A - aA = ( l - A)(A - a) + A(A - z) D (1 - A)(A - a),

and so, taking into account Eq (1.4), we have

cone(A - A) D cone(A - a\) D cone ((1 - A)(A - a)) = cone (A — a) = cone (A — A)

Therefore cone(A — A) = cone(A — a^)- Since a\ € A, from Eq (1.4) we obtain that a\ G qriA The proof of Eq (1.5) is complete

Let a £ qriA; then, by Eq (1.4), a - A C cone(A — a), and so

Ta - T{A) = T(a-A)cT (cone(A - a)) C T (cone(A - a))

= cone (T(A) - Ta), which shows that Ta € qriT(A)

Assume now that qriA ^ and fix a £ qriA It is sufficient to show Eq (1.6); then the equality qriA = A follows immediately from the last inclusion in Eq (1.6) for T = Idx and from Eq (1.5)

Let y e i(T'(A)); using Eq (1.1), there exists A > such that (1 + X)y

-XT a € T(A) So, (1 + X)y - -XTa = Tx for some x £ A It follows that y = Txx, where xx ~ (1 + A)_1(Aa + x) But, using Eq (1.4), x\ qriA,

The second inclusion in Eq (1.6) was already proved, while the third is obvious So, let y G T(A); there exists x G A such that y = Tx By Eq (1.5), (1-X)a + Xx G qriAfor A G]0,1[, and so (l-X)Ta + Xy G T ( q r i A ) for A G ]0,1[ Taking the limit for A —> 1, we obtain that y G T(qri A)

When Y is separated and finite dimensional we have (as already ob-served) that i(T(A)) = qriT(A); then Eq (1.7) follows immediately from

Eq (1.6) • The notion of quasi relative interior is related to that of united sets Let

X be a locally convex space and A,B C X he nonempty convex sets; we

say that A and B are united if they cannot be properly separated, i.e if every closed hyperplane which separates A and B contains both of them

The next result is related to the above notions

Proposition 1.2.8 Let X be a locally convex space, A,BcX be

non-empty convex sets and x G X

(i) A and B are united O- cone(A-B) is a linear subspace •& (A — B)~

is a linear subspace

(ii) Assume that cone (A — x) is a linear subspace Then x G c\A

Moreover, i/aff A is closed and rint A ^ then ~x G rint A

Proof, (i) Assume that A and B are united but C := cone(A — B) is not a linear subspace Then there exists XQ G (—C) \ C By Theorem 1.1.5 there exists x* G X* such that {x0, x*) < < (z, x*) for every z G C Then

0 < (x - y,x*) for all x G A, y G B, and so (x,x*) < X < (y,x*) for all

x G A, y G B, for some A G M Therefore Hx* t\ separates A and B It

follows that (x,x*) = X = (y,x*) for all x G A, y G B, and so < (z,x*) for every z G C Thus we have the contradiction (x0,x*) = Therefore

cone (A — B) is a linear subspace

Let C := cone (A — B) be a linear subspace and (x,x*) < X < (y, x") for all x G A, y G B, for some x* G X* and A e i Then < (z,x*) for every

z G A — B, and so < (z, x*) for z € C Then, since C is a linear subspace,

0 = (z,x*) for every z G C which implies immediately that Hx*,\ contains

A and B Therefore A and B are united

The other equivalence is an immediate consequence of the bipolar the-orem (Thethe-orem 1.1.9)

Suppose now that aSA is closed and rint A / Without loss of gen-erality we suppose that x = By what was shown above we have that

x = € clA C cl(affA) = aff A Thus X0 := aS A is a linear space

Assuming now that £ intx0 A ^ 0, we obtain that {x} and A can be

properly separated (in XQ, and therefore in X) using Theorem 1.1.3 This

contradiction proves that x € rint A •

From the preceding proposition we obtain that when X is a locally convex space and A C X is a nonempty convex set, the quasi relative interior of A is given by the formula

qri A = A (~1 {x € X \ {x} and A are united}

The next result shows that the class of convex sets with nonempty quasi relative interior is large enough

Proposition 1.2.9 Let X be a first countable separable locally convex

space and A C X be a nonempty cs-complete set Then qri A ^

Proof Since X is first countable, by Proposition 1.1.11, the topology of

X is determined by a countable family = {pn \ n N} of semi-norms

Without loss of generality we suppose that pn < pn+i for every n € N Since

Ty is semi-metrizable, the set A is separable, too Let A0 = {xn | n € N} C

A be such that A C cl A0 Consider j3n € ]0,2~n] such that /3npn(xn) < 2_ n

The series ^n > 1 Pnxn is Cauchy (since for m > n and p e N w e have that PniT^Z^Xk) < ET=mPkPn(xk) < ET=Z^Pk(xk) < 2"™+!) Taking ^n '•= (Z)n>i Pn) 1Pn, ]Cn>i ^ n1" *s a Cauchy convex series with elements

of A Because A is cs-complete, the series ^n > 1 Xnxn is convergent and

its sum x € A Suppose that x ^ qri A Then there exists XQ € (—C)\C, where C := cb~m(A — x) Using Theorem 1.1.5, there exists x* € X* such that (xo,x*) < < {z,x*) for all z € C In particular ( *) > for every x £ A But En> i ^» ( ^ — ^ ^ o ) = 0- Since {xn — X,XQ) > and

A„ > for every n, we obtain that (x - x, x*) = for every x € AQ Since

AQ is dense in A and x* is continuous, we have that {x — x, x*) = for every x € A, and so (z,a;*) = for every z E C Thus we get the contradiction

1.3 Open Mapping Theorems

Throughout this section the spaces X, Y are topological vector spaces if not stated otherwise We begin with some auxiliary results

Lemma 1.3.1 Let X,Y be first countable topological vector spaces, 31 :

X =$ Y be a multifunction and XQ G X Suppose that grIR satisfies condi-tion (Hwa;) Then

where Nxl^o) ** the class of all neighborhoods of XQ

Proof Let y0 G C\u£7fx(x0)int (clft(t/)) Replacing grft by gv%

-{x0,y0) if necessary, we may assume that (xo,yo) = (0,0) Let U G

Nx-Since X is first countable, there exists a base (Un)n>i C N x of

neighbor-hoods of such that Un + Un C Un-i for every n > 1, where [7o := ?/•

Because G f " ) ^ ^ int (cl#([/)), there exists (Vn)n>i C Ny such that

V„ C int (cl3i([7n)) for every n > Since Y is first countable, we may

suppose that (Vn)n>i is a base of neighborhoods of G Y and, moreover,

yn +i + Vn+i C KJ for every n >

Consider j / ' G int (clCR.(C/i)); there exists fi G]0,1[ such that y := (1 —

lJ)~1y' £ cl3?([/i) There exists (£1,2/1) e g r $ such that x\ G C/i and

2/ - 2/i G M^2- It follows that /x-1(y - j/i) G V2 C cl!R(t/2)- There exists

(^2,2/2) G grR such that xi G L/2 and /x_1(2/ _ 2/i) — 2/2 G M^3- It follows

that /U~2y — fJ-~2yi — H~lyi V3 C c\"R{Uz) Continuing in this way we

find ({xm,ym))m>1 C grR such that xm e Um and n~m+xy - n~m+1yi

-M-m+22/2 - ••• - ym G fiVm+i for every m > Therefore um := y

-2/1-/^2/2 At",_12/m G ^mK i + i C Vm +i for every m > Moreover,

Mm_12/m = vm-i -vm e \im~xVm - fimVm+1, and so ym £ Vro - fiVm+1 C

Vm + Vm C Vm_i for m > It follows that ( xm)m> i -> 0, (ym)m>i ->

and (um)m>i -> 0, whence J2m>i nm~lym has sum y Taking Am :=

( l - / i ) ^m _ 1, we have that (Am)m>i C R)., Em> i A™ = *> Em> i Am2/m h a s

sum (1 - ju)y = y', the sequence ((xTO,ym)) is bounded (being convergent)

and, because £ m t f „+i Ama;TO € C/„+i + Un+2 + ••• + Un+P C f7„+1 + Un+1 C

f/n for every n > 1, the series Zm > Amxm is Cauchy By hypothesis there

exists x' £ X, sum of the series ]Cm>i Am#m> such that (x',y') £ g r X Let

x := (1 - n)~lx'- We have that YZi=i Hm~lxm G t/i + U2 + • • • + Un C

U Thus y' £ "R{U) Therefore int (cl3l(l7i)) C R(U) In particular

0 € int%(U) The proof is complete •

Note that we didn't use the fact that X or Y is separated Observe also that it is no need that x0 £ dom"R when y0 £ f\ue^x(xo) *n t (c^ ( ^ ) ' but,

necessarily, xo £ cl(domD?) and yo £ int(ImlR) in our conditions Note also that condition (Hwa;) may be weakened by asking that the sequence

((xm,ym)) C A be convergent instead of being bounded

In the case of normed spaces one has the following variant of the result in Lemma 1.3.1 Having the normed vector space (nvs) (X, ||-||), we denote by Ux the closed unit ball {x £ X \ \\x\\ < 1}, by Bx the open unit ball

{x € X | ||a;|| < 1} and by Sx the unit sphere {x X | ||a;|| = 1}

Lemma 1.3.2 Let (X, || • ||), (Y, || • ||) be two normed linear spaces and

let 31: X =4 Y be a multifunction Suppose that condition (Hwa;) holds and (x0,y0) &X xY If

yo + nUy C cl (^(XQ + pUx)),

where rj,p > 0, then

yo + nBy C%(x0 + pUx)

Proof We may take (a;o,2/o) = (0,0) One follows the same argument as in the proof of the preceding lemma, but with Un := pUx and Vn := nBy

for n > Consider y' nBy and take p, £]0,1[ such that y :— (1 —

p) y' £ c\"R(pUx)- We find the sequence {{xn,yn))n>l C grft such that

(xn) C pUx and v„ := y - 2/1 - py2 ^ "_ 12 /n £ pnr)BY for n >

Hence (un) -> and /x"_12/„ = u„_i - vn, whence \\yn\\ < 77(1 + p) for

n > Taking An := (1 — p)pn^1 > for n > 1, X)n>i ^ " = 1> *n e s e rie s

S « > i ^n%n is Cauchy and the series X3n>i ^n2/n is convergent with sum y'

Since 51 satisfies the condition (Hwi), we obtain that the series J2n>i ^nXn

is convergent with sum x' and (x',y') £ gr!R Of course, x' £ pUx- Hence

nBy C npUx)- • L e m m a 1.3.3 Let X,Y be topological vector spaces, 3? : X =4 Y be a

Proof Let y0 G f)u£Nx(x0)int (c l%(U))- Replacing gr51 by g r R - { x0, y0)

if necessary, we may assume that (xo,yo) = (0,0) Let us first show that

V i £ ] , l [ , VU,U' £Kx • tc\3l(U)c 31 {t{U + U')) (1.9)

Fix t G ]0,1[ and take tn := t2 for n > Of course, lim£„ • • • t\ = t

Consider U' G Kx- There exists a base of neighborhoods (t/„)neN C K x

such that U\ +U\ C U' and Un+i + Un+i C Un for every n € N Let

Un := (1 - i „ )- 1i „ • tyUn- For every n G N there exists Vj( Ky such

that V^ C c l # ( [ / ' „ ) Consider V„ := ^ ( l - t„)Vn

Let £/ G K x and j/ G cl %{U); we intend to show that ty G 31 (*(£/ + £/'))• We construct a sequence x = XQ G U, X\ G t/i, , xn G J 7n, with

the property:

V n G N : tn .hy G clK (i„ .tx (x + ••• + xn-i +Un)) (1.10)

Since (t/ - Vi) n 3?([7) ^ 0, there exist yx G Vi and x G J7 such that

y — yi G 3£(a:); hence

tij/ = *i(y - yi) + (i - *0 ((i - hyhm) G t&ix) + (l - W C ti3i(a;) + (1 - t i ) c l K ( ^ ) C c\5t{h(x + Ui))

Suppose that we already have x £ U, xi £ Ui, ,xn-i G £/n-i with the

desired property Since

(i„ tiy - Vn+i) n 31 (i„ h(x + • • • + x„_i + Un)) + 0,

there exist yn+i G Vn+i a nd xn G C/n such that

tn -hy-yn+i G i ( tn t i ( H F x „ - i + xn))

Therefore

tn+l •••hy = tn+l(tn- hy — 2/n+l) + ( - < n + l ) ( ( 1 _ tn+1)~~ tn+1yn+i)

£ tn+i$.(tn ti(x-{ ha;„_! +xn)) + (1 -tn+1)V„+1

C i„+i^(*n t i ( i + • • • + a;n-i + a;„)) + (1 - t„+i) d l ( t / ;+ 1)

C cl3t(tn+i h(x-\ hx„_i +xn + Un+i))

So the desired sequence is obtained Since xn+\ H h x „+ p G Un+i + • • • +

Un+P C Un for all n , p € N, the series 2n> i x™ 's convergent Denote by

x1 its sum Since i i + - + i » E i i + f/i and Ui is closed we have that

Let now U" E Nx and V" E Ny be arbitrary There exists n E N such that tn h(x-\ \-xn-i+Un) C t(x+x')+U" andtn .hy G ty+V" By

Eq (1.10) we obtain that tn txy E cl!R(t(x + x') + U") It follows that

(ty + V")nJi(t{x + x') + U") ^ Hi, and so there exist x" E U" and y" G V"

such that ty + y" E $.(t(x + x')+x"), i.e (t(x + x'),ty) + (x",y") € g r # It follows that (t{x + x'), ty) E cl (gr 3V) = grft Therefore ty E ft (t(U + U'))

To complete the proof, let U E Nx(0) Then there exists U' E Nx such that [/' + [/' C 1U From Eq (1.9) we obtain that | c l # ( E / ' ) C 3? ( | ( [ / ' + [/')) C #(*/) Since E C\ue^x{xo) int (cl3i(U)), we have that

0G i n t £ ( [ / ) ° D

Note that Lemma 1.3.3 is a particular case of Lemma 1.3.1 when Y is first countable; otherwise these results are independent

Corollary 1.3.4 Let Y be a first countable topological vector space and

C CY be an ideally convex set Then i n t C = int(clC)

Proof Consider X an arbitrary Frechet space (for example X = R) and ft : X =t Y denned by IR(0) = C, R(x) = for x ^ Of course condition (Hwir) is satisfied Taking y0 E int(clC) and xo = we have that

2/o G f)ue^x(xo)int (c l i(C 7))' a n d s o 2/o € int C O

Theorem 1.3.5 (Simons) Let X and Y be first countable Assume that

X is a locally convex space, "R : X =$ Y satisfies condition (Hwa;), yo E

lb(ImJi) and x0 E 3l_1(j/o)- Then yQ E intaff(ims) R(U) for every U E

N x ( z0) In particular i6(ImIR) = rint(Imft) if i 6(ImK) ^

Proof Once again we may consider that (xo,yo) = (0,0); so Y0 :=

aff(ImCR) — lin(Im3i) Replacing, if necessary, Y by YQ, we may suppose that Y is barreled and G (ImIR)\ Let U E Ncx; since grft is a convex set

and R(U) = Pry (gr ft n U x Y), %(U) is convex, too "R{U) is also absorb-ing Indeed, let y E Y Because Imft is absorbing, there exists A > such that Xy E Imft Therefore there exists x E X such that (x,Xy) E grft Since U is absorbing, there exists fi G]0,1[ such that fix E U As grft is convex we have that (fj,x,fj,Xy) = fx(x,Xy) + (1 — /x)(0,0) G grft, whence /xAj/ G R(U) Therefore $.(U) is also absorbing It follows that cl (£([/)) is an absorbing, closed and convex subset of the barreled space Y Therefore G int (clft(C/)) Using Lemma 1.3.1 we obtain that G intft(Z7) for every neighborhood U of G X The last part is an immediate consequence of

Corollary 1.3.6 Let X be a Frechet space, Y be first countable and

51 : X =$ Y be li-convex Assume that yo € t6(Im!R) and XQ 5i~1

(yo)-Then y0 £ intaff(im3j) 5l(U) for all U E Nx(zo)- In particular *b(Im5l) =

rint(Im^) provided i 6(ImK) ^

Proof There exist a Frechet space Z and an ideally convex multifunction

S : Z x X =4 Y such that grIR = P r xxy ( g r S ) - Then § verifies condition

(Hw(z,i)) by Proposition 1.2.6 (ii) Of course, there exists ZQ Z such that yo £ S(zo,zo)- Since ImS = Imft, by the preceding theorem, j /0 €

intaff(imX) #(E0 = intaff(imK) HZ x ^ ) for every U G Kx(^o)- •

Theorem 1.3.7 (Ursescu) Let X be a complete semi-metrizable locally

convex space and 51 : X =£ Y be a closed convex multifunction Assume that yo £ t6(ImCR) and xo € 5l~1(yo)- Then yo € intaff(Im^) 5l(U) for every

U £ X j ( x o ) In particular i 6( I m ^ ) = rint(ImK) if ib{Im 51) ^

Proof If Y is first countable it is obvious that the conclusion follows

from Simons' theorem Otherwise the proof is exactly the same as that of Simons' theorem but using Lemma 1.3.3 instead of Lemma 1.3.1 •

An immediate consequence of Theorem 1.3.5 is the following corollary

Corollary 1.3.8 Let Y be a first countable barreled space and C CY be

a lower ideally convex set Then Cl = i n t C

Proof Let yo G Cl There exist a Frechet space X and an ideally convex

multifunction 51: X =3 Y such that C = Im 51 The conclusion follows from

Theorem 1.3.5 taking again U — X O In fact the conclusion of the above corollary holds if C is the projection

on Y of a subset A of X x Y with (a') X is a first countable locally con-vex space and A satisfies condition (Hwx) or (b') X is a semi-metrizable complete locally convex space and A is closed and convex

Putting together Corollaries 1.3.4 and 1.3.8 we get the next result

Corollary 1.3.9 Let Y be a first countable barreled space and C C Y be

an ideally convex set Then Cl = i n t C = int(clC) = ( c l C ) \ •

In normed spaces the following inversion mapping theorem holds T h e o r e m 1.3.10 (Robinson) Let (X, ||-||) and (Y, ||-||) be normed vector

that y0 + rjUy C 3\{XQ + pUx) for some n,p > Then

d f c K -1^ ) ) < P+\\x-x0\\ ,d( ^( a.)) VarGX, Vy£y0 + r,BY

v-\\y-yo\\

Proof Replacing g r # by g r # — (xo,yo), we may assume that (xo, yo) =

(0,0) Let x E X and y G nBy The conclusion is obvious if x £ domIR or y € &(z), so suppose that neither is true Choose > and find ye € 3£(:c) such that < \\ye - y\\ < d[y, R(x)) + 6; define a := n - \\y\\ > and take

e G ]0, a[ Consider

ye :=y + {a-e)\\y-ye\\~1 (y - ye);

thus ||y£|| < \\y\\ + (a — s) =n — e, and so ye G nBy- Therefore there exists

xe G pUx with y£ G R(x€) Define A := \\y -yg\\(a- e + \\y - y ^ l )- G

]0,1[ Then

2/ = (1 - X)ye + Xys G (1 - A)tt(a:) + A#(x£) C E((l - A)z + Aa;£)

Thus (1 - A)z + Xx£ G 3£_1(j/), whence d ( x , ^ -1^ ) ) < A | | x - xe| | As

ll^ — ^ell < \\x\\ + ll^ell < P + INI a nd A < (a — e )_ 1 ||y — yg\\, we obtain

that

d (x^iy)) <(P+ \\x\\) (a - e ) '1 (d(y, X(x)) + 9)

Letting 6, e —> 0, we obtain the conclusion • Combining the preceding result and Lemma 1.3.2 we obtain the

follow-ing important result in normed spaces The implication (i) =$> (ii) is met in the literature as the Robinson-Ursescu theorem

Theorem 1.3.11 Let (X, ||-||), (Y, ||-||) be two normed vector spaces and

31 : X z t be a multifunction Suppose that Y is a barreled space, g r # verifies condition (Hwx) and (xo,yo) G gr3i Then the following conditions are equivalent:

(i) yoeQmX)'; (ii) 2/0 e int:R(xo + Ux)\

(iii) 3n > 0, VA G [0,1] : 2/0 + XnUy C IR(a;o + At/X);

(iv) 37,77 > 0, Vz G z0 + r?t/X; Vy G y0 + nUy : d ( ^ S T ^ y ) ) <

7-d(y,K(a;));

(v) r?> , V y G y o + r?[/x, V x G X : d(x,-R-\y)) < ^ E f f f •

Proof The implications (hi) =>• (ii) =>• (i) are obvious The implication (i) =$> (ii) follows immediately from Simons' theorem, while the implication (ii) =$> (v) is given by the preceding theorem

(iv) => (iii) Let 7' > 7; we may assume that 777' < 1, because, in the contrary case, we replace 77 by 77' := I / ' < 77 Let y £ yo + A?7?7y, y 7^ j/o, w i t h A e ] , l ] T h e n d ( a ; o , ^ -1( y ) ) < • d{y,3L(x0)) < 7' ||y - yQ\\ Hence

there exists x € 5l~1(y) such that ||x —io|| < 7'|l2/~2/o|| < 7'Ar/ < 77

Therefore y0 + A^C/y C R(x0 + \UX

)-(v) => (iv) Taking a; € XQ + § t / x , 2/ G j/o + f^y' w e obtain that

d (x, ft"1 (y)) <i-d(y, %(x)) with := + 77/2 D

Important consequences of the Ursescu theorem are: the closed graph theorem, the open mapping theorem and the uniform boundedness princi-ple; we state the first two of them in Frechet spaces

Theorem 1.3.12 (closed graph) Let X, Y be Frechet spaces and T :

X —> Y be a linear operator Then T is continuous if and only if gr T is closed in X x Y

Proof It is obvious that gr T is closed if T is continuous (even without being linear) Suppose that g r T is closed and consider the multifunction

51 := T_ 1 : Y =$ X It is obvious that grlR is closed and convex (even

linear subspace) Moreover ImO? = X So we can apply Theorem 1.3.5 for

(Tx0,x0) G Y x X Therefore

V V e N H T i o ) : R(V)=T-1(V)£Nx(x0),

which means that T is continuous at XQ •

Corollary 1.3.13 (Banach-Steinhaus) Let X, Y be Frechet spaces and

T : X —> Y be a bijective linear operator Then T is continuous if and only if T_ 1 is continuous; in particular, if T is continuous then T is an

isomorphism of Frechet spaces

Proof Apply the closed graph theorem for T and T- 1, respectively D

Theorem 1.3.14 (open mapping theorem) Let X, Y be Frechet spaces

and T G L(X,Y) be onto Then T is open

Proof Of course, T is a closed convex relation and Im T = Y Let D C X be open and take 7/0 G T{D)\ there exists XQ G D such that 7/0 = Tx0

Applying the Ursescu theorem for this point, since D is a neighborhood of

An interesting and useful result is the following

Corollary 1.3.15 Let X, Y be Frechet spaces, A C X and T : X ->• Y

be a continuous linear operator Suppose that I m T is closed Then T(A) is closed if and only if A + ker T is closed

Proof Replacing, if necessary, F by I m T and T by T" : X -> I m T ,

T'{x) := T(x) for x G X, we may suppose that T is onto Consider T : Xj kerT —> Y, T(x) := T(x), where x is the class of x It is easy to verify

that T is well defined, linear and bijective Let q : X —> M be a continuous semi-norm; since T is continuous, p := qoT is a continuous semi-norm, too For all x € X and u € ker T we have that (<? o T)(x) = g (T(z + u)) = p(x + u), whence g o T < p, where p(x) := inf{p(a; + u) \ u £ kerT} Therefore

T is continuous Since X is a Frechet space, X/keiT is a Frechet space,

too Using now the preceding corollary we have that T is an isomorphism of Frechet spaces It is obvious that T(A) = T(A), where A :— {x \ x € A} Taking IT : X —> X/keiT, ir(x) := x, we obtain that

T(A) is closed & n(A) - A is closed <$• IT'1 (A) = A + ker T is closed,

and so the conclusion holds • As an application of the Simons and Ursescu theorems we give the

fol-lowing two interesting results, useful in studying optimal control problems We recall that a process is a multifunction C : X => Y whose graph is a cone; when the graph of the process C is convex or closed one says that e is a convex process or closed process, respectively The adjoint of the process is the w*-closed convex process 6* : Y* =3 X* whose graph is the set {(y*,x*) £ Y* x X* | (-x*,y*) € ( g r e ) + )

Theorem 1.3.16 Let X, Y, Z be Banach spaces, : X =$Y be an ideally

convex process and T G £(•£, Y) Consider the following statements:

(i) ImT d m ;

(ii) 3P l> , V(»*,*•) GgrC* : \\T*y*\\ < Pi\\x*\\;

(hi) 3P2>o : T{UZ) c P2e(uxy,

(iv) 3p3>0 : T(Uz)Cp3c\(e(Ux))

Then (i) ôã (hi), (ii) ãÊã (iv) with p\ = p3 and (hi) => (iv) with p3 = p2

Moreover, ifT is onto then (iv) => (hi) with (any) p2 > p3

Proof The implications (hi) =$• (i) and (iii) => (iv) (with p3 = p2) are

(i) => (hi) Let 3? := T~1oQ; one obtains immediately that 3J is an ideally

convex process with ImIR = Z Applying the Simons theorem (Theorem 1.3.5) for (a;o,2/o) = (0)0); there exists p > such that pUz C 3£(t/x),

which means that pT(Uz) C G(Ux)- Taking P2 := p"1, (hi) holds

(iv) => (ii) Let (y*,x*) G grC*, i.e (-z*,y*) (grC)+ Then

| | T V I I = sup ( z , - T V > = sup ( T z , - y * ) < sup fo.-y*)

||z||<i z&z yep3c\(e(ux))

= p3 sup (y, -y*) < p3 sup{(a;, -x*) \ (x,y) G grC, ||a;|| < 1}

yee(ux) <P2\\x*\\

because (y, —y*) < (x, —x*) for (x,y) G grC So (ii) holds with p\ = p3

(ii) =*• (iv) Suppose that (iv) does not hold and take z G Uz such that Tz $ /93cl(C([/x))- Since C(f/x) is convex, using Theorem 1.1.5, there exist y* G Y* and A G E such that

%T*T) = (Tz,y*) < A < (p3y,y*) Vy G G(UX)

It follows that A < 0, and so we can take A = -p3 Hence —p3 > (z, T*y*) >

- | | r * y * l l and - < (y,^) = (x,0) + (y,y*) for (x,y) G grC with x G Ux,

whence ||T*y*|| > p3 and

(0, y*) G (gr n Ux x ) ° = «;* - cl ((gr 6)+ + tf* x {0})

= (gre)+ + ?7x' x { }

because Ux* x {0} is W*-compact (we have used the Alaoglu-Bourbaki

theorem) Therefore there exists x* G Ux* such that (y*,x*) G gr(2* (<S> ( - r ,j/*) G (grC)*) Since \\T*y*\\ > p3 > p3 \\x*\\, (ii) does not hold

for pi= p3

(iv) =^ (iii) when T is onto Since T is onto and Z, Y are Banach spaces, T is open (see Theorem 1.3.14) It follows that int(T(Uz)) = T{BZ) But

G(Ux) is cs-closed Indeed, by Proposition 1.2.2, A := grCnUx x Y is

cs-closed, and so, X being complete, A satisfies condition (Hz) (see Propo-sition 1.2.6(h)); by PropoPropo-sition 1.2.6(v) we have that Pry(A) = G(Ux) is cs-closed Using Corollary 1.3.4 we obtain that intC([/x) = iat(c\G(Ux))-So, from (iv) we obtain that T(BZ) C p3Q(Ux), and so T{UZ) C p<i£(Ux)

for every p2 > p3 D

T h e o r e m 1.3.17 Let X,Y,Z be normed spaces, T e £>{X,Y) and :

X =$ Z be a convex process Consider the following statements:

(i) I m T ' C l m e * ;

(ii) p > , V(x,z) S g r C : ||Ta;|| < p||;z||; (iii) p > : T*(UY*)CPe*(Uz>)

Then (i) O- (iii) and (ii) O- (iii) with the same p

Proof The equivalence of (i) and (iii) follows from the equivalence of (i)

and (iii) of the preceding theorem; just apply it for the Banach spaces X*,

Y*, Z*, the continuous linear operator T* and the closed convex process

e*

(iii) =>• (ii) The proof follows the same lines as the proof of (iv) =£• (ii) in the preceding theorem, so we omit it

(ii) => (iii) Even if the proof is similar to (ii) =>• (iv) in the preceding theorem, we give the details So, suppose that (iii) does not hold and take

y* e Uy* such that T*y* £ pG*(Uz>) We may assume that y* £

Sx-(otherwise replace y* by ||2/*||_1y*) Using the fact that Uz* is w*-compact

and gre* is u>*-closed it follows easily that G*(Uz*) is uAclosed; being also convex, we can apply Theorem 1.1.5 in (X*,w*) Therefore there exist

x £ X and A G E such that

(Tx,y*) = {x,T*y*) > A > (x,px*) Vx* € e*{Uz>)

Hence A > 0, and we can take X = p Hence ||Tx|| > p and > (x, x*) for every x* € e*(Uz>), i.e - < (x,x*) + (0,z*) for (x*,z*) (grC)+ with

z* £ Uz* • It follows that

(af,0) € ( ( g r Q + f l l x Uz.)° = cl ((gre)++ + {0} x UZ)

= cl(gve+{0}xUz)

Hence there exist ((xn,zn)) C grC and (z'n) C Uz such that (xn) -> x

and (wn) := (zn + z'n) ->• It follows that (||Ta;„||) ->• \\Tx\\ and \\zn\\ =

\\wn — z'n\\ < + ||iwn|| for every n, whence plimsup \\zn\\ < p < \\Tx\\ <

liminf ||Ta;„|| Therefore there exists n0 such that ||Ta;n|| > p||z„|| for

1.4 Variational Principles

A very important result in nonlinear analysis is the "Ekeland variational principle."

Theorem 1.4.1 (Ekeland) Let {X,d) be a complete metric space and

f : X —» M be a proper lower semicontinuous and lower bounded function Then for every xo G dom / and e > there exists xe G X such that

f(xE) < f(x0) -ed(xQ,xE)

and

f{xe)<f{x)+ed{xe,x) V x l \ { x£}

Proof Let XQ G dom / and e > be fixed For every x G X consider the set F(x) := {y G X \ f{y) + ed(x,y) < f(x)} Note that the conclusion is equivalent to the existence of an xe G F(xo) such that F(xs) = {xE}

Since the function X y *-> f(y) + ed(x, y) G E is lower semicontinuous (lsc for short), F(x) is closed Note that x G F(x) C d o m / for every

x G d o m / and F(x) = X for x G X \ d o m / Also note that F(y) C F(x)

for every y G F(x) The inclusion is obvious for x ^ d o m / So, let

x G d o m / , y G F(x) and z G F(y) Then

f{z)+sd(y,z)<f(y), f{y)+sd(x,y) < f{x), d(x,z) < d(x,y) + d{y,z)

Multiplying the last relation by e, then adding all three relations we obtain that f(z) + ed(x,z) < f(x), i.e z G F(x)

Since / is bounded from below, s0 := inf{/(x) | x G F(xo)} G E; take

X\ G F(xo) such that / ( ^ i ) < SQ + 2~1 Then consider si := inf{/(z) | x G

F(xi)} G E and take x2 G F(a;i) such that ffa) < s\ + 2~2 Continuing

in this way we obtain the sequences (sn)n>o C M and (xn)n>o C X such

that

sn = inf{/(a;) | x G F(xn)}, xn+1 G F(xn), f{xn+1) < sn + 2-"-

for all n > Because i^(a;n+i) C F(xn), s„+i > s„ for n > Moreover,

as xn+i G F ( i „ ) ,

ed(a;ri+i,a;n) < f(xn) - f(xn+1) < f{xn) - sn < f(xn) - s„_i < 2~n

for n > 1, whence d(x„+ p,a;„) < e~121~n for n , p > This shows that

xm G F{xn) for m > n, we have that xE € clF(a;n) = F(xn) for every

n > In particular xE £ F{XQ). Let a; F(ar£); then x € F(xn) for every

n > 0, and so, as above, ed(x,xn) < f(x„) — f(x) < f(x„) - sn < 2~n,

which shows that (xn) —>• x As the limit is unique, we get x = xe, which

shows that F{xe) = {xe} The proof is complete •

In applications one often uses the following variant of the Ekeland variational principle In the sequel by infx / , or simply inf/, we mean

mi{f(x) | x € X}, where / : X -> I

Corollary 1.4.2 Let (X,d) be a complete metric space and f : X -» R

be a bounded below lower semicontinuous proper function Let also e > and xo S d o m / be such that f(xo) < mix f + £• Then for every A > there exists x\ S X such that

f(x\) < f(x0), d(xx,x0) < A

and

f(xx) < f{x) + eX-idfaxx) Va; £ X \ {xx}

Proof Applying the preceding theorem for xo and eA_ 1, we get x\ € X

satisfying the second relation of the conclusion and f(x\)+e\~1d(xo,x\) <

f(xo) Hence f{x\) < f{xo) Moreover, because f(x0) < inf^ / + £ <

f(x\) + e, we get also that d{x\, XQ) < A D

A good compromise is obtained by taking A = -Jz in the preceding result An interesting application of the Ekeland theorem is the following result

Corollary 1.4.3 Let (X, ||-||) be a Banach space and f : X -> R be a

lower semicontinuous function Assume that f is Gateaux differ•entiable and bounded from below If (xn)n^ is a minimizing sequence for f, i.e

(f(xn)) -> infjt / , then there exists a minimizing sequence (xn) for f such

that (\\xn -xn\\) -> and (||V/(x„)||) -»

Proof Consider en := f(xn) — infx f £ K+- If f{xn) = infx f we

take xn := xn; otherwise, applying the preceding corollary for xn, en and

A = y/e„~ we get xn e X such that f(x„) < f(xn), \\x - xn\\ < ^fe^ and

f(xn)<f(x)+^/£n~\\x-xn\\ MxeX

tu) — f{xn)) Taking the limit for t —¥ we get — y/e^\\u\\ < (u,S7f(xn))

for all u £ X It follows that ||V/(z„)|| < y/ệ The conclusion follows • The next result is met in the literature as the smooth variational

prin-ciple This name is due to the fact that, at least in Hilbert spaces, the

perturbation function is smooth (for p > 1) The result will be stated in a Banach space (X, ||-||) even if it holds in any complete metric space (with the same proof) Before stating it let us observe that, when (un)n>o C X

is bounded, (nn)n>o C M+ is such that Yln>o ^n — 1 a n^ P £ [1J °°[> the

function

Qp : X -+ E, Qp(x) := ^2n>0^n ^ ~ Un^ ' (L 1)

is well defined and Lipschitz on bounded sets; in particular 0P is continuous

Theorem 1.4.4 (Borwein-Preiss) Let (X, ||-||) be a Banach space, f :

X —> M be a proper lower semicontinuous and bounded from below function and p [1, oo[ If xo £ X and e > are such that f(xo) < mix f + £ then for every A > there exists a sequence (wn)n>o C B(xo,X) converging to

some u € X such that

f(u) < mix f + e, \\xo - u\\ < A and

f(u) + e\-pQp{u) < f(x) + e\-pQp{x) VxeX, (1.12)

where Qp is defined by Eq (1.11)

Proof Fix A > and consider 7, S, n, /i, > such that

f(x0) - mixf < V < < e, H<l-ie~\ < M( l - (7?7~1)1 /T>

(1.13) and := (1 - fi)e\~p Let uo := #o and /0 := / Taking fi(x) := fo(x) +

5 \\x - u0\\p for a; £ I , we have that fi(u0) = /o(«o) > mix h- Hence there

exists u\ X such that f\(u\) < 6f0(u0) + (1 — 6) mix fi- Taking f2(x) :=

fi(x) + 5/i\\x — ui\\p for £ I , we have that /2(ui) = fi(ui) > mix

fi-Hence there exists u2 £ X such that f2(u2) < 6f\(u\) + (1 — 0)infx /2

Continuing in this way we obtain a sequence (u„)n>o C X and a sequence

of functions (fn)n>o such that

and

fn+l{Un+l)<Ofn(un) + (l-e)miXfn+l V n > (1.15)

It is clear that / „ < fn+i; taking sn := infj*:/n, (s„)„>o C E is a

non-decreasing sequence Let an := / „ ( un) ; because fn+i{un) = fn(un) >

i n f x / n + i , from Eq (1.15) we obtain that an+\ < an for every n >

Using again Eq (1.15), we get

sn < s„+i < an+i < 0an + (1 - 9)sn+i < an,

and so

a-n+i - sn+i < Qan + (1 - 6)sn+i - sn+i = 6(an - sn+i) < 0(an - s„),

whence

an-sn<9n(a0-s0) Vn > (1.16)

It follows that lims„ = lima„ € E Taking x := un+i in Eq (1.15), we get

fln > a«+i = fn{un+i) + Sn" \\un+1 - un\f >sn + 6p,n ||u„+i - un\\p ,

which, together with Eqs (1.13) and (1.16), yields

Sfi" | | «n + 1 - u „ f <an-sn< 6n(a0 - s0) < 6nr) Vn >

It follows that | | un + 1 - un\\ < (77/<5)1/J ,(6'/^)"/P, whence

\\un+m - u „ | | < (v/8)1/p(0/l*)n,p{l ~ Wltf1')'1 V n , m >

Since < #//z < 1, the sequence (un)„>o is Cauchy, and so it converges to

some u X Moreover, the inequality above and (1.13) imply that

I K - unll < (V/S)1/P(vhr1/P = (l/S)1/p < (e/Sy/p(l - fi)1^ = A

(1.17) for all n,m > In particular (un) C B(xo,X)- Letting m —> co in the

above inequality we get \\u — un\\ < A for n > 0, and so \\u — xo\\ < A

Consider n„ := fin(l - fx) > 0; hence J2n>of1n = 1- Let 0P be defined by

Eq (1.11) From Eq (1.14) we obtain that

and so f(x) + eX pQp(x) — lim/„(a;) > lims„ Using Eq (1.17) we get

f(un) + e\~pQp(un) = fn(un) + e\~p Y] ^k \\un - uk\\p

fikX" = an + e} ak

K=n+1 *—/k=n+l

for every n > Taking into account the continuity of 0P and the lower

semicontinuity of / , the preceding inequality yields f(u) + eX~pQp(u) <

lima„ = lims„ Therefore (1.12) holds From Eq (1.17) we obtain that

Qp{x0) < Y " ^Un I K - Un\\P < J~] HkjS"1 = fJ.jS-1 < fJ,XP,

^—'n>l ^ — ' n > l

and so, by Eq (1.12),

f{u) < f(x0) + eX~pep{x0) < f(x0) +sfi< infx / + + £/* < infx f +

£-The proof is complete D Note that |©i(:r) -Qi(y)\ < \\x — y\\, and so Eq (1.12) becomes f(u) <

/(a;)+£A-1||a;—w|| for every 2; € Xwhenp= So, under the slight stronger

condition f(x0) < mix / + £ one recovers the conclusion of Corollary 1.4.2,

but the condition f(x\) < f(xo)

Although not directly related to what follows, we give the next two dual interesting results which have not been published by their author, C Ursescu

Theorem 1.4.5 Let (X,d) be a complete metric space and (F„)neN be a

sequence of closed subsets of X Then

c l(U„e N i n t F")=d i n t(U„e N^)- d-18) Theorem 1.4.6 Let X be a complete metric space and (Dn)ne^ be a

sequence of open subsets of X Then

i n t(n„6 N c i^)= i n t c i(n„e N^)- (L I 9)

Note that Theorem 1.4.5 can be obtained immediately from Theorem 1.4.6 by taking Dn = X\Fn (and Theorem 1.4.6 is obtained from Theorem

1.4.5 by taking Fn = X \ Dn) We give only the proof of Theorem 1.4.6

Then there exists ri G]0,r[ such that D{xi,r\) C f]n£Nc\Dn;

there-fore D(xi,r\) C clD„ for every n £ N It follows that B(x\,ri) n D\ is an open nonempty set There exist X2 € X and r2 ] , r i / ] such

that D(x2,r2) C B(xi,n) D £>i It follows that D f o , ^ ) C D(a;i,ri) C

cl£>2) and so B{x2,r2) fl £>2 is an open nonempty set Continuing in

this way we find the sequences (a?n)neN C X and (rn)n£N C F such that

(r„)n£m ->• and D(xn+\,rn+i) C B ( x „ , r „ ) n £>„ for every n In

partic-ular D(a;n+i,rn+i) C D(xn,r„) for every n Since X is a complete metric

space, using Cantor's theorem, (~}neND(xn,rn) = {x'} for some x' € X It

follows that x' £ £>„ for every n Since x' D(xi,r{) C B(x,r), we have that B(x,r)nf\n€JiDn As r > is arbitrary, x £ cl (finer*-^M- Therefore

int ( f |n e Nc l D „ ) C cl ( f |n € NA i ) , whence the inclusion "C" in Eq (1.19)

holds, too • From Theorem 1.4.5 we obtain immediately the famous Baire's

theo-rems:

Theorem 1.4.7 (Baire) Let (X,d) be a complete metric space Then any

countable intersection of dense open subsets of X is dense

Proof Let A = C\n€N Dn with Dn open and dense for every n £ N From

Eq (1.19) it follows that X = int(clA), and soc\A = X • Remind that an intersection of a countable family of open subsets of

the topological space (X, r ) is called a Gs set

Theorem 1.4.8 (Baire) If (-Fn)ngN is a sequence of closed subsets of the

complete metric space (X,d) such that X = \Jne^Fn, then at least one of

Fn 's has nonempty interior

Proof The conclusion is immediate from Eq (1.18) •

1.5 Exercises

Exercise 1.1 (Caratheodory) Let X be a linear space of dimension n N and

let A C X be nonempty Prove that

c o A = { Y^ill^ | M i e W T C R+, (xi)ieT^+T C A, Y^ill^ = ! } •

Exercise 1.2 Let X be a finite dimensional normed space and A C X be a

nonempty convex set Prove that 'A =£ 0, "(cl A) — 'A, cl(*A) = cl A and for every

E x e r c i s e 1.3 Let X be a separated locally convex space, H C X be a closed hyperplane and A C X be a nonempty convex set If int# (A O H) ^ and

A <£ H prove that int ^4 ^ Moreover, if M C X is a closed affine set with finite

codimension and intM{A f~l M) ^ 0, prove that rint A ^

E x e r c i s e 1.4 Let X be a separated locally convex space and A, C C X be non-empty convex sets with int C ^ Prove that int(A + C) = A + int C Moreover, if A n int C ^ 0, then cl(A DC) = cl yl l~l cl C In particular, if C is a convex cone with nonempty interior, then cl C + int C — int C

E x e r c i s e 1.5 Let X be a separated locally convex space, Xo C X be a linear subspace, a i , , ap G X, ipi, , tpk G X" and a i , , a* G R, where p, A; G N

Prove that:

(a) if Xo is closed and C = { E L i - ^ I Vi G L7p : A, > 0}, then X0 + (7 is

a closed convex cone In particular C is a closed convex cone;

(b) Xo + {x £ X | Vi 1, k : (x, ipi) < at} is a convex closed set

E x e r c i s e 1.6 Let (X, (• | •)) be a Hilbert space, xo £ X \ {0} and a G [0,7r/2] Prove that

P ( a ) := {x £ X | Z(a;, x0) < a}

is a closed convex cone, where Z-(x,y) := Arccos " Moreover, P(a)° = P ( r / - a )

E x e r c i s e 1.7 Let n £ N \ {1}, p > and a i , , a „ £]0,1[ be such that a i + + an = Consider

ATp := { ( x i , , x „ , xn +i ) G Rn + 1 | xi, ,x„ > 0, | xn+ i | < px"1 •• -a;""} •

Prove that (Kp)+ = Ry, where p' := (pa"1 • • • a " " )- 1 In particular, Kp is a

closed convex cone E x e r c i s e 1.8 Prove that

P := {(x,t/,z) £ R3 | x,z > , x z > t/2}

is a closed convex cone and P+ = P (P is called the "ice cream" cone.)

E x e r c i s e 1.9 Let X be a normed space, ip G X* \ {0} and < a < \\(p\\ Prove that the set C := {x G X | <p(x) > a\\x\\} is a pointed (C -C = {0}) closed convex cone with nonempty interior and C+ = R+ • ((p + aU*) = R+ • D((p, a)

E x e r c i s e 1.10 Let X, Y be topological vector spaces and 31 : X =t Y be a convex multifunction Assume that there exists x G X such that int 3^(5:) ^ Prove that for every (xo,j/o) G grft with yo G (Imft)1 there exists u G X such

E x e r c i s e 1.11 Let X, Y be two separated locally convex spaces and : X =4 Y be a multifunction whose graph is a closed linear subspace of X x Y Prove that domC is dense if and only if C*(j/*) is a singleton for every y* G dome* In particular, dom (2 is dense and C(x) is a singleton for every x G dom G if and only if domC* is w*-dense and C*(y*) is a singleton for every y* € dom6*

E x e r c i s e 1.12 Let (X, d) be a metric space Prove that (X, d) is complete if for any Lipschitz function / : X —• R+ there exists x € X such that f(x) <

f(x) + d(x,x) for every x £ X This shows that the completeness assumption in

Ekeland's variational principle is essential

E x e r c i s e 1.13 Let (X, d) be a complete metric space and / : X —)• R be a proper lsc and lower bounded function Suppose that for every x G X such that

f(x) > inf / , there exists x G X\{x} such that f(x) + d(x,x) < f(x) Prove that

argmin / ^ and d(x, argmin / ) < f(x) — inf / for every x E X

Exercise 1.14 Let (X, d) be a complete metric space, / : X —• R be a proper

lsc lower bounded function and 3£ : X =t X Assume that for every x G X there exists y G 3l(x) such that d(x,y) + f(y) < f(x) Prove that 31 has fixed points,

i.e there exists xo S X such that xo €

3£(xo)-E x e r c i s e 1.15 Let (X, ||||) be a normed space and / : X -> R Prove that (i) limn^n^oo f(x) = oo if and only if [/ < A] is bounded for every A G R (ii) Assume that X = Rfc, / is lsc and limnsn^oo f(x) = oo (i.e f is coercive)

Prove that there exists x G R* such that f(x) < f(x) for every x G Rfc

1.6 B i b l i o g r a p h i c a l N o t e s

S e c t i o n 1.1: For the notions and results on topology and topological vector spaces not recalled in this section one can consult many classical books (see [Kelley (1955); Willard (1971); Bourbaki (1964); Holmes (1975)]) The proofs of the results mentioned in this section can be found in [Holmes (1975)] or [Bourbaki (1964)]

S e c t i o n 1.2: Ideally convex sets were introduced by Lifsic (1970), cs-closed sets by Jameson (1972), cs-complete sets by Simons (1990), lower cs-closed sets by Amara and Ciligot-Travain (1999), while condition (Hx) was used by Zalinescu (1992b) The properties stated in Proposition 1.2.1 are mentioned in [Kusraev and Kutateladze (1995)] All the results concerning lcs-closed sets, as well as Proposi-tion 1.2.2 (for cs-closed sets), are from [Amara and Ciligot-Travain (1999)] The set *CA, introduced by Zalinescu (1987), is also introduced by Jeyakumar and

Wolkowicz (1992) under the name of strong quasi relative interior of A The notation lbA was introduced in [ZalinescuN(1992b)] but the condition G lbA

stated in [Borwein and Lewis (1992)] for A a cs-closed set and X a Frechet space Joly and Laurent (1971) introduced the notion of united sets, but Proposition 1.2.8 is mainly established by Moussaoui and Voile (1997)

S e c t i o n 1.3: Lemma 1.3.1 is proved by Amara and Ciligot-Travain (1999) for

X a metrizable lcs, Y a Frechet space and ft cs-closed, while the statement and

proof of Lemma 1.3.3 are those of Ursescu (1975); Corollary 1.3.4 (for Banach spaces) is due to Lifsic (1970) The statement of Theorem 1.3.5 is very close to that of [Kusraev and Kutateladze (1995), Th 3.1.18]; when X, Y are Frechet spaces our statement is slightly more general For X, Y metrizable locally convex spaces and CR satisfying (Ha;) Theorem 1.3.5 is equivalent to the open mapping theorem of Simons (1990) Theorem 1.3.7 is obtained by Ursescu (1975) for

Y0 := aff(Imft) = Y It is established in [Robinson (1976)] for X, Y Banach

spaces and Yo = Y; in this case the above result is met in the literature under the name of Robinson-Ursescu theorem Corollary 1.3.8 was obtained by Amara and Ciligot-Travain (1999) for lcs-closed sets Theorem 1.3.10 is due to Robinson (1976), while Theorem 1.3.11 can be found in [Li and Singer (1998)] Theorems 1.3.16 and 1.3.17 are due to Carja (1989) (with different proofs)

Convex Analysis in Locally C o n v e x Spaces

2.1 Convex Functions

We begin this section by introducing some notions and notations For a function / : X —¥ E and A g l consider:

d o m / e p i / e p is/

[/<A] [/<A]

= {x e x | f(x) < +00},

= {(x,t) eX x E I f(x) < t},

= {(x,t)\f(x)<t}, = {x X I f{x) < A},

= { i £ l | f(x) < A};

the sets d o m / , e p i / and epis / are called the domain, the epigraph and

the strict epigraph of the function / , respectively, while the sets [/ < A] and [/ < A] are the level set and strict level set of / at height A One says that the function / is proper if d o m / 7^ and f(x) > — 00 for every

x £ X It is evident that d o m / = P r y ( e p i / )

Let X be a real linear space and / : X -> E; we say that / is convex if

Var.l/GX, V A g [ , l ] : /(Arc + (1 - \)y) < \f(x) + (1 - X)f(y), (2.1)

with the conventions: (+oo) + (—00) = +00, 0-(+oo) = +00, 0-(—00) = Note that when x = y, or A £ {0,1}, or {a;, y} (£ dom / , the inequality (2.1) is automatically satisfied Therefore the function / is convex if

V x , j / e d o m / , x ^ , V A G ] , l [ : /(Aar + ( l - \ ) y ) < \f(x) + (1 - \ ) f ( y ) (2.2)

If relation (2.2) holds with " < " instead of " < " we say that / is strictly

convex Similarly, we say that / is (strictly) concave if —/ is (strictly)

convex Since every property of convex functions can be transposed easily to concave functions, in the sequel we consider, practically, only convex functions

To avoid multiplication with 0, taking into account the above remark, we shall limit ourselves to A G ]0,1[ in the sequel

In the following theorem we establish some characterizations of convex functions

Theorem 2.1.1 Let f : X —>• E The following statements are equivalent:

(i) / is (strictly) convex;

(ii) the functions tpxy : E —)• R, (px,y(t) := f((l — t)x+ty), are (strictly)

convex for all x, y G X (x ^ y);

(hi) dom f is a convex set and

Vx,yedomf, V A e ] , l [ : f(\x + (1 - X)y) < Xf(x) + (1 - X)f(y) (the inequality being strict for x ^ y);

(iv) V n € N, \/xi, ,xn E X, VAi, ,A„ G]0,1[, A i + - - - + A„ = :

f(XlXl + • • • + Xnxn) < X1f{x1) + ••• + Xnf(xn) (2.3)

(the inequality being strict when Xi, , xn are not all equal);

(v) epi f is a convex subset of X x E; (vi) epis / is a convex subset of X x E

Proof The implications (i) => (ii), (ii)=^(i), (i) =>• (hi), (iii) =4> (i) and (iv) => (i) are obvious

(i) =£• (v) Let {xi,ti),(x2,t2) e p i / and A e]0,1[ Then x\,x2 G

d o m / , f(xi) < ti and f(x2) < t2 Since / is convex, from Eq (2.1) we

obtain that

f{Xxx + (1 - X)x2) < Xf{Xl) + (1 - X)f(x2) < Xtx + (1 - X)t2,

whence X{xi,t\) + (1 - X)(x2,t2) € e p i / Therefore e p i / is convex

(v) =^ (iv) Let k N, A!, ,Af c <E]0,1[, AI + • • - + Xk = 1, xu ,xk e

X If there exists i such that f{x{) = oo, then Eq (2.3) is obviously true

(2.3) is verified Finally, suppose that / ( x , ) < oo for every i and that there exists io with f(xi0) = — oo; consider i; G E such that (x;,£i) e p i / for

every i ^ io', since ( i j0, —n) € e p i / for every n N, we have

/(AiXiH hAfcXfc) < AitiH hAj0_i£j0_i+Ai0(-n)+Ai0 +iii o +i+ +\ktk

for n G N Letting n -»• oo in the above inequality we get f (J2i=i^ixi) =

—oo; hence Eq (2.3) is verified

The proofs for (i) => (vi) and (vi) => (iv) are similar to those of (i) =>• (v) and (v) => (iv), respectively

The proof for the "strictly convex" case is similar •

Note that in (v) and (vi) there are no counterparts corresponding to / strictly convex

Proposition 2.1.2 If f : X —> E is sublinear then f is convex Moreover,

f is sublinear if and only if e p i / is a convex cone with (0, —1) £ e p i /

Proof Let / be sublinear It is obvious that / is convex, and so e p i / is a convex set which contains (0,0) If (x,t) G e p i / and A > then

f(Xx) = \f(x) < Xt, and so X(x,t) G e p i / ; hence e p i / is a convex cone

Since /(0) = > - , we have that (0, - ) g e p i /

Assume now that e p i / is a convex cone with ( , - ) ^ e p i / It is immediate that / is convex ( e p i / being convex) and f(Xx) = Xf(x) for A > and x G X So, for x,y E X we have that /(a; + y) = / ( | x + | y ) <

fix) + fiy)- I f /(0) < then (0, -t) € e p i / for some t > 0, whence the

contradiction (0, —1) e p i / Therefore /(0) = 0, and so / is sublinear •

The indicator function of the subset A of X is

iA:X->R, iA(x) := j 0 if x G A,

+oo if x e X\A

Note that dom iA = A and epi t,A = A x R+ From the preceding theorem

we obtain that LA is convex if and only if A is convex

If / is convex the sets [/ < A] and [/ < A] are convex for every A G E The converse is generally false A function / with the property that " [ / < A] is convex for every A G E" is said to be quasi-convex So, / : X -» E is quasi-convex if and only if

The notion of convex function is extended naturally to mappings with values in ordered linear spaces Let Y be a real linear space and Q C Y be a convex cone; Q generates an order relation on Y: y\ < 2/2 (or 2/1 <Q 2/2 if there is any risk of confusion) if 2/2 — 2/1 £ Q- By analogy with E, consider

Y* : = U {00}, where 00 ^ Y; we put 2/ < 00 for all y £ F and A • 00 = 00

for every A £ P (note that y < 00 if y ^ 00) The element - 0 is introduced similarly To point out that Y is ordered by Q we write (Y, Q) or (Y, < )

Let (Y, Q) be an ordered linear space and H : X —>• Y'; the operator H is Q-convex if

Va;,2/eX, V A e ] , l [ : H{Xx + (1 - X)y) < XH{x) + (1 - X)H{y)

The operator H : X ->• F U {-00} is Q-concave if - i J : X -» F* is Q-convex

If A £ L(X, Y) then ^4 is Q-convex for every convex cone Q C Y As in the case Y = E, the domain and the epigraph are defined by: domi? := {x £ X | #(2;) < 00} a n d e p i i ? := {(x, y) eX xY \ H(x) < y} The characterizations of convex functions given in Theorem 2.1.1 are valid for a Q-convex operator

A function / : (Y, Q) —> E is Q-increasing if 2/1 <Q 2/2 => f(,Vi) < /(2/2)• For such a function we consider that /(oo) = +00 It is obvious that every function is {0}-increasing, and that a linear functional ip : Y —> E is Q-increasing if and only if (p(y) > for every y £ Q One defines similarly the Q-decreasing functions

In the next result we mention some methods for deriving new convex functions from known ones For a, (3 £ E we set a V /? := max{a,/?} and

a A P := min{a, /3}

Theorem 2.1.3 Let X,Y be linear spaces and Q CY be a convex cone

(i) / / fi : X —> E is convex for every i € / (/ ^ 0) then supi gj fi is

convex Moreover, e p i ( s u pi € // , ) = P |i /e p i / j

(ii) / / / , /2 : X -> E are convex and X £ E+, tften /1 + fi and A/i are

convex, where • /1 := tdom/i • Moreover,

dom(/i -1-/2) = dom fi n dom fi, dom(A/i) = dom /

(hi) If fn : X —t M is convex for every n £ N and f : X —> E is such

(iv) If A C X xR is a convex set then the function ipA is convex, where

ipA:X^R, <pA(x) := inf{t \ (x, t) € A} (2.4)

(v) / / F : X x Y —> E is convex, then the marginal function h

asso-ciated to F is convex, where

h:Y^% h(y):=mix€XF(x,y) (2.5)

(vi) Let g : Y —> E be a convex function If H : X —> Y' is Q-convex

and g is Q-increasing, then g o H is convex; this conclusion holds also if H : X —> Y U {—oo} is Q-concave and g is Q-decreasing In particular, if A L(X, Y) then go A is convex

(vii) Let f : X —>• E, g :Y —> K 6e proper convex functions, and let

$,V:XxY->R, <!>{x,y):=f(x) + g(y), ¥(x,y) := f(x)V g(y) Then $ and \& are convex and proper Moreover, d o m $ = d o m * =

dom / x dom g, inf $ = inf / + inf g and inf $ = inf / V inf g (viii) If f : X —> R is convex and A G L(X, Y) then the function

Af:Y^R, (Af)(y):=mi{f(x)\Ax = y}, is convex Moreover dom(Af) = A(domf) and inf Af = inf/

(ix) If / i , f2 : X —> R are convex and proper, their convolution and

max-convolution, defined by

f1Df2 : X -> K, ( / i D /2) ( i ) := inf {/i(a;i) + /2( z2) | an + x2 = x} ,

/1O/2 : X -> E, (f10f2){x) := inf { / i ( n ) V /2( z2) I Zi + *2 = z} ,

are convex Moreover dom(/1D/2) = dom^O./^) = dom/x + d o m /2,

inf fiDfi = inf /1 + inf /2, inf f10f2 = inf /1 V inf f2,

epi4(/iD/2) = epis /1 + epi5 /2 (2.6)

and

VA E : [AO/2 < A] = [A < A] + [f2 < A] (2.7)

Proof, (i) It is clear that epi(supiejr fi) = f)ieIepifi- Since fi is convex

for every i, using Theorem 2.1.1, we have that epi/i is convex, and so epi(supi € / fi) is convex The conclusion follows using again Theorem 2.1.1

(iii) Let A s]0,1[ and x,y € d o m / Since limsup/„(x), limsup/„(y) < oo, there exists n0 £ N such that fn(x), fn(y) < oo for every n > no- Since

/ „ is convex, we have

fn(Xx + (1 - X)y) < \fn(x) + (1 - X)fn(y)

Taking the limit superior we obtain

f(Xx + (1 - X)y) < Xf(x) + (1 - X)f(y)

Therefore / is convex

(iv) Let (xi,ti), (x2,t2) £ episipA and A £]0,1[ Then there exist

si,S2 £ ffi such that (xi,si),(x2,s2) € ^4 and si < £i, S2 < t2- Since A is

convex, A(:ci,si) + (1 - A)(z2,S2) = {Xx\ + (1 - A)x2,Asi + (1 - A)s2) £ A Therefore, (PA(XXI + (1 — A)x2) < Xsi + (1 — X)s2 < Aii + (1 — A)i2, and so

X(xi,ti) + (1 — A)(a;2,t2) £ epis y>A- Hence ip^ is convex

(v) Note that

epish = P ry x R( e p isF ) , dom h = Pry (dom F) (2.8)

The conclusion follows from Theorem 2.1.1(vi)

(vi) We observe that ( / o H)(x) = infy 6y F(x,y), where F(x,y) :=

g(y) + t,ep\H(x,y), because / is (J-increasing Since obviously F is convex,

by (v) we obtain that (/ o H) is convex, too

The other case is proved similarly The second part is immediate taking into account that A is {0}-convex and g is {0}-increasing

(vii) One obtains immediately that dom $ = dom $ = dom / x dom g and that $ and \? are convex The formulas for inf $ and inf * are well-known

(viii) We have that (Af){y) = mixZX F{x,y), where F(x,y) := f(x) +

>-gTA(x,y)- It is obvious that F is convex From (v) we obtain that Af

is convex As d o m F = {(x,Ax) \ x £ d o m / } , we have that dom(Af) —

A(domf) The formula for inf Af is obvious

(ix) Let us consider the functions $, \P : X x X -y R defined by

*(xux2) := fiixx) + f2(x2), *(zi,x2) := h{xi)y f2{x2),

and A : X x X -> X, A{x1,x2) := z i + x2 Then ( ^ D ^ ) ^ ) = (A$)(x)

and (fiQf2)(x) = (A^!)(x) By (v) we have that $ and $ are convex; using

domains, epigraph and level sets follow by easy verification The formulas for inf / i D /2 and inf /1O/2 follow immediately from (vii) and (viii) •

The formulas (2.6) and (2.7) motivate the use of "epi-sum convolution" and "level-sum convolution" for the convolution and max-convolution, re-spectively The convolution and max-convolution are extended in an ob-vious way to a finite number of functions; from Eqs (2.6) and (2.7) one obtains that these operations are associative

The convex functions which take the value —00 are rather special

Proposition 2.1.4 Let f : X —»• E be a convex function If there exists

XQ € X such that f(xo) = —00 then f{x) = —00 for every x € ' ( d o m / ) In particular, if f is sublinear and S ' ( d o m / ) , then f is proper

Proof Let x £ ' ( d o m / ) ; since xo £ d o m / , there exists /J, > such that y := (1 + fi)x — \ix§ € d o m / Taking A := (p, + )_ 1 e]0,1[, x =

(1 — \)XQ + Ay, and so

/ O r ) < ( l - A ) / ( x0) + A/(2,) = - o o

The case of / sublinear is immediate from the first part • In the sequel we denote by A(X) the class of proper convex functions

defined on X

Let now | / C c I and / : C -> M; we say that / is convex if C is convex and

Vx,y€C, V A € ] , [ : /(Ax + (1 - X)y) < Xf(x) + (1 - \)f(y)

It is easy to see (Exercise!) that the function / above is convex if and only if

?-X^W ?M-( f{x) if X e C>

is convex in the sense of the definition given at the beginning of this section Of course, we can proceed in the opposite direction; so a proper function / : X —> ffi is convex if and only if /|dom/ is convex in the above sense The consideration of (convex) functions with values in M has certain advantages, as we shall often see in the sequel

Theorem 2.1.5 Let f € A(E) be such that int(dom/) ^

(i) Let t\,ti € d o m / , t\ < £2 Suppose that there exists XQ €]0,1[

such that / ( ( l - A0)ii + X0t2) = (1 - A0)/(*i) + A0/(£2)- Then

V A e [ , l ] : / ( ( l - A ) t1+ A t2) = ( l - A ) / ( * i ) + A / ( *2) , (2.9)

h - t s t - t

* - t i

(ii) Let to d o m / The function

V t € [ t i , t2] : /(*) = r—rf(ti) + i—±m) (2.10)

t2 — t\ 12 — T\

Vt0 : dom/ \ {i0} -»• K, <*„(*) == / (*1 f (*o),

t — to is nondecreasing; if f is strictly convex then tpt0 is increasing

(iii) Let to e d o m / Tfte following limits exist:

t4.to I — to *>to I — to

r.(tt):-ta!StJM.mpmzmet, (2.12)

*t*o t — to t<t0 t — to

and

/K*o)</;(to); (2.13)

moreover /l(£o),/+(*o) € K whenever t0 € i n t ( d o m / ) Therefore f is left

and right derivable at any point o / i n t ( d o m / ) Moreover

r e [ / - W , / | ( t o ) ] n l « v t e i : T(t-t0) < f(t) - f{t0) (2.14)

/ / / is strictly convex, in relation (2.14) the inequality is strict for t ^ to (iv) Lei f i , i2 £ d o m / , £1 < t2 Then f\.(h) < f'_{ti); if f is strictly

convex then f\_{t\) < /!_(£2) Therefore the functions f'_ and f'+ are

non-decreasing on d o m / Furthermore, f is strictly convex if and only if f'_ [resp f'+] is increasing on int(dom/)

(v) The function f is Lipschitz on every compact interval included in i n t ( d o m / ) , and so f is continuous on i n t ( d o m / ) ; moreover, for every to E int(dom/) we have:

(vi) The function / is monotone on int(dom/), or there exists to & int(dom/) such that f is nonincreasing on ] — oo,£o] H d o m / and

non-decreasing on [io,oo[n d o m /

(vii) Let to € d o m / and A 6]0,1[ Then the mapping : d o m / —> R,

if>(t) := \f(t) — f(to + \(t — to)), is nonincreasing on Ii :=] — oo,to\Hdom/ and nondecreasing on Ir := [io,oo[ndom/ / / / is strictly convex then ip

is decreasing on I\ and increasing on Ir

Proof To begin with, let ti,t2,t3 € d o m / be such that t± < t2 < t3

Then t2 = {1 - X)tx + \t3 with A = (t3 - t2)/(t3 - h) € ]0,1[; therefore

m)<^-m) + ^ m ) , (2.15) the inequality being strict if / is strictly convex Subtracting successively

f(h), f(h), f(t3) from both members of Eq (2.15), multiplying then by

t£u> it,-tl)(tl-t2) a n d A > respectively, we obtain

m)-m) </(*3)-/(ti)> m)-w) </(f3)-/(*2))

t2 — t\ ~ t3 — tl t\ —t2 ~ t3 —12

f(tl) - /(*3) < / ( * ) - / ( * )> ( )

*i — *3 ~ *2 — i3 these inequalities being strict if / is strictly convex

(i) Let ti,t2 € d o m / and Ao G]0, l [ b e such that / ( ( l - A0)ii + A0i2) =

(1 — A0)/(*i) + X0f(t2) Suppose that there exists A e]0,1[ such that

/ ( ( l - \)ti + Xt2) < (1 - X)f(h) + Xt2; assume that A < Ao (one proceeds

similarly for A > A0) Taking := (1 - Ao)/(l - A) e]0,1[, we have that

A0 = dX + (1 - B) • and (1 - A0)*i + A0*2 = 0((1 - A)*i + Xt2) + (1 - 6)t2;

so we get the contradiction

/ ( ( l - X0)h + Xt2) < Of((l - X)h + Xt2) + (1 - 9)f(t2)

<6[(l-X)f(t1) + Xf(t2)} + (l-6)f(t2)

= (l-X0)f(t1) + Xof(t2)

Therefore Eq (2.9) holds Taking t€ [h,t2] and A = ( t - * i ) / ( i2- * i ) £ [0,1]

in Eq (2.9), we obtain Eq (2.10)

(ii) Let t0 £ d o m / and tx,t2 £ d o m / \ {t0}, h < t2 Considering

successively the cases t\ < t2 < to, h < t0 < t2, to < t\ < t2, from Eq

(2.16) we obtain that (ft0 is nondecreasing; if / is strictly convex, ipt0 is

(iii) To begin with, let to € int(dom/) Taking into account that

ipt0 is nondecreasing on everyone of the nonempty sets d o m / n ] i0, oo[ and

d o m / n ] — oo,t0[, the following limits exist:

, ,,x , /(*) - /(to) • f /(«) ~ /(to) ^

hm(pto(t) := hm —- = inf -—- < oo,

*4-*o t\.to t — to t>to t — to

,. m r fit) ~ /(to) fjt) - f(t0) ^

1™ V«o(*) := im — : — r — = SUP — : — : > -°°>

tjto tfto t — to t<t0 t — to

and so /+(to) and / i ( t o ) exist Since ipto is nondecreasing on dom / \ {to},

the inequality (2.13) holds What was proved above shows that fL{to),

f'+ito) G K

If to = max(dom/) then f(t) = oo for t > t0, whence /+(to) = oo; of

course f'_ito) < oo (the existence of / i ( t0) follows from the monotonicity

of <Pt0)- In the case to = min(dom/) we have / i ( t o ) = — oo <

/+(io)-Let now r € [ / i ( t0) , / | ( t0) ] n E and t < t0 < t'; from Eqs (2.11) and

(2.12) we have that

m-fM < f (to) < r < /;(to) < im^lM

t-t0 - •/-1-u^ - - J + 1" ' - t'-to

Note that the first inequality, in the above relation, is strict if the first quantity is finite and the function / is strictly convex; similarly for the last inequality and the last quantity From these inequalities we have immedi-ately Eq (2.14), with strict inequality if / is strictly convex Conversely, assume that r l and r(t —10) < fit) — /(to) for every t € K; dividing by

t — to in each of the cases t > to and t < to and taking the limit for t —> to, we obtain that r

[/L(to),/+(to)]-(iv) Let ti,t2 £ d o m / , t\ < t2 and consider t € ] t i , t2[ ; using Eqs

(2.11), (2.12) and (2.16) we obtain that

r+{h) < /(')-/(«*> < m)-m) = /(«o-/(«a) < , ( }

the inequalities being strict if / is strictly convex Using Eq (2.13) we get that f'_ and f'+ are nondecreasing, even increasing if / is strictly convex

Suppose that / is not strictly convex; then there exists ti,t2 G d o m / ,

ti < t2, and A0 6]0,1[ such that / ( ( l - A0)ti + A0t2) = (1 - A0)/(ti) +

A0/(t2)- Therefore Eq (2.10) holds, whence

vte]tut2[ f'-(t) = fW = fit?-{itl)

Because ]ii,i2[C int(dom/), we obtain that f'_ and f'+ are not increasing

on int(dom/)

(v) Let ti,t2 £ int(dom/), ti < t2 and t,t' e]ti,t2[, t < t' From Eqs

(2.11), (2.12) and (2.16) we get

/;(tl) < IMzM < m^M < /('»)-/(«) < r (ta)>

J+K ; - t i - t - f - t - t2-t -j-^^'

whence \f(t') - f(t)\ < M\t' - t \ , where M := max{|/^(*i)|, \fL(t2)\} G R

Therefore / is Lipschitz on ]ti,<2[- Since every compact sub-interval of int(dom/) is contained in an interval ] i i , ^ [ with [ti,t2] C int(dom/), we

get the desired conclusion

Let now to G d o m / be such that / is left-continuous at to (therefore

to > inf(dom/)); for example to € int(dom/) We already know that

V t e d o m / n ] - o o , t0[ : /!(*) < f+(t) < /l(*o) € ] - oo.oo]

Let A E, A < / i ( i0) ; from the definition of the least upper bound and

relation (2.12), there exists *i d o m / , ii < to, such that A < (f(h) —

f{to))l{t\ - t0) By the left-continuity of / at t0, there exists t2 £]ti,t0[

such that A < (/(ti) - f{h))/{h - t2) Let £ €]*2,*i[- Using again Eq

(2.16) we get

A fit) - f(t2) = f(t2) - f{t)

t-t2 t2-t -J~{>

-Therefore \im^t0 f'-it) — u mtt<o/+(^) = /-(*<))• The other formula is

obtained similarly

(vi) To begin with, note that / is nondecreasing (resp increasing) on [io,oo[n d o m / if /+(t0) > (>)0 Indeed, if h,t2 E d o m / , t0 < h < t2,

0(<) < f'+ito) < f'+ih) < ifih) - fih))Ht2 - h) Similarly, if f_(t0) <

(<)0 then / is (decreasing) nonincreasing on ] — oo, to] l~l dom /

So, from the above arguments, if /+(£) > for every t £ int(dom / ) then / is nondecreasing on int(dom/); if /+(£) < for every t G i n t ( d o m / ) , by Eq (2.13), fL(t) < for every t S int(dom/), whence / is nonincreasing on i n t ( d o m / ) If none of these conditions is satisfied, there exist t\,t2 G

int(dom/), h < t2, such that f'+(ti) < < f'+(t2) Let t0 := inf{« €

d o m / | f'+(t) > 0} €]h,t2] It follows that f'_(t) < for every t €

dom / , t < t0, whence / is decreasing on ] - oo, t0] C\ dom / By (v) we have

(vii) Let t i , t2 G Ir be such that ti < t2 Then

to + A(*i - t0) < h < t2 and t0 + A(tx - t0) < t0 + A(t2 - i0) < t2

So, from the convexity of / , we have

m) < ^ffii^m + \(tl - to)) + t3%xl§;Xm),

f(to + X(h - to)) < ^ f f S T ^ / f a + A(ti - h)) + I-^^L_f(t2),

the second inequality being strict if / is strictly convex Multiplying the first inequality by A, then adding them, we get

A/(*i) + /(to + A(t2 - t0)) < / ( t0 + A(ti - t0)) + A/(t2),

i.e V'(ti) < ipfo), the inequality being strict if / is strictly convex The

proof is similar on /( •

The preceding theorem shows that /|dom/ is continuous on int(dom/) when / : K -¥ K is a proper lower semicontinuous convex function We have even more

Proposition 2.1.6 Let / G A(R) Then /|ci(dom/) JS upper

semicontinu-ous (on c l ( d o m / ) j Moreover, if f is lower semicontinusemicontinu-ous then /|ci(dom/)

is continuous

Proof As mentioned above, / is continuous at any to € i n t ( d o m / ) Let

to G d o m / If d o m / = {to} it is nothing to prove In the contrary case

we can take to = inf(dom/) and some t € d o m / with t > to Let (t„) C d o m / \ {to} converge to to We may assume that tn <t for every n N

Suppose first that t0 € d o m / ; then, by Eq (2.15), / ( t „ ) < =E^/(*o) +

V-T/" / ( t ) for every n Passing to limit superior we get lim s u p / ( t „ ) < / ( t0)

If to ^ d o m / the preceding inequality is obvious Hence / is use at to- If / is lsc then /|ci(dom/) is lsc, too Hence /|ci(dom/) is continuous •

The next theorem furnishes a useful representation of lsc convex func-tions on E This result will motivate the following convenfunc-tions for a function / G A(K):

/ l ( t ) : = / + ( t ) := - c o V f G E \ d o m / , t < i n f ( d o m / ) ,

Theorem 2.1.7 Let ip : E —> E be a non-decreasing function and a G E

fee SMC/I t/m£ V'C0) € E Then

/ : E ^ E , / ( * ) : = J > ( s ) d s ,

is a proper lower semicontinuous convex function with I:— {s G R | </?(s) € R} C dom f Cell and

f'-=V-<V<V+ = f+, (2-17) where the integral is taken in Lebesgue sense and

<p-(t) := sup{<p(i') | t' < t}, ip+(t) := 'mi{<p{t') | t' > i } ,

for t G E Moreover, if g : E —>-R is a proper lower semicontinuous convex function such that g'_ < ip < g'+ then g — f + a for some a G E

Proof As usual, Ja ip(s) ds := — Jb ip(s) ds if a > b The statement is

obvious if / is a singleton So we assume that i n t i ^ If t G E \ c l / it is obvious that f(t) = oo Hence I C d o m / C c l / If </?(6) G E, the integral may be taken in Riemann sense, while for t = sup / G E we have

f(t) = l i m ^ f(t), and similarly for t = mil Therefore / |ci / is continuous,

whence / is lower semicontinuous Let to,ti G / , to < t\ and A G]0,1[ Taking t\ := (1 — A)io + Ati, we have that

/ ( * A ) - /(*o) = / £ ¥>(*) ^ < (tA - t0)v(*A) = A(ti - *O)V(*A),

f(ti) - f(tx) = Jtl <f(s) ds > (h - tx)<p(tx) = (1 - A)(ti - toMtx)

Multiplying the first relation by (1 - A) and the second one by -A, then adding them, we obtain that f(t\) < (1 — A)/(to) + A/(*i) Since f\cu is

continuous we obtain this inequality also for to,ti G d o m / and A G]0,1[; hence / is convex Let to G d o m / n c l / and t E, t > to; because

ip(s) > ip+(to) for s > to, we have that

V(t) > f{t)-f{to) = - J - / % ( s ) c f e > <p+(t0),

t — t0 t — to j t o

and so / ' ( t0) = ip+(t0) Similarly, fL(to) = ip-{to) Since these relations

are obvious for t0 $ d o m / n c l / , Eq (2.17) holds

Let now g : E —> E be a proper lsc convex function such that g'_ <

ip < g'+ It follows that int(domg) = i n t / Taking into account Theorem

and so g'_(t) = fL{t) and g'+(t) — /+(£) for every t € int I Taking h :

int I —> K, h(t) := f(t) — g(t), we have that h is derivable and h! = It

follows that ft is a constant function Therefore there exists a G E such that g(t) = f(t)+a for all t € int I Using the preceding proposition it

follows that g = f + a •

The following consequence will be used several times

C o r o l l a r y 2.1.8 Let f : E —• E be a proper convex function Then

\/t,t' e int(dom/) : /(*') - f(t) = / / ' /;(*) ds = / / ' / i ( s ) da

Moreover, if f is lower semicontinuous, then the preceding equalities hold for all t,t' € d o m /

Proof Assume that int(dom/) ^ When t := sup(dom/) e l o r t := inf(dom/) € E, we replace, if necessary, f(t) by limt.j.j/(£) and f(t)

by lini£4.(/(£); then / is lower semicontinuous Applying the preceding

theorem for ip = f'+ and cp — f'_ the conclusion follows D

The following result is used frequently to show that a function of one variable is convex

T h e o r e m 2.1.9 Let I C E be a nonempty open interval and f : I —> E

be a derivable function The following statements are equivalent:

(i) / is convex;

(ii) V t , a e / : / ' ( * ) • ( * - * ) < / ( * ) - / ( * ) ;

(iii) V t , s € J : (f'{t) — f'(s)) • (t — s) > 0, i.e / ' is nondecreasing; (iv) ( ' i / / is twice derivable on I) 'it £ I : /"(£) >

Proof, (i) =$> (ii) Let / be convex Since / is derivable we have that f'(s) = / i ( s ) = /+(s) for every s e I The conclusion follows from Eq

(2.14) taking r = f'(s)

(ii) =$• (iii) Let s,t £ I By hypothesis we have that

(hi) =4> (i) Let tuh € I, h < t2, A €]0,1[ and tx := (1 - A)fi + Xt2 €

}tut2[ Then

(1 - X)f(t1) + Xf{t2) - f(tx)

= (l-\)[f(t1)-f(tx)] + X(f(t2)-f(tx))

= (1 - A J / ' f a X t ! - tA) + A/'(r2)(i2 - tx)

= A(l - A ) / ' ( n ) ( t i - i2) + A(l - A)/'(r2)(i2 - tO

= A ( l - A ) ( i1- i2) ( / ' ( r1) - / ' ( r2) ) > ,

with Ti &]ti,t\[, r2 €.]tx,t2[ (obtained by using the Lagrange theorem); of

course, we have used the fact that / ' is nondecreasing

Suppose now that / is derivable of order on J In this case (iii) •&

(iv) by a known consequence of the Lagrange theorem •

Theorem 2.1.10 Let I C R be an open interval and f : I —> R be a

derivable function The following statements are equivalent:

(i) / is strictly convex;

(ii) Vt,8el,t?8 : /'(*) • ( * - « ) < f(t) - f(s);

(iii) Vt,s £ I, t ^ s : (/'(*) — f'(s)) • (t — s) > 0, i.e f is increasing; (iv) (if f is twice derivable on I) Vt E I : /"(*) > and {t e I \

f"(t) — 0} does not contain any nontrivial interval

Proof The proof is completely similar to that of the preceding theorem;

just replace, where necessary, the inequalities by strict inequalities and, of

course, use the properties of increasing functions • Similar characterizations to those of the above theorems can be

imme-diately formulated for concave and strictly concave functions

As immediate applications of the above two theorems we obtain that: (i) / i : R -» R, f\(t) := \t\p, where p £]l,oo[, is strictly convex; (ii) /2 :

R+ —> R, /2(£) := tp, where p e]0,1[, is strictly concave and increasing;

(iii) /3 : P -> R, f3(t) := tp, where p e R!_, is strictly convex and

de-creasing; (iv) fi : R -> R, fi(t) :- if t > 0, /4( t ) := if t < 0, where

p e]l, oo[, is convex and nondecreasing; (v) /s : R -> R, /s(£) := exp(i), is

strictly convex and increasing; (vi) /6 : P -> R, /6(i) := lnt, is strictly

con-cave and increasing; (vii) f7 : R+ -> R, f7(t) := tint if t > 0, /r(0) := 0,

is strictly convex; (viii) f% : R ->• R, f%(t) := V l + 12, is strictly convex

T h e o r e m 2.1.11 Let D C (X, ||.||) be a nonempty, convex and open set,

and / : £ ) - > E be a Gateaux differentiable function (on D) The following statements are equivalent:

(i) / is convex;

(ii) \/x,yED : (y-x,Vf(x))<f(y)-f(x); (hi) Vx,y£D : (y - x, Vf(y) - V/(x)> > 0;

(iv) Va; G £>, Vu G X : V2/ ( : E ) ( M , U ) > (when f is twice Gateaux

differentiable on D)

Proof For x,y G D consider Ix<y := {t | (1 - t)x + ty € D}\ one

proves without difficulty that Ix<y is an open (since D is open) interval

(since D is convex) and that [0,1] C IXtV Let us consider the function

(t):=f((l-t)x + ty) (2.18)

It is well-known that

¥>*,»(*) = V/((l-t)a:+*y)(j/-a:), < y W = V2 f{{l-t)x+ty)(y-x,y-x)

(2.19) for every t G / x ^ ; of course, the second formula is valid when / is twice Gateaux differentiable on D

(i) => (ii) Let x,y € D By Theorem 2.1.1 we have that <px,y is convex;

using then Theorem 2.1.9 we obtain that (p'xy(Q)(l — 0) < <px,y0-) — <Px,y{Q),

whence the conclusion

(ii) => (iii) Writing the hypothesis for the couples (x,y) and (y,x) we obtain immediately the conclusion

(iii) => (i) Let x,y € D and t, s G 7x>y, s < t; then

0<(t-s)(<p'(t)-<p'(s))

= ((1 - t)x + ty-(l- s)x - sy, V / ( ( l - t)x + ty) - V / ( ( l - s)x + sy))

Using Theorem 2.1.9 we obtain that ipX:y is convex, whence, by Theorem

2.1.1, / is convex

Suppose now that / is twice Gateaux differentiable on D

(i) =*> (iv) Let x € D and u € X Taking into account that D is open,

D - x is absorbing; hence there exists a > such that y := x + au G D

From the hypothesis we have that ipx>y is convex, whence by Theorem 2.1.9, <Px,y(t) > for every t G Ix>y In particular

Therefore the conclusion holds

(iv) => (i) Let x,y € D For every t G Ix,y, using Eq (2.19), we have

<p'x<y{t) > Applying again Theorem 2.1.9 we obtain that ipx,y is convex,

and so / is convex • In the next result we give characterizations for strictly convex functions

T h e o r e m 2.1.12 Let D C (X, \\ • ||) be a nonempty, convex and open set,

and f : D -» E be Gateaux differentiable (on D) Then (i) <$• (ii) •£> (hi) •£=

(iv), where

(i) / is strictly convex;

(ii) Vx,yeD,x?y : (y - x, V / ( z ) ) < f{y) - f(x);

(iii) Vx,y£D,x^y : (y - x, V/(y) - V/(x)) > 0;

(iv) \/x E D, V w £ l \ {0} : S72f(x)(u,u) > (when f is twice

Gateaux differentiable on D)

Proof Of course, using Theorem 2.1.1, we know that / is strictly convex if and only if the function <px>y given by Eq (2.18) is strictly convex for all

x,y G D, i / j / The proof is similar to that of the preceding theorem,

using Theorem 2.1.10 instead of Theorem 2.1.9 •

A remarkable property of convex functions is given below

T h e o r e m 2.1.13 Let f G A(X) and x G d o m / Then for every u e X

there exists

f(x, u) := lim ^ + fa)-^} = inf & + tu) ' ™ G I , (2.20) J v ' no t t>o t K '

and

VueX : f'(x,u)<f{x + u)-f(x), (2.21) the inequality being strict if f is strictly convex, u ^ and f'(x,u) < oo

Moreover fix, •) is sublinear and dom f'(x, •) = cone(dom/ — x) If x G

4( d o m / ) then f'(x,-) is proper, while if x £ ( d o m / )1 then f'(x,u) G K for

every u € X

Proof Let u £ X and rp : R -> R, tp(t) := f(x+tu) Taking into account that VJ — ifiXtX+u (<px,y being constructed in Theorem 2.1.1), is convex

2.1.5 we obtain the existence of

^(Q)=lim^-f^inf^-f),

i.e Eq (2.20) holds Using again Theorem 2.1.5 we obtain Eq (2.21), with

the corresponding variant for / strictly convex and u ^

It is obvious that f'{x,0) — and that f'(x,Xu) = Xf'(x,u) for every

u G X and A > Let now u,v £ X We have

f(x + t(u + v)) = f ( | ( x + 2tu) + \{x + 2tv)) < \f(x + 2tu) + \f{x + 2tv),

and so

f(x + t(u + v)) - f(x) f(x + 2tu)-f(x) f(x + 2tv)-f(x) t ~ 2t 2t '

Letting t ] we obtain

V « , « £ l : f'(x,u + v) < f'(x,u) + f'(x,v)

Therefore f'(x, •) is sublinear

Note that d o m / ' ( x , •) = E+ • ( d o m / - x) = cone(dom/ — a;) If a; € *(dom/) then dom f'(x,-) = lin(dom/ — x), whence € l (domf'(x, •))

From Proposition 2.1.4 we have that f'(x, •) is proper If a; € (dom / ) * then dom f'(x, •) = X and f'(x, •) is proper, which proves that f'(x,u) G M for

every u £ X • The number f'(x, u) G 11 is called the directional derivative of / at x

in the direction u It is possible that f'(x,-) take the value — oo Consider the function / : R -> E, /(*) := - ^ - i2 if |i| < 1, / ( i ) := oo if |i| > 1;

we have / ' ( — l , i ) = - o o for every t > (Exercise!)

The preceding result can be extended in a significant way

Theorem 2.1.14 Let / G A(X), x G d o m / anrf e G M+ Then the

function

fe{x,) : * - > ! , ^ , « ) : = j n f/ ( a ; + to)t-/(3;)+e,

is sublinear,

and

ft(x,u)<f{x + u)-f(x)+£ V « l , (2.23) f'(x,u) = l i m fe{x,u) = inf fe{x,u) Vw £ X

Furthermore, if x £ *(dom/) £/ien f'e{x,-) is proper, while if x £ ( d o m / )1

i/ien f'e{x,u) € E /or every u £ l

Proof From the definition of f'e{x, •) it is clear that f'E(x,u) < oo if and

only if there exists t > such that x + tu dom f, i.e u £ cone(dom f — x) Hence Eq (2.22) holds

It is obvious that f^(x,0) = and for A >

fs{x,Xu) = inf — A = Xfe(x,u)

Let now u,v e X and s,£ > Then

/ (x + ^ ( « + « ) ) = / ( ^ ( s + *u) + ^ ( z + tvj)

<^rtf(x + su) + ^rtf(x + tv)

It follows that

(f(x+£i(u + v))-f(x)+s)/£ +t

< f(x + su)- f{x)+e f(x + tv)-f(x)+e

~ s t

Therefore, for all s, t > we have

/(a! + g u ) - / ( g ) + g , / ( a + fa)-/(aQ+e

Taking the infimum in the right-hand side, successively with respect to s and t, we obtain that

£ ( x , u + i ; ) < / ; ( z , u ) + £(£,</)

Letting t = in the definition of f'e(x,u) we get Eq (2.23) On the other

hand, observing that for < e\ < £2 < 00 the inequalities

hold, we obtain

i- tu \ • f tit \ • f- f f(x + tu)-f(x)+£ limf i u = inf f'(x,u) = mi inf

-= inf inf / ( * + * " ) - / ( * ) + £ = i n f f(x +t u) ~ f(x)

t>0£>0 t t>0 t

= f'(x,u)

for every u £ X The other conclusions are proved like in the preceding

theorem • The number f'e{x,u) € K is called the £-directional d e r i v a t i v e of /

at x in the direction u

We end this section with a characterization of convex functions in ar-bitrary topological vector spaces using a generalized directional derivative We shall give other characterizations in Section 3.2 using generalized sub-differentials So, let X be a topological vector space and / : X -> M a proper function We define the upper Dini directional derivative of / at x X in the direction « £ l b y

Df(x u\ _ / l i m s u pt 4.0t_ 1( / ( a ; - l - t u ) - / ( x ) ) if x € d o m / ,

\ — oo otherwise When / is convex and x G d o m / from Theorem 2.1.13 we have that

Df(x,u) = f'(x,u) for all x € d o m / and u € X

For the proof of the announced characterization we need the following auxiliary result which is interesting in itself, too

Lemma 2.1.15 Let a, b M, a < b, and (p : [a,b] —>• E be a lower

semi-continuous proper function with <p(a) < <p(b) Then there exists c G [a,b[ such that Dtp(c, 1) >

Proof By contradiction, assume that Dtp(t, 1) < for every t [a, b[

Fix a > 0; we have that tp(t) < <p(a) + a(t — a) for all t [a,b] Indeed, because Dip(a, 1) < < a, there exists t' E]a,b] such that ip(t) < ip(a) +

a(t - a) for all t € [a, t'[ Let

t := sup {Ie]a,b] \ <p(t) < <p(a) + a(t - a) V* [a,I[} e]a,b]

ip(a) + a(i — a) + a(t — i)= ip(a) + a(t — o) for t £ [M"[, contradicting

the choice of t Hence t = b, and so desired inequality is proved As the inequality tp(b) < tp(a) + a(b — a) holds for all a > 0, we obtain the

contradiction ip(b) < f(a) The conclusion holds •

Theorem 2.1.16 Let X be a topological vector space and f : X ->• R be

a proper lower semicontinuous function Then f is convex if and only if Df{x,y-x)+Df(y,x-y)<0

for all x, y £ X for which the sum makes sense

Proof Assume first that / is convex If x £ d o m / then Df(x,y — x) = —oo, and so Df(x, y — x)-\- Df(y, x — y) equals — oo or does not make sense Assume that x,y £ d o m / Then

Df(x,y- x) = f'(x,y- x) < f(y) - f(x), Df(y, x-y) = f(y, x - y) < f(x) - f(y);

summing the inequalities side by side we get the conclusion

We prove the sufficiency by contradiction Assume that / is not con-vex Then there exists x,y € d o m / , x ^ y, and A g]0,1[ such that / ( ( l - X)x + \y) > (1 - A)/(z) + Xf(y) Let <p : [0,A] -¥ S and V : [0,1 - A] -> I be defined by

<p{t) := / ((1 - t)x + ty) - (1 - t)f(x) - tf(y),

</>(*) := / (tx + (1 - t)y) - tf(x) - (1 - t)f(y)

Of course, <p and ijj are lsc, / being so Then <p(0) = < <f(X), V"(0) = < I/>(1 - A) = ifi(X), and

D<p(t, 1) = Df((l - t)x + ty,y-x) + f(x) - f{y) V t e [0, A[,

^V(* 1) = Df(tx + (1 - t)y, x - y) + /(y) - f(x) Vt [0,1 - A[ Prom the preceding lemma we get t\ € [0, A[ and ti £ [0,1 — A[ such that ~Dtp(ti,l) > and 'Dip{t2,1) > Taking u := (1 - i i ) z + ^ y and

v := ^2^+(1 — ^2)2/, we have that u — v = s(x — y) with s := —ti—*2 S]0,1]

Moreover,

Df(u, v — u) + Df(v, u - v)

= s-1 (p<p(tu 1) - / ( z ) + f(y)) + s -1 (^V(*2,1) + f(x) - f(y))

a contradiction The proof is complete •

2.2 S e m i - C o n t i n u i t y of Convex F u n c t i o n s

In this section X is a separated locally convex space if not stated explicitly otherwise We begin with some characterizations of lower semicontinuous convex functions

Theorem 2.2.1 Let f : X —> R The following conditions are equivalent:

(i) / is convex and lower semicontinuous; (ii) / is convex and w-lower semicontinuous; (iii) epi / is convex and closed;

(iv) epi / is convex and w-closed

Proof It is well-known that: / is lsc •£> e p i / is closed in I x I «

[/ < -M is closed VA £ The equivalence of conditions (i)-(iv) follows

immediately applying Theorem 2.1.1 D In the sequel we shall denote by T(X) the class of proper lower

semi-continuous convex functions on X

The following criterion for convexity is sometimes useful

Theorem 2.2.2 Let f : X —> K be a proper function satisfying the

fol-lowing conditions: (i) /(0) = 0, (ii) V i £ l , VA P : /(Ax) = Xf(x),

(iii) / is quasi-convex Suppose that either (a) or (b) holds, where (a) V i £

X : f(x) > 0, (b) d o m / C cl{a; e X | f(x) < 0} and f is lsc at every point x G d o m / Then f is sublinear

Proof Note first that f(x + y) < f{x) + f(y) if x,y E d o m / and either f(x) • f{y) > or = f{x) < f(y) Indeed, if f{x) • f{y) > then

whence f(x + y) < f(x) + f(y) If = f(x) < f(y), for n £ N we have that

J £ T / ( * + V) = f(nTTnx + n^iV) < max{f(nx),f(y)} = f(x) + f(y)

Suppose that (b) holds Let A := {x G X \ f(x) < 0} C d o m / If

x € d o m / \ ^ , by hypothesis, there exists a net (a;i)iej C A, with (xi) —>• a;

Moreover,

0 < f(x) < \iminfieif(xi) < limswpieIf(xi) < 0,

whence (/(a;,)) —> /(a;) = Therefore f(x) < for every x € d o m / Let x,y E d o m / Since the roles of x,y are symmetric, we have to consider the following three situations: a) x,y £ A, ft) x,y G d o m / \ A and 7) x G d o m / \ A and 2/ G A In the case a ) we have that f(x)-f(y) > 0, and so f(x + y) < f(x) + f(y) In the case /3) we have that f(x) = / ( y ) = and a; + y G d o m / , whence f(x + y) < = / ( x ) + / ( y ) In the case 7) there exists a net (xi)iei C A such that (xi) —> a;; then (/(a:;)) —>• Since

a; + y G d o m / and /(a^ + y) < f(xi) + f(y) for every i e J [by a)], taking

the limit inferior we obtain that f(x + y) < f(x) + f(y) • Corollary 2.2.3 Let f : X —> R be a quasi-convex, proper, positively

homogeneous and Isc function, / + := / V and g := f + LC\A, where A :=

{a; G X I f{x) < 0} U {0} Then f+ and g are sublinear and f = /+ A g

Proof It is obvious that /+ and g are quasi-convex, lsc and positively

homogeneous The subhnearity of / + and g follows applying the preceding theorem Because f(x) < for x G cl-A, we have also that / — / + A g •

A result of the same type is given by the following corollary

C o r o l l a r y 2.2.4 Let f : X —> E be a quasi-convex and positively

homo-geneous function If f is upper bounded on a neighborhood of the origin or

d i m X < 00, then f+ is sublinear

Proof Suppose that there exists Ao > such that [/ < Ao] is a

neigh-borhood of Since [/ < A] = ^ [ / < A0] for every A > 0, we have

0 G int[/ < A] for every A > Let < A < fi and x G cl[/ < A] Since 0 G int[/ < A], from Theorem 1.1.2 we have that ^x G int[/ < A] Hence

x G f int[/ < A] = int (f [/ < A]) = int[/ < p] C [/ < »]

Therefore

VA > : cl[/ < A] C f | \f </!] = [ / < A]

Hence [/ < A] is a closed set for every A > Since [/ < 0] = HM>o[^ — lA>

[f+ < A] — for every A < 0, it follows that /+ is lsc Taking into account

that / + is quasi-convex, /+ is sublinear (applying Theorem 2.2.2) Suppose now that d i m X < oo By hypothesis, [/ < 1] is convex and absorbing Since d i m X < oo, as observed at the end of Section 1.1, we obtain that

0 G int[/ < 1] The conclusion follows as above •

Note that, under the conditions of the preceding theorem, / + is contin-uous (see Theorems 2.2.9 and 2.2.21)

A result similar to that of Proposition 2.1.4 is the following

Proposition 2.2.5 Let f : X —» E be a lsc convex function If there

exists XQ G X such that f(xo) — — oo, then f(x) = —oo for every x G dom / In particular, if f is sublinear then f is proper

Proof Suppose that there exists x G d o m / such that f(x) =: t G E Then (x, t) G epi / and (xo,t — n) G epi / for every n G N It follows that

V n G N : ±(x0,t - n) + (1 - ±){x,t) = (±x0 + *^x,t - l) G e p i / ;

therefore (x,t — 1) G cl(epi/) = e p i / , whence the contradiction f(x) <

f{x) - •

Propositions 2.1.4 and 2.2.5 motivate the consideration, in the sequel, of (almost only) proper convex functions

It is useful to consider the lower semicontinuous envelope or lower

semicontinuous regularization / := </>ci(ePi/) of the function / : X —> E

(see Eq (2.4)) Because cl(epi/) is closed we have that e p i / = cl(epi/)

Theorem 2.2.6 Let f : X —> K be a convex function Then

(i) / is convex;

(ii) if g : X —> E is convex, lsc and g < f then g < f;

(iii) the function f does not take the value —oo if and only if f is bounded

from below by a continuous affine function;

(iv) if there exists xo G X such that f(xo) = — oo (in particular if

f(xo) = — ooj then f(x) = —oo for every x G d o m / D d o m /

Proof, (i) Since e p i / = cl(epi/) and e p i / is convex, we have that e p i / is convex; hence / is convex

(iii) Suppose that / does not take the value — oo If f(x) = oo for every

x € X then f(x) > (x, 0) + for every x € X Consider dom / ^ and let x € dom / Then (x, t) fi epi / , where t := f(x) — Since epi / is a convex,

closed and nonempty set, using Theorem 1.1.5 we obtain (x*,a) € X* x E such that

V (a;, t) € epi / : {x,x*) + at < (x, x*) + at

Taking x = x and t = f(x) +n, n £ N, we obtain that a < Dividing, by —a > 0, we can suppose that a = — Thus

Vz e d o m / D d o m / : f{x)>f(x)>(x,x*)+j, where •y :—t— {x,x*} Therefore / is bounded from below by a continuous affine function

Conversely, if f(x) > (x,x*) + =: g(x), where x* G X* and £ R, then g is convex and lsc; by (ii) we have that g < f Therefore / does not take the value —00

(iv) If f(xo) = - 0 , using the preceding theorem, f(x) — —00 for every

x d o m / It is obvious that d o m / D d o m / •

Proposition 2.2.7 Assume that f : X —> ffi is sublinear Then f is

sublinear o- / is lsc at •£> / is proper

Proof Because e p i / = cl(epi/) is a convex cone, the first equivalence follows by Proposition 2.1.2 If / is sublinear, by Proposition 2.2.5 we have that / is proper Conversely, if / is proper then > /(0) > —00 which

implies that 7(0) = • To an arbitrary function / : X —> E one associates, naturally, a lower

semicontinuous and convex function; this function is denoted by c o / and has the property that epi(co/) = cl(co(epi / ) ) and is called the lsc convex

hull It is obvious that c o / < / < /

An application of the lsc convex hull of a function is the property in the next example used in the study of Hamilton-Jacobi equations

Example 2.2.1 Let X be a locally convex space, / € r ( X ) , g : X -> E be

a proper function and a, > Then co(co(f+ag)+/3g) =co(f+(a+/3)g) (See the solution of Exercise 2.19 for details.)

f is upper bounded (by a real constant) on a neighborhood of xo; therefore xo G int(dom/) 7^ In the sequel we prove that the converse is true for

every convex function

Let us first establish the following auxiliary result

Lemma 2.2.8 Let f G A(X) and x0 G d o m / Suppose there exist U G

J4CX and m G ffi_|_ such that

VxGx0 + U : f(x) < f(x0) + m (2.24)

Then

Vxexo + U : \f(x)-f(x0)\<mpu(x-x0), (2.25)

where pi; is the Minkowski gauge ofU In particular f is continuous atXQ

Proof Replacing / by g : X ->• R, g(x) = f(x0 + x) — f{x0), we may

suppose that xo = and f(xo) = Therefore f(x) < m for every x € U Prom Proposition 1.1.1 we have that pu is a continuous semi-norm and

U = {x G X I pu{x) < 1} Let x U Suppose first that t :— pu{x) >

Then y :— t~1x € U Since / is convex and t < 1, we obtain that f(x) <

tf{y) + (1 - t)f(0) < tm = mpu{x) Suppose now that pu{x) = Then nx G U for every n € N It follows that f(x) < ±f(nx) + ^ / ( O ) < \m

for every n N Therefore, once again, f(x) < mpu(x) Since = / (\x' + \{-x')) < \f{x') + y(-x'), we obtain that -f(x') < f(-x') for every x' e X Using this inequality we obtain that \f(x)\ < mpu(x) for

every x eU Therefore Eq (2.25) holds • Using the preceding lemma we get the following important result

Theorem 2.2.9 / / the convex function f : X —> K is bounded above on a

neighborhood of a point of its domain then f is continuous on the interior of its domain Moreover, if f is not proper then f is identically - c o on

i n t ( d o m / )

Proof Suppose that there exist Xo £ d o m / , V G N(xo) and m G M such that f(x) < m for every x G V Then V C d o m / , and so xo G int(dom / ) 7^ If / takes the value - 0 then, by Proposition 2.1.4, f(x) = —00 for every x G int(dom/) Therefore the conclusion holds in this case Let / be proper and x G int(dom/) Then there exists /J, > such that

xi := (1 + fi)x - nx0 G dom / Taking A := (1 + p,)~l G ]0,1[, we have that

for every x € V But V\ := Xxi + (1 — A)V £ N(x) Applying the preceding

lemma we obtain that / is continuous at x •

Corollary 2.2.10 Let f : X —>• E be a convex function Then f is

continuous on int(dom/) if and only i/int(epi/) is nonempty in X x E

Proof Suppose that / is continuous at some XQ £ i n t ( d o m / ) Then, for m &]f(xo),oo[, there exists U £ N x such that f(x) < m for every

x e xo + U So (xo, m + 1) G int(epi/) because (xo + U) x [m, oo[ C e p i /

Conversely, assume that (xo,M £ hit(epi/); then there exist U £ N x and e > such that (xo + U) x [t0 — e, t0 + e[ C epi / , and so f(x) < m := t0 + e

for every x xo + U By Theorem 2.2.9 we have that / is continuous on

int(dom/) ( ^ 0) •

Under the conditions of Lemma 2.2.8 we have a stronger conclusion

Theorem 2.2.11 Let f € A(X) and XQ £ d o m / Suppose that there

exist U € J^cx and m E+ suc/i t/iat condition (2.24) is satisfied Then for

every p e]0,1[,

Vx,y £x0+pU : |/(cc) - f(y)\ < m- pu(x-y)

1-p

Proof As in the proof of Lemma 2.2.8, we may assume that xo = and /(a;o) = Let p e]0,1[ and consider x,y pU We consider two cases: a) pu{x — y) ^ and b) pu{x — y) = In the proof we shall use the fact that pu is a continuous semi-norm, U = {x \ pu{x) < 1} and intf/ = {x | pu{x) < 1} (see Proposition 1.1.1)

a) Let <p : [l,oo[—» M, <p(t) = pu(tx + (1 — t)y); since pc/ is continu-ous, if is continucontinu-ous, too Moreover, (p(t) > tpu{x — y) — pu(—y), and so limt_,.oo ¥>(*) = oo As (p(l) = pu(x) < P < 1, there exists t > such that ¥>(£) = Then := te+(l-t)j/ £ U It follows that x = (l-i~1)y + i-1z,

and so f(x) < (1 — i~1)f(y) + i~1f(z) From Lemma 2.2.8 we have that

\f(y) I < mpu(y) < wp Therefore

/ ( * ) - f(v) < ^ (m - f(y)) < *_ 1( m + mp)

But i{x-y) - z-y, whence ipu(x-y) - pu(z-y) >Pu(z)-pu(y) > 1-p The preceding inequalities imply that

fix) - f(y) < m\^-Pu{x - y) (2.26)

b) In this case we have, using again Lemma 2.2.8, that f(t(x — y)) = for every t £ i So,

W G ] , [ : f(tx)=f(ty+(l-t)J^(x-y))<tf(y)

Since x G int U C int(dom / ) , by Theorem 2.2.9, / is continuous at x From the above inequality we obtain that f(x) < f(y) taking the limit for t —> Hence (2.26) also holds in this case

The conclusion follows from Eq (2.26) changing x and y •

Corollary 2.2.12 Let (X, ||-||) be a normed space and f G A(X) Suppose

that XQ G dom / and for some p > and m > 0,

Vz G D(x0,p) : f(x) < f(x0) + m

Then

Vp'e]0,p[,Vx,yeD(xQ,p') : \f(x) - f(y)\ < - • P-^ • \\x - y\\

p p-p'

Proof Consider U := D(0;p) — pUx- Then pu{x) = p_ 1||x|| for any

x X The conclusion is immediate from the preceding theorem •

The conclusion of Theorem 2.2.11 says, in fact, that / is Lipschitz on a neighborhood of xo • We say that the function / : X —> K is Lipschitz on a set A C X if / is finite on A and there exists a continuous semi-norm p on X such that \f(x) — f(y)\ < p(x — y) for all x,y E A; we say that / is

locally Lipschitz on A if for every x G A there exists a neighborhood V of

a: such that / is Lipschitz on V

Corollary 2.2.13 If f £ A(X) is bounded above on a neighborhood of a

point of its domain then f is locally Lipschitz on the interior of its domain

Proof Let x G int(dom/) Applying Theorem 2.2.9 we obtain that / is continuous at x, and so / is bounded above on a neighborhood of a; Apply-ing now Theorem 2.2.11 we obtain that / is Lipschitz on a neighborhood

of a; • Recall that in Theorem 2.1.5 we have already proved that a function

/ G A(E) is locally Lipschitz on int ( d o m / ) , while in Proposition 2.1.6 we proved that /|dom/ is continuous if / is, moreover, lsc

Corollary 2.2.14 Let ft A{X) forl<i<n and set f := ftD • • • D / „ ,

g := / i • • • 0/n- -V / i *s continuous at a point of its domain then

int(dom / ) = int(domg) = int(dom ft) + dom ft + (- dom / „

and either f (resp g) is identically —oo on int(dom/) (resp int(dom<7),);

or f (resp g) is proper and continuous on int(dom/) (resp int(domg)^.D

Recall that ftOfa and /1O/2 are defined in Theorem 2.1.3

Prom Theorem 2.2.9 (or Corollary 2.2.10) we obtain that g is continuous on int(domp) if f,g are convex, g < f and / is continuous on int(dom) (supposed to be nonempty) A similar result is true for a larger class of convex functions

The convex function / : X —• E is said to be quasi-continuous if aff(dom/) is closed and has finite codimension (i.e its parallel linear sub-space has finite codimension), rint(dom/) 7^ and /|aff(dom/) ls

contin-uous on rint(dom/) The set A C X is contincontin-uous if LA is quasi-continuous; it follows that A is quasi-continuous exactly when aff A is a closed affine set of finite codimension and rint^4 7^ The following result holds

Proposition 2.2.15 Let f,g : X —>• E be convex functions such that

g < f If f is quasi-continuous, then g is quasi-continuous, too

Proof Without loss of generality we assume that S dom / and /(0) < 0 Then aff (dom / ) is a linear subspace and Y0 := aff (epi / ) = aff (dom / ) x

E Of course Y0 has finite codimension and is closed Moreover, by

Corol-lary 2.2.10, we have that inty0(epi/) 7^ Since e p i / C epig, we have that

inty0(y0nepip) ^ It follows (see Exercise 1.3) that rint(epi5) 7^ Since

aff (epi g) = aff (dom g) x E, using again Corollary 2.2.10, we obtain that

g |aflf(domp) is continuous, and so g is quasi-continuous •

Applying the preceding result to indicator functions one obtains

Corollary 2.2.16 Let A C B C X If A is quasi-continuous then so is

B D

There are several classes of convex functions, larger than the class of lower semicontinuous convex functions, which will reveal themselves to be useful in the sequel We introduce them now

Let / : X —> E We say that / is cs-convex if

En

, Afc/(xfc)

whenever J2n>i ^nxn is a convex series with elements of X and sum x € X

Of course, if / is cs-convex then / is convex, while if / is lsc and convex, / is cs-convex (Exercise!) Also, we say that / is ideally convex,

bcs-complete, cs-closed, cs-bcs-complete, li-convex or lcs-closed if epi / is

ideally convex, bcs-complete, cs-closed, cs-complete, li-convex or lcs-closed, respectively Of course, taking into account the relationships among these notions for sets and Proposition 2.2.17 (i) below, we have:

/ lsc, convex =>• / sc-convex

a-f cs-complete =>• / cs-closed =>• a-f lcs-closed

^ ^ ^ / bcs-complete =>• f ideally convex => / li-convex => f convex, the reversed implications being not true, in general

Taking into account Proposition 1.2.3 and that for / : X -» R we have [/ < A] = Fix ( e p i / n (Xx } - oo, A]),

[ / < A] = P rx ( e p i / n ( X x ] - co, A[),

the sets [/ < A] and [/ < A] are li-convex (lcs-closed) for every A € R if / is li-convex (lcs-closed)

Let A C X; since epi LA = A X R+ and ffi is a Frechet space, we have that A is ideally convex (bcs-complete, cs-closed, cs-complete, li-convex, lcs-closed) if and only if i& is so

Remark 2.2.1 When <p : X —> R is a continuous affine functional (that is <p = x* + a for some x* £ X* and a e l ) , / : X -> R is cs-convex (ideally

convex, cs-closed) if and only if / + </> is so (Exercise!)

Proposition 2.2.17 Let f, g : X —>• R have nonempty domains

(i) / / / is cs-convex then f is cs-closed Conversely, if f is cs-closed

and is minorized by a continuous affine functional then f is cs-convex

(ii) / / / and g are ideally convex (resp cs-closed) and are minorized by

continuous affine functionals then f + g is ideally convex (resp cs-closed) Moreover, if g is bcs-complete (resp cs-complete) then f + g is bcs-complete (resp cs-complete)

(i) The proof of the first part is immediate Suppose that / is cs-closed and / > Let X)n>i ^nXn be a convergent convex series with elements of

X and sum x G X Since / is convex ( e p i / being convex), we may suppose

that An > for every n £ N; moreover, we may assume that xn G dom /

for every n Because f(xn) > 0, there exists r := l i m n - ^ X)*!=i ^kfi^k) G

EU{oo} If T < oo, because (xn,f(xn)) € e p i / and J2n>i ^n{xn,f{xn)) =

(X,T), and / is cs-closed, we have that (x,r) £ e p i / , whence f(x) < r If

T = oo, the preceding inequality is obvious Hence / is cs-convex

(ii) We prove only the second part of the "ideally convex" case, the rest of the proof being similar

So, let / be ideally convex, g be bcs-complete and / , g > Let

J2n>i ^n(xn,rn) be a Cauchy b-convex series with elements of epi(/ + g)

Then rn = sn + tn with (xn,sn) G e p i /n and (xn,tn) G epig for

ev-ery n Because < sn,tn < rn, we have that (sn), (tn) are bounded, s := ^2n>i ^nSn € + , t := 2n > 1 Xntn G M+ and r = s + t Because g

is bcs-complete we obtain that the b-convex series ^2n>1Xn(xn,tn) with

elements of epi g is convergent with sum (x,t) £ epig for some x G X Hence the b-convex series Yln>i ^n{xn,sn) with elements of e p i / is

con-vergent with sum (x, s) G e p i / Therefore (x,r) G epi(/ + g) Thus f + g

is bcs-complete • The classes of li-convex and lcs-closed functions have good stability

properties Let us begin with the following characterizations

Proposition 2.2.18 Let / : X -» K have nonempty domain Then the

following statements are equivalent:

(i) / is li-convex (resp lcs-closed); (ii) epis / is li-convex (resp lcs-closed);

(iii) there exist a Frechet space Y and an ideally convex (resp cs-closed)

function F : X x Y -» E such that f(x) = infyCy F(x,y) for every x G X

(i.e f is the marginal function associated to an ideally convex (resp cs-closed) function)

Proof We prove the "li-convex" case, the proof for the other case being similar

(i) => (ii) Taking % § : X =} E such that gr 31 = epi / and gr S = X x P, we have that "R and S are li-convex multifunctions Since E is a Frechet space, using Proposition 1.2.5 (iv), epis / — e p i / + {0} x P = gr(IR + §) is

(ii) =$• (i) Since epi,, / is li-convex, as above, the set An := epis / +

{0}x ] — i , o o [ is li-convex for every n N Since e p i / = f]n&NAn, from

Proposition 1.2.4 (i) we obtain that e p i / is li-convex

(i) => (iii) Since / is li-convex, there exist a Frechet space Y and an ideally convex set A C Y x X x E such that e p i / = PIXXR(A). Consider the function F: XxYxR-^R defined by F(x,y,r) := r + iA(y,x,r) Then

f(x) = i n f( a.i r)e e p i /r = i n f ^ ^ ^ r = inf(j,) r)eyx RF(a;,j/,r)

for every x £ X Since A is ideally convex, so is LA] using Proposition 2.2.17 (ii) and Remark 2.2.1 we obtain that F is ideally convex The con-clusion follows because Y x E is a Frechet space

(iii) =>• (ii) Let f{x) = infy ey F(x, y) for every a; € X, where Y is a

Frechet space and F is an ideally convex function From (i)=^(ii) it follows

that epis F is li-convex Then, by Eq (2.8), epis / is li-convex •

Other useful properties of li-convex and lcs-closed functions are collected in the following result

Proposition 2.2.19 (i) / / / „ : X -> E is li-convex (resp lcs-closed) for

every n E N , then s u pn e N / „ is li-convex (resp lcs-closed)

(ii) If fi,h '• X -» E are li-convex (resp lcs-closed) functions and A € E+, then / i + /2 and A/i are li-convex (resp lcs-closed)

(iii) If F : X xY ^ R is li-convex (resp lcs-closed) and X is a Frechet

space, then h : Y —»• R, h(y) := inixex F(x,y), is li-convex (resp

lcs-closed)

(iv) Let Y be a Frechet space and g : Y —> R If Q C Y is a convex

cone, H : X —>• (Y',Q) has li-convex (resp lcs-closed) epigraph and g is li-convex (resp lcs-closed) and Q-increasing, then g o H is li-convex (resp lcs-closed) In particular, if A : X —> Y is a linear operator with li-convex (resp lcs-closed) graph and g is li-convex (resp lcs-closed), then g o A is li-convex (resp lcs-closed)

(v) / / X is a Frechet space, A : X —> Y is a linear operator with

li-convex (resp lcs-closed) graph and f : X —>• E is li-li-convex (resp lcs-closed), then Af is li-convex (resp lcs-closed)

(vi) If X is a Frechet space and / i , /2 : X -> E are li-convex (resp

Proof Again, we treat only the "li-convex" case

(i) Because epi (supn € N /„) = rineNePi/™' *n e conclusion follows using

Proposition 1.2.4 (i)

(ii) Taking % : X =t R, g r # i := epi/j (i = 1,2), we have that epi(/i + /2) = gr(3?i +3?2)- The conclusion follows from Proposition 1.2.5 (iv) Also,

epi(A/i) = T ( e p i / i ) for A > 0, where T : X x E - > X x R i s the isomorphism of topological vector spaces given by T(x, t) = (x, Xt); hence A/i is li-convex in this case If A = 0, A/i = tdom/i- But d o m / i = P r j t ( e p i / i ) , and so d o m / i is li-convex by Proposition 1.2.3, whence 0/i is li-convex

(hi) We have, by Eq (2.8), that epis / = PryXR(epis.F) Since episF

is li-convex and X is a Frechet space, we get from Proposition 1.2.3 that epis / is li-convex

(iv)-(vi) Using the same constructions as in the proofs of Theorem

2.1.3 (vi), (viii) and (ix), the conclusions follows from (iii) • If X is a barreled space, the lower semi-continuity of a convex function

(even weaker conditions) ensures its continuity on the interior of its domain

Theorem 2.2.20 Let X be a barreled space and f : X —> E be convex

Suppose that either (a) X is first countable and f is li-convex or (b) / is lower semicontinuous Then ( d o m / ) ' = int(dom/) and f is continuous on

i n t ( d o m / )

Proof, (a) By Proposition 2.2.18, there exist a Frechet space Y and a li-convex function F : X xY -> R such that f(x) = infy €y F(x, y) for every

x £ X Consider the multifunction X : x l r j I with gr3? := {(y,t,x) \ (x,y,t) G e p i F } Of course, Ji is ideally convex and I m R = d o m / Let xo € (dom f)1 (if this set is nonempty) and {ya,tQ) € 3l_1(a;o) Applying

Simons' theorem (Theorem 1.3.5), we have that U := R(Yx ] - 00, t0 + 1[)

is a neighborhood of x0 So, for every x € U there exist y Y and

t < to + =: m such that f(x) < F(x,y) <t < m So / is bounded above

on the neighborhood U of x0, whence x0 £ int(dom/) and / is continuous

at

xo-The case (b) follows similarly (taking Y = {0} for example) and using

Ursescu's theorem instead of Simons' theorem • In finite dimensional linear spaces the preceding result becomes:

Corollary 2.2.21 Let X be a finite dimensional linear normed space and

f € A(X) be such that (dom / ) ' 7^ Then f is continuous on int(dom / ) =

Proof By Proposition 1.2.1 we have that e p i / is cs-closed, and so / is lcs-closed The conclusion follows from the assertion (a) of the preceding

theorem •

The application of Corollary 2.2.12 and Theorem 2.2.20 yields the fol-lowing uniform boundedness principle for convex functions

Theorem 2.2.22 Let X be a Banach space and C C X be an open convex

set Consider {fi)i^i o, nonempty family of continuous convex functions from C into E If (fi{x)).j is bounded for every x £ C then for every x £ C there exist rx,Lx > such that Ux := x + rxllx C C and \fi(y) —

fi(z)\ < Aclly - z\\ for all y,z £ Ux and all i £ I (i.e (fi) is locally

equi-Lipschitz on C)

Proof By hypothesis, for every x £ C there exists Mx £ E+ such that

\fi(x)\ < Mx for all i £ J Let r'x > be such that U'x := x + r'xUx C C

and consider Fi : X -> E be denned by Fi{y) := fi(y) for y £ U'x and

Fi(y) := oo otherwise Then Ft £ T(X) Consider F := supieI Ft E T(X)

The hypothesis shows that d o m F = Ux By Theorem 2.2.20 we obtain

that F is continuous on int(dom.F) — x + rxBx- Therefore there exist r'x £]°><[ and M > such that F(y) < M for all y G x + r'J,Ux Then

for y G x + r'J.Ux and i G / we obtain that ft(y) - fi(x) < M — (-Mx) =

M+Mx = : L'x Using now Corollary 2.2.12 we have that fi is Lipschitz with

constant Lx :~ ZL'x/r'^ o i i x + rxUx, where rx := r'^/2 The conclusion

follows •

Taking X a Banach space, Y a normed space, and {Xi | i G 1} C £ ( X , Y) (I 7^ 0) with {TiX \ i £ 1} bounded for every x £ X, the conditions of the preceding theorem are satisfied by the family of functions (fi)iei, where U{x) := ||Ti(a:)||, and C := X Hence \\Ti(x)\\ = \h(x) - /4(0)| <

L||x|| for x £ rllx and i £ I, for some r,L > 0; hence \\Ti\\ < L for every i £ I So we obtained the classic uniform boundedness principle in Functional Analysis

From the preceding result we obtain that pointwise convergence implies uniform convergence on compact sets

Corollary 2.2.23 Let X be a Banach space and C C X be an open

convex set Consider (/„) a sequence of continuous convex functions defined on C with values in E such that (fn(x)) -> f(x) for every x £ C with

(or, equivalently, (fn{xn)) -> f(x) for every sequence (xn) C C converging

to x E C)

Proof First of all observe that / is convex (by Theorem 2.1.3) and locally Lipschitz (by the preceding theorem)

Let x,xn E C (n E N) be such that (x„) -> x By Theorem 2.2.22 there

exist r, L > such that Ux := x + rUx C C and \fn{y) - fn(z)\ <L\\y — z\\

for all n E N and y,z E Ux- Since (xn) -> x, xn E £/x for n > nx for some

nx E N Then

|/n(*n) - f(x)\ < \fn(xn) - fn(x)\ + \fn(x) - f(x)\

<M\\xn-x\\ + \fn(x)-f(x)\

for n > nx, whence (fn(xn))n -*• f(x)

Assume now that there exists some compact subset K of U such that (/„) does not converge uniformly to / on K Then there exist e > 0, P C N an infinite set and a sequence (xn)nep C K such that \fn(xn) — f(xn)\ > £•

Since K is compact, we may assume that (xn) -> x K C C Then, as

shown above, {fn{xn))p -> f{x) Since / is continuous, we have also

that (f(xn)) p —>• f(x), which yields the contradiction > e •

The next result corresponds to a known theorem for continuous linear operators

Proposition 2.2.24 Let X be a Banach space and C C X be an open

convex set Assume that / , / „ : C —> M (n € N) are continuous convex functions Then (fn(x)) —> f(x) for every x C if and only if (a) (fn{x))

is bounded for every x G C and (b) (fn(x)) -> f{x) for every x € D for

some dense subset D of C

Proof The necessity is obvious Assume that conditions (a) and (b) above are satisfied but (fn{x)) does not converge to f(x) for some x E

C Hence there exist e > and P C N an infinite subset such that \fn(x) - f(x)\ > e for every n E P By Theorem 2.2.22 we find r,L >

such that Ux := x + rUx C U and / , / „ (n E N) are L-Lipschitz on Ux

Since D is dense, there exists (xk) C D converging to x; we assume that

Xk E Ux for every fcgl Then for n E P and k E N we have

Fixing k G N such that \\xk - x\\ < e/(4L), we get |/„(xfc) - f(xk)\ > e/2

for n G P, contradicting that (fn(xk) - n G N) converges to f(xk) •

Let / G T(X); we call the recession function of / the function /oo :

X —>• IK whose epigraph is ( e p i / ) ^ Let XQ G d o m / ; taking into account

formula (1.3), we have that

(u, A) G epi/oo < » V i > : ( x0, / ( x0) ) + £(u, A) G e p i /

/(x0 + fa) - /(x0) <

^ J i m /(X0 + fa) - / ( x0) = g u p /(Xp + fa) - / ( X Q ) < A

t-S-OO £ ( >0 t ~

Therefore

V t i S l : /oo(u) = lim ; (2.27)

t—>oo r

thus /oo(u) > - o o for every u G X and /oo(0) = Because epi/oo is a closed convex cone we have that f^ is a lsc sublinear functional

The preceding relations show that

f(x + u) < / ( x ) + /oo(u) V x G d o m / , Vu G X, (2.28)

[/ < A]^ = {u e X | /«,(«) < 0} V A G E w i t h [ / < A ] # , (2.29)

when / € T(X) Taking into account the discussion on page 6, relation (2.29) shows that the function / € T(X) has bounded level sets when d i m X < oo This result is no longer true when d i m X = oo; take f.i / := PA, the Minkowski functional associated to the set in Example 1.1.1 (/oo = / in this case)

Remark 2.2.2 Note that, for / G F(X) and x G X, the mapping t >->•

/ ( x + fa) is nonincreasing from E into E when /oo(u) < 0; in particular d o m / + E+u = d o m / Moreover, if /<»(«) < and /oo(~w) < then

f(x + tu) = f{x) for all x G X and t G E

Indeed, let ti < t2 If / ( x + t\u) < oo then

/ ( x + t2u) - f(x + tiu) _ f(x + tin + (t2 - h)u) - f(x + tiu)

and so f(x + t2u) < f(x + t\u); the inequality is obvious if f(x + tiu) = oo

It follows that d o m / + R+u = d o m / When /«>(«) < and /oo(-w) < we can apply the previous result for u and — u and obtain that the mapping

t H-> f(x + tu) is constant

2.3 Conjugate Functions

In this section X and Y are separated locally convex spaces if it is not

stated explicitly otherwise Let / : X —> M; the function

f*:Xm-+% f*{x*):=suv{(x,x*)-f(x)\xeX}, (2.30)

is called the conjugate or Fenchel conjugate of / Note that if there exists

XQ X such that /(xo) = ~°o then f*(x*) — oo for every x* € X*, while

if f(x) = oo for every x then f*{x*) = - o o for every a;* £ X* When / is proper (but also for / not proper, using the convention inf = oo), we have

f*(x*) = sup{(x,a;*) - f(x) | x € d o m / }

The conjugate of a function h : X* -> K is defined similarly:

/i* : X - > K , /i*(a;) := sup{(a;,a;*) - h{x*) \ x* e X*}

In fact, considering the natural duality between X and X*, it is the same definition; this distinction is useful in the case of normed spaces The above remark concerning /* is also valid for h*

In the following theorem we establish several simple properties of con-jugate functions

Theorem 2.3.1 Let f,g : X -> I , h : Y ->• , k : X* -)• I and

A€l(X,Y)

(i) /* is convex and w*-lsc, k* is convex and Isc;

(ii) the Young-Fenchel inequality below holds:

V x e l V i ' e r : f{x) + f*(x*)>(x,x*);

(iii) f<9^9*<f*;

(v) if a > then (af)*(x*) = af'ia^x*) for every x* G X*; if/3 ^ tfien (f(l3-))*(x*) = f'ifi-^x*) for every x* G X*;

(vi) if g(x) = f(x + x0) for x G X, then g*(x*) = f*(x*) - {x0,x*) for

every x* G X*;

(vii) if x*0 G X* then (/ + x*0)*{x*) = f*(x* - x*0) for every x* G X*;

(viii) if f, h are proper and $ : I x -> , $(x, y) := f(x) + h(y), then

$*(x*,y*) = f*(x*) + h*{y*) for all (i*,y*) E X* x Y*;

(ix) {AfY =f*oA* and (fDg)* = f* + g*

Proof, (i) If / is not proper, we have already seen that / * is constant,

and so / * is convex and w*-continuous If / is proper we have that / * =

suPzedom/ *Px, where ipx : X* -*• R, ipx(x*) := {x,x*) - f(x) It is obvious

that for every x G d o m / , (px is affine (hence convex!) and u>*-continuous

(hence w*-lsc!) Therefore / * is convex and u;*-lsc For the statement about

h we use the same arguments

(ii) By Eq (2.30) we have

VxeX,Vx* eX* : / ' ( a ; * ) > < a : , a : * ) - / ( i ) ,

which gives immediately the Young-Fenchel inequality

(iii) is an immediate consequence of the definition and of the relation f<9- _ _

(iv) We already remarked that c o / < f < f, whence, using (iii), we get / * < / * < (co/)*- Let x* G X* and a G R be such that f*(x*) < a Then

(x,x*) — f(x) < a for every x G X, whence ip(x) := (x,x*) — a < f(x)

for every x Since G T(X), and epi(p D e p i / , we have that epiy> D epi(co/) = co(epi/) Therefore (f(x) < (cof)(x) for every x G X, and so

(x,x*) — cof(x) < a for every x; hence (co/)*(a;*) < a Thus / * = / =

(co/)*-(v), (vi), (vii) and (viii) are immediate (ix) We have

(Afy(y*) = sup{(y,y*) - (Af)(y)] = sup ((j,,**) - inf f(x)) y£Y y£Y V {x\Ax=y} J = sup{(y,y*) - f{x) \(x,y)£X x Y, Ax = y}

= sup{(x,A*y*)-f(x)\x£X}

= f*(A*y*) = (roA*)(y*)

I the sequel we denote by T*(X*) the class of those functions in A(X*) which are w*-lower semicontinuous The discussion at the beginning of this section and the preceding theorem yields the next result

Corollary 2.3.2 Let f : X -> E and h : X* - • E Then

(i) /* € T*(X*) <S> d o m / ^ and 3x* G X*, a G K, Vrr G X :

f(x) > (x,x*) + a ;

(ii) h* G r ( X ) <S> dom/i / and i I , a e E, Va:* £ X* :

/i(:r*) > (x,a;*) + a • The following result is fundamental in duality theory

Theorem 2.3.3 (of the biconjugate) Let f e T(X) Then f* € T*{X*)

and f **:=(/*)* = f

Proof Applying Theorem 2.2.6 we get x$ G X* and a G ffi such that

V i e l : / ( i ) > (x,x*0)+a (2.31)

Thus, by the preceding corollary, we have that /* G T*(X*) Moreover, Theorem 2.3.1 shows that /** < /

Let x € X be fixed and consider t G E such that t < f(x); therefore

(x, t) ^ e p i / Applying Theorem 1.1.5 for {(x,t)} and e p i / , there exist (x*,a) G X* x E and A G E such that

V ( z , i ) G e p i / : (x,x*) + ta < A < (x,x*) + ta (2.32)

Taking (x, t) = (x, f(x) + n), n G N, with x G d o m / , we obtain that

V n G N : (x,x*) + af(x) + na < A < (x,x*) + ta

Letting n -> oo we obtain that a < Take first a < Dividing eventually by —a > 0, in Eq (2.32) we can suppose that a = —1 Thus

(x,x*) - f(x) < A V i £ d o m /

Hence /*(#*) < A < (x,af) - <, and so

t<(x,x*)-r(r)<r*(x)

Take now a = 0; using relation (2.32) we get c > such that

This together with Eq (2.31) yields

f(x) > (x, XQ) + a > (x, XQ) + a + t(x, x*) + tc — t(x, x*)

for all x € dom / and all t > 0; this implies successively:

MxeX, V i > : -tc + t(x,x*) - a> (x,XQ+tx*) - f(x),

V £ > : -tc + t{x,x*) -a > f*(x*Q + tx*),

V i > : f**(x) > (x,x*0+tx*)-f*(x*0+tx*) > a+ tc+(x,x*0)

But there exists t > such that a + tc + (x, a;*,) > t; hence f**(x) > t in this case, too Thus we obtained f**(x) > f{x) Therefore /** = / •

The preceding theorem shows that for any function / : I -> we have ((/*)*) = /* (for the duality (X,X')) which shows that there is no interest to consider conjugates of order greater than It also shows that the conjugation is an isomorphism between T(X) and T*(X*) More precise information on the biconjugate of an arbitrary function is furnished by the following result

Theorem 2.3.4 Let f : X —>• ffi have nonempty domain

(i) / / co/ is proper, then /** = c o / ; if c o / is not proper, then /** = —oo

(ii) Suppose that f is convex If f is Isc atxG d o m / , then f(x) = /**(x); moreover, if f(x) € R, then /** = / and f is proper

Proof, (i) The function c o / is convex and lsc If c o / is proper, using the

preceding theorem and Theorem 2.3.1(iv), we have that

co/= (co/r = (/•)* = /**•

If c o / is not proper, since dom(co/) D dom / / 0, c o / takes the value - c o ; hence /* = (co/)* = oo, and so /** = —oo

(ii) Taking into account that / is convex, co/ = / Since / is lsc at x, we have that f(x) = f(x) It is obvious that f**(x) = f(x) if f(x) = - o o Let f(x) £ M; then f(x) € E, and so / is proper From the first part we

have that /** = J, whence f**(x) = J(x) = f(x) •

Corollary 2.3.5 Let f,g £ A(X) If fUg is proper, then (/* + g*)* =

fOg = fOg~, while in the contrary case (/* + </*)* = —oo Furthermore, if f is continuous at a point of its domain then

Proof From Theorem 2.3.1 we have that

uogr = r+g* = T+r = (f^9)*\

since /•<? is convex, the conclusion of the first part follows from Theorem 2.3.4

If / is continuous at xo £ d o m / , then / is upper bounded on a neigh-borhood of £o, whence fOg is upper bounded on a neighneigh-borhood of xQ+yo,

where yo € doing Applying Corollary 2.2.14 we get that fOg is continuous on int(dom(/Dg()), and so it is lsc on this set The conclusion follows now

from Theorem 2.3.4(h) D Consider O ^ A c X ; the support function of A is defined as being

sA:X*^W, sA(x*) :=sup{(x,x*)\x &A}; (2.33)

for ^ B C X* the support function SB : X —> E is defined similarly It is obvious that SA is w*-lsc and sublinear while SB is lsc and sublinear Furthermore, if C C X is another nonempty set then SA+C — $A + $c and

SAUC — SA V sc- Note that

(IA)*(X*) = sup(a;,a;*) = sA(x*) = sup (x,x*) = (LCOA)*(X*) = ScoA

-x£A X € C O J

Moreover we have that icoA — cotyi and

dom SA = dom(tJ4)* = {x* € X* | x* is upper bounded on A};

therefore d o m s ^ = X* if and only if A is u;-bounded In the next section we shall see that LA is useful in determining the normal cone of C

2.4 The Subdifferential of a Convex Function

We have already seen in Section 2.1 that if the proper convex function / : (X, || • ||) -> M is Gateaux differentiable a t i G int(dom / ) then

V z e d o m / ( V a ; e X ) : (x -x, Vf(x)) < f(x) - f(x)

Taking into account this relation, it is quite natural to consider the elements

x* E X* which satisfy the inequality

VxEX : (x-x,x*)<f(x)-f(x), (2.34)

In this section the spaces under consideration are separated locally con-vex spaces if not stated otherwise

Let / : X -¥ E and x £ X be such that f(x) G E An element x* G X* is called a subgradient of the function / at x if relation (2.34) is satisfied; the set of all the subgradients of the function / at x is denoted by df(x) and is called the subdifferential or Fenchel sub differential of / at x We consider that df (x) = if f(x~) £ E; of course we can have df(x) = even if f(x) G E Thus we obtain a multifunction df : X =4 X* By the preceding considerations we have that domdf C d o m / We say that / is

subdifferentiable at x G X if df(x) ^

Note that if x* G df(x), the afflne function : X —• R, (p(x) :=

(x,x*) — (x,x*) + f(x) minorizes / and coincides with / at x; this proves

that

V ( x , i ) € e p i / : (x,x*) — t < a := (x,x") — f(x),

which shows that the hyperplane {(x,t) € X x R \ (x, x*) — t • = a] is a, non vertical (since the coefficient of t is ^ 0) supporting hyperplane (see p for the definition)

Recall that in Theorem 2.1.5 we have already determined the sub-differential of a function / A(M) at to £ dom / :

0/(*o) = [/:(*o),/;(to)]nR

In the sequel we shall establish properties of the subdifferential and methods for computing it also for X ^ E

In the next result we collect several easy properties of the subdifferential

Theorem 2.4.1 Let f : X -> and x X be such that f(x) E Then:

(i) df(x) C X* is a convex and w*-closed (eventually empty) set (ii) Ifdf(x)^<bthen

(co/)(x) = J ( i ) = f(x) and d(cof)(x) = dj(x) = df(x);

in particular /** = c o / and f is proper and Isc at ~x

(iii) / / / is proper, dom f is a convex set and f is subdifferentiable at

every x £ dom / , then f is convex

Proof, (i) Let x\, x*2 e df{x) and A G ]0,1[ Then

Multiplying the first relation with A > 0, the second with - A > 0, then adding them, we obtain

V x £ l : (x-x,Xx*1+(l-X)x*2)<f(x)-f(x),

whence Ax* + (1 — \)x% G df(x)

Let x* G X* \ df(x) Then there exists XQ G X such that (xo — x,x*) > /(xo) - fix) Let a G E be such that {x0 — x,x*) > a > f(x0) — f{x)

Then V := {x* \ (x0 -x,x*) > a} is a neighborhood of x* for the topology

w* = a(X*,X) It is obvious that V n df(x) - So df(x) is w*-closed

(ii) We already know that co/ < f < f- Let x* G df(x) and

y>:K->K, ip(x):=(x-x,x*) + f(x);

(fi is convex, continuous and <p < f Therefore < c o / < / < / Since ip(x) = f(x), we have that cof(x) = f(x) = f(x) This relation proves

that the functions / , / , c o / are proper and / is Isc at x Prom the above inequality we obtain that d(cof)(x) D df(x) D df(x) If x* G d(cof)(x), then

V x £ l : (x-x,x*) < (cof){x) - (co/)(z) < f{x) - f(x),

whence d(cof)(x) C df(x) Therefore d(cof)(x) = df(x) = df(x) (iii) By (ii) we have that f(x) = cof(x) for every x G d o m / Since c o / is a convex function and dom / is convex, it is obvious that / is convex •

The property (ii) of Theorem 2.4.1 justifies why we consider only proper convex functions when discussing subdifferentiability (in the sense of convex analysis!)

In a similar way we introduce the subdifferential of a function h : X* —> l a t a point x* G X* with h{x*) G E:

dh(x") = {x € X | Vx* G X* : {x,x* - x*} < h(x*) - h(x*)}

(As for conjugation, this distinction is important only when working with normed spaces; in the case of locally convex spaces we always consider the natural duality between X and X*, when the dual of X* is X.)

In practice (for example for solving numerically problems using com-puters) it is possible to determine the subgradients only approximately In this sense the following notion of subgradient reveals itself to be useful Let / : X ->• , x G X with f(x) G M and e G ! + ; the element x* G X* is called an e-subgradient, of the function / at x~ if

V x e l : (x-x,x*) <f(x)-f(x)+e; (2.35)

the set of e-subgradients of / at x is denoted by def(x) and is called the

e-subdifferential of / at ~x As for the subdifferential, if f(x) ^ M we

consider that def(x) = 0; we obtain a multifunction dEf : X =£ X* with

dom(<9E/) C d o m / Note that / is proper if d£f{x) ^ for some e > 0; if

0 < £i < £2 < oo then

Of(x) = d0f(x) C 3ei/(af) C 5e a/ ( i )

Moreover

V e G l ^ : 9e/ ( i ) = f l a , / ( i )

The £-subdifferential of a function /t: I * -> K at ? € I * with /i(x*) G E is introduced similarly

In the following theorem we collect several properties of the subdifferen-tial and of the £-subdifferensubdifferen-tial Before stating this theorem, let us intro-duce some notions: we say that the set M C X x X* is monotone if

V(x,x*),(y,y*)€M : (a: - y,x* - y*) > 0; M is strictly monotone if

V{x,x*), (y,y*)eM, x^y : (x - y,x* - y*) > 0;

M is maximal monotone if a) M is monotone and b) M' C X x X*

monotone and M C M' imply M = M', i.e if M is a maximal element of the class of monotone subsets ordered by inclusion

We say that the multifunction T : X =3 X* is monotone, strictly

monotone or maximal monotone if gr T is monotone, strictly monotone

or maximal monotone, respectively Of course, when X = E and T is single-valued (i.e T(a;) is a singleton for every x G domT), T is (strictly) monotone if and only if the function T|do mT is nondecreasing (increasing)

Theorem 2.4.2 Let f,g : X ->• R, h : Y -> R be proper functions,

A L(X,Y), x G domf n domg, y G dom/i and e E M+ T/»en:

(i) dsf(x) is a convex w* -closed set

(ii) x* G 3e/ ( z ) ôã / ( x ) + f*(x*) <(x,x*)+e^x£ def*(x*)

(iii) a;* G df(af) «• / ( x ) + /*(a;*) < (x,x*) & f(x) + f*(x*) = (x,x*) =>

xedf*(x*)

(iv) dom(<9£/) C d o m / , lm(dsf) C d o m / * and df is monotone

(v) 3/(3;) ôã /(3f) = m a xa.e x ((x,x*) - f*{x*))

(vi) de(f + x*)(x) = x* + def(x) for x* E X*; 0e(A/)(3:) = \ds/xf(x)

and d(Xf)(x) — Xdf(x) for X > 0; if g(x) = f(x + XQ) for x G X then dEg(x) = dEf(x + x0)

(vii) Assume that y = Ax; then A* (deh(y)) C de(h o A)(x)

(viii) U^6[0,e] (dvf(*) + 9S-Vg(x)) C 3£( / + g)(x)

(ix) Suppose X is a normed space, f is Isc, £j > and (xi,x*) G

grdEif for every t £ If (ei)ig/ —> e < oo and either (a) (xi)iei —• a;,

(x*)iei -^-> a;* and (x*)ie/ is norm-bounded or (b) (x»)ie/ -^-> £, (a;j)jej is

norm-bounded and (x*) —t x*, then (x,x*) G grdsf In particular grdef

is closed in X x X* (for the norm topology)

Proof, (i) The fact that dEf(x) is convex and u>*-closed is shown similarly

to the first part of Theorem 2.4.1 (ii) We have that

x* ed£f(x) o V x G X : (x - x,x*) < f{x) - f(x) +e

» V i £ l : {x,x*)-f(x)<(x,x*)-f(x) + e

&f*(x*)<(x,x*)-f{x)+e

&f{x) + f*(x*)<(x,x*)+e

Assume now that x* G def(x) Then, by what precedes,

f"(x) + /*(**) < f(x) + fix*) < (x,x*)+e,

and so x € d£f*(x*)

(iii) We already know that (37, x*) < f(x) + f*(x*) (the Young-Fenchel inequality!); the equivalences follow now from (ii) taking e =

(iv) It is obvious that dom(<9£/) C d o m / , while the inclusion Imdef C

Let (x,x*), (y,y*) £ gr<9/ Then x,y £ d o m / and

{y - x, x*) < f(y) - f(x), (x - y,y*) < f(x) - f(y)

Adding these relations we get (y — x,x* — y*) < 0, and so df is monotone (v) If x* € df(x), taking into account (iii) and the Young-Fenchel inequality, we have

V x ' e l * : (x,x*)-r(x*)=f(x)>(x,x*)-r(x*),

and so the implication "=^" is true The converse implication is an imme-diate consequence of the equivalences of (iii)

(vi)-(viii) follow easily from the definition of the e-subdifferential (ix) Suppose that X is a normed space Let (e'j)ie/ -> e < oo and

((xi,x*))i€l have the properties from (a) or (b) Taking into account that

\X{, 3Jj j \Xy X / — \X{ Xj X^ J ~\- \X)

X* -X*) = (Xi,X* -X*) + (Xi-X,X*)

for every i, we get ((y — Xi,x*)) —t (y — x,x*) for every y X in both cases But

V i e / , Vy£X : f(xi) + (y-xi,x*)<f(y)+ei;

taking the limit inferior, we obtain x* G def(x) O

In the preceding theorem we obtained that df is a monotone multi-function In fact df is cyclically monotone One says that T : X =} X* is

cyclically monotone if for all n S N and ((xi,x*))._ C g r T one has

Y" (xi+1-Xi,x*)<0, (2.36)

* — ' i =

where xn+i := XQ Taking n = in Eq (2.36) we obtain that {x\ - x$, x0') +

(x0 — xi,x\) < 0, i.e (xi —xQ,xl — XQ) > 0, for all (X0,XQ), (XI,XI) £

g r T Hence every cyclically monotone multifunction is monotone When / : X -> R is a proper function, df is a cyclically monotone multifunction Indeed, let n e N and ((xi,a;*))"=0 C gr<9/; then (xi)™=0 C d o m / and

(xi+i -xt,x*) < f(xi+i) - f(xi) Vi, < i < n,

where, as above, xn+\ := XQ Adding these relations side by side for i =

Proposition 2.4.3 Let T : X =4 X* be a cyclically monotone

multi-function and (icc^o) £ gr^ - Consider fr : X —>• M defined by

h{x) := sup f (a; - zn, < ) + X ) =Q (a:*+1 ~ ^ ^ ) ) > (2-3 7)

where the supremum is taken for all families ((x,,x* ) ) "= 0 C g r T withne

N T/ien /T e T(X), /T(XO) = and T(x) C dfT(x) for every x eX

Proof Because fr is a supremum of continuous afHne functions, fa is lsc, convex and nowhere - o o Note that domT C d o m / y Indeed, let (x,x*) G g r T Taking k G N, ((a:i,xj))*=0 C g r T , n := fc + and ( x „ , < ) := ( i , ? ) ,

from Eq (2.36) we obtain that

Eit-i _

{Xi+i - Xi, X*) + (X- Xk,X*k) + (X0 - X, X*) < 0,

1=0

whence fr{x) < (x — x0,x*) Hence x G d o m / y In particular the

preced-ing inequality shows that / T ( ^ O ) < Taking n = and (xi,x*) = (XO,XQ)

in the definition of fr(xo), we get fr(xo) > Hence / T ( ^ O ) = Therefore h € T(X)

Let now (x,x*) G g r T and x G X Consider an arbitrary it < fr{x) +

(x — x,x*) Then there exist k G N and ((XJ,X*)J C g r T such that

_ r—\fc-i

/ x - ( x - x , x * ) < {x-xk,x*k) + ^ = fci+i

-a;»,a;*>-Taking n = k + and (xn,a;*) := (x,x*), the preceding inequality yields

En—1 i=o (Xi+1 ~XuX^ -

-^W-Letting (i —¥ fr (x) + (x — x, x*), we obtain that fr (x) + (x — x, x*) < fa (x),

and so x* G / r ( S ) D In the next theorem we use the convexity of the function /

Theorem 2.4.4 Let f G A(X), x G d o m / and e G ffi+ T/»en:

(i) def(x) = df'e(x, -)(0); moreover, ifx£ ( d o m / )1 and / is Gateaux

differentiate at x then df(x) = { V / ( x ) }

(ii) 7/ / is strictly convex then df is strictly monotone

(hi) / is lsc atx if and only ifd£f(x) ^ for every e > 0; d£f*(x*) ^

if f (x*) eRande>0

Proof (i) Ii x* £ dsf(x), replacing x by x + tu, with t > 0, in Eq (2.35),

then dividing by t we obtain

V « € l : (u, x ) < inf — ^ - ^ — fe{x,u),

and so a;* £ df^(x, -)(0) The converse inclusion follows from the inequality

f'e(x, u) < f(x + u) — f(x) + e, valid for every u £ X

If / is Gateaux differentiable at x then f'+{x, •) — V/(aT); the conclusion

is obvious

(ii) Let x,y £ d o m / , x / y, and x* € df(x), y* £ df(y) From (i) we have that

(y-x,x*) < f+(x,y-x) < f(y)-f(x), & - y, y*) < f+(y, x-y)< f(x) - f(y),

because / is strictly convex Adding the above two relations we obtain that

(y — x, x* — y*) < This shows that df is strictly monotone

(iii) Suppose that / is lsc at x and let us take e > 0; using Theorem 2.3.4 we have

f(x) = r(x) = sup{(z,z*) - f*(x*) I x* G X*} > f(x)-e

Therefore there exists a;* £ X* such that {x, x*)—f*(x*) > f(x)—s, whence

x* £ def(x)

Conversely, suppose that dEf(x) ^ % for every e > Then

V e > , 3x*£X* : f(x) - e < {x,x*} - f*(x*) < f(x),

whence f(x) < f**(x) Since / * * < / < / , it follows that J{x) = f(x), i.e

f is lsc at x

Let x* £ X* be such that f*(x*) £ R and e > By the definition of /*, there exists x £ X such that f*{x*) < (x,x*) — f(x) + e, whence

(x,x*-x*) < (x,x*) - f(x) - f*(x*)+e< f(x*) - f*(x*)+e, i.e.x£def*(x*)

(iv) Since / is lsc at x, we have that f**(x) = f{x); the conclusion is

immediate using Theorem 2.4.2 (ii) •

Remark 2.4-1 Note that for / A(X), x £ d o m / and U £ Mx{x) we

different for e > 0, i.e df is a local notion while dsf (with e > 0) is a

global one for convex functions

Indeed, since f'(x, u) — (f + t[/)' (x, u) for every u £ X, the first remark follows from assertion (i) of the preceding theorem For the second remark let us consider / € A(R) given by f(x) = x2 and U = [—1,1] Then

ds{f + iu)(0) = def{0) = [-2y/i,2y/e\ for e € [0,1] and de{f + ta)(0) =

[-1 -£,l + e]£ [-2y/i,2^/i\ = 0e/(O) for e >

Generally, the formula d(\f)(x) = Xdf(x) is not true for A = 0; this formula, however, is true if x £ (dom/)1 In assertions (vii) and (viii) of

Theorem 2.4.2 we have equality only under supplementary conditions called generally "constraint qualification conditions;" in Section 2.8 we shall give such conditions

Let A C X be a nonempty convex set and a e A; then

diA{a) = {x* eX*\Vx£A : (x-a,x*)<0}

= {x* | V x G A : (x,x*) < (a,x*)}

= {x* | Vx cone(A - a) : (x,x*)<0}

= —(cone(A — a ) )+ = —cone(A — a)+

The set <9M(O) is denoted by N(A;a) and is called the normal cone of

A at a S A, while the set cone (A — a) is denoted by C(A;a) (even if A is

not convex); evidently, the set C(A;a) is a closed (convex if A is so) cone Taking into account the relation established above and the bipolar theorem (Theorem 1.1.9), we have that

N{A;a) = -(C(A;a))+ and C(A;a) = -(N(A;a)) +

It is obvious that N(A; a) = {0} if a £ A* We observe that x* € N(A; a) \ {0} if and only if Hx.^a^x*) is a supporting hyperplane to A at a and

A c / / - , , ,,

The set df(x) may be empty even if / is lsc at x For example, the function / : R -> E, f(x) := -Vl - x2 for \x\ < 1, f(x) := oo for |a;| >

1 (already considered in Section 2.1) is lsc, finite at 1, but df(l) = Moreover ( - / ) (1) = R_

Corollary 2.4.5 Let fi : Xi —> R for i E \,n be proper functions and

x = (xi, ,xn) E Yli=i d o m / j Let us consider the function

nn — • ^n

Xi-tR, <p{xi, ,xn) := > fi(xi),

ande E K+• Then

de<p{x) = (J | n ^ i 9 ^ ^ ) £i - °> £l H + e« = e j •

/n particular

nn

dfi(xi)

Proof We have already seen in the preceding section (for n = 2, but the extension is immediate) that y*{x\, ,xn) = Y12=ifi(xi)- Therefore, by

Theorem 2.4.2 (ii),

x* E de<p(x) &(p(x) +ip*(x*) < (x,x*) +e = (xi,xl)-\ 1- ( z „ , < ) + e

O e\, , en > 0, ei + + en = e,

Vi G T~n : /i(xi) + /*(z*) - (xi,x*) < et,

O 3 £i, , 6n > 0, £! + + £„ = £,

Vi G l , n : x* € dEifi(xi),

whence the conclusion •

Corollary 2.4.6 Let f : X -» R be a proper function and A E &(X, Y)

If x E d o m / and y E Y are such that y = Ax and (Af)(y) = f(x), then for every e E R+ we have

de(Af)(y)=A'-1(def(x))

Proof Using Theorem 2.4.2 (ii) we have that

y* E d£(Af)(y) o (Af)(y) + (A/)*(j,*) < (y,y*) +e

& f(x) + f*(A*y*) < (Ax,y*) + e = (x,A*y*) + e *> A*y* G def(x) &y*E A*'1 (def(x))

Corollary 2.4.7 Let / i , , / „ : X -> R (n € N) be proper functions

Suppose that for every i € l , n there exists x~i £ Aora.fi suc/t i/iaf

(/lD • • • • / „ ) ( ! ! + + ! „ ) = / i ( l i ) + • " • + /„(x„) (2.38)

Then for every e € E+

ae( / i D D / „ ) ( i i + ••• + ! „ )

= [ J { ^ e i / l ( ^ l ) n - - - n 9e„ / „ ( l „ ) | £ l , , e „ > 0, £X + + £ „ = £ }

In particular

0(/iD n/n)(xi + • • • + xn) = a/i(ii) n • • • n 5/„(in)

Conversely, if dfi(x~i) Pi • • • (~1 dfn(xn) ^ 0, i/ien relation (2.38) is vafo'd

Proof We apply Corollary 2.4.5 to the functions fi,- • • ,fn and Corollary

2.4.6 to / : Xn ->• E defined by / ( x i , ,£„) := / i ( x i ) + • • • + fn(xn) and

to the operator A : X " —> X defined by A(xi, , xn) := X\ + • • • + xn

If x* £ / i ( ô i ) (~1 ã ã ã n dfn{xn), then

Vi £ l , n , V i j G-X" : (xi-Xi,x*)<fi(xi)-fi(x~i); hence / I ( X J ) H h fn{xn) < / i ( x i ) ^ 1- fn{xn) for all x i , , xn e X

such that xi + • • • + xn = x~i + • • • + xn, and so Eq (2.38) holds •

The next result shows that the convolution has a regularizing effect

Corollary 2.4.8 Let / i , /2 e A(X), xt e d o m / i for i e {1,2} and

x — xi +X2- Assume that (fiDf2)(x) = /i(x"i) + f2(x2) and fiDf2 is

subdifferentiable at x If f\ is Gateaux differentiable at x\ then f\Of2 is

Gateaux differentiable atx and V(fiDf2){x) = V / ^ x j ) Moreover, if X

is a normed vector space and / i is Frechet differentiable at xi then /1D/2 is Frechet differentiable at x

Proof By the preceding corollary we have that d(f\Of2) (x) = df\ (x~i) n 0/2(z2)- Let x* £ d(f1nf2)(x) Then

(u,x*) < (haf2)(x + u) - (AD/aXaf)

< fl(xi +u) + /2( x2) - fi{xi) - /2( x2)

< / i ( S i + « ) - / i ( ^ i ) (2-39) for every u € X If fi is Gateaux differentiable at x\ then, from the

Using (2.39) we obtain that (fiOf2)'(x,u) = (w, V/i(a;1)) for every u € X,

which shows that / i D / is Gateaux differentiable at x and V(/iD/2)(x) =

V / ! ( l i )

Assume now that X is a normed space and f\ is Frechet differentiable at x\ From the preceding situation we have that /1IH/2 is Gateaux differ-entiable at x and V(/iD/2)(5;) — V/i(afi) Using again Eq (2.39), we have

that

0 < (/id/2)(af + u) - (fiDf2)(x) - ( u , V / i ( i i ) >

< fi{xi + u) - / i ( i i ) - (u, V/i(ii)> Vu G X;

hence /1CH/2 is Frechet differentiable at 2; and V(/iD/2)(af) = V/i(aFi) D In Section 2.6 we shall extend the results of Corollaries 2.4.6 and 2.4.7 to the case where the infimum is not attained in y and x, respectively

The following result establishes a sufficient condition for the subdifferen-tiability of a convex function; this result is of exceptional importance

Theorem 2.4.9 Let f G A(X) If f is continuous at a; G d o m / , then

def(x) is nonempty and w* -compact for each e G M+ Furthermore, for

every e > 0, f'e(x, •) is continuous and

V « l : f'e{x,u) = max{(«,x*) | x* G d£f(x)} (2.40)

Proof Suppose that / is continuous at x G d o m / (from Theorem 2.4.4

we already know that def(x) ^ for e > 0!)

Let rj > Since / is continuous at x, there exists V G K x such that

Vxex + V : f{x) <f{x)+r] (2.41)

Therefore (aT + V) x [f(x) + n, oo[C e p i / , whence int(epi/) ^ The set e p i / being convex and (x, f(x)) £ int(epi/) (because (x,f(x) — 5) $ e p i / for every S > 0), we can apply a separation theorem; so, there exists (a;*,a) G X* x E \ {(0,0)} such that

V ( z , f ) € e p i / : (x,x*) + at < (x,x*) + af(x) (2.42)

(a;*,a) = (0,0) Therefore a < 0; so we can consider a = - in Eq (2.42) This relation becomes

V x e d o m / : (x,x*) - f(x) < (x,x*) — f(x),

i.e x* e df(x)

Let now e > For every x* G dsf(x) = df£(x, -)(0), using Eq (2.41),

we have

V i i g y : (u,x*) <fs{x,u) <f(x + u)-f(x)+£<r) + e, (2.43)

whence {u,x*) > - for every u £ (r] + e)~1V (recall that V is symmetric)

which proves that

def(x) C (fa + e ) -1 V)° = (r, + e)Va (2.44)

By the Alaoglu-Bourbaki theorem (Theorem 1.1.10) we have that V° is w*-compact; since def(x) is w*-closed, it follows that d£f(x) is w*-compact

Taking into account that x £ int(dom/), from Theorem 2.1.14, we have that domf^x, •) = X; hence, using Theorem 2.2.13 and Eq (2.43), the function fg(x,-) is continuous Furthermore, using again Eq (2.43), we have that

f'e(x,u) > sup{(u,a;*) | x* £ def(x)}

We intend to prove that in the above inequality we have equality and the supremum is attained For this end let u € X \ {0} Consider XQ := Eu and ip : Xo -> E, p(tu) := tf'e(x,u); it is obvious that ip(u') < f'E(x,u') for

every u' Xo, and so, applying the Hahn-Banach theorem, there exists

x* X' such that (u,x*) = tp(u) = f^{x,u) and (u',x*) < f'e(x;u') for

every u' € X Since f^(x,-) is continuous, x* is continuous, too; hence

x" e df'£{x, -)(0) = dEf(x) The proof is complete •

Using the preceding result we obtain the following criterion for the Gateaux differentiability of a continuous convex function

Corollary 2.4.10 Let f € A(X) be continuous atxe domf Then f is

Gateaux differentiate at x~ if and only if df(x) is a singleton

— (—u,x*) = (u,x*), we have that / is Gateaux differentiable and Vf(x) =

x* D

Related to Frechet differentiability of convex functions see Theorem 3.3.2 in the next chapter

When / is not continuous at x £ d o m / , relation (2.40) may be false; see Corollary 2.4.15 and Exercise 2.30 However the following result holds

Theorem 2.4.11 Let f £ T(X), x £ dom / , and e £ ]0, oo[ Then

Vu£X : f'e(x,u)=sup{{u,x*)\x* £d£f(x)} (2.45)

Therefore f'e(x, •) is a Isc sublinear function Furthermore, for every u £ X,

f'(x,u) = lim (swp{{u,x*) | x* £ def(x)})

= inf (sup{(u,a:*) | x* £ def(x)}) (2.46)

Proof In Theorem 2.4.4(i) we have seen that dEf(x) — df^.(x,-)(0),

whence

V / 3E/ ( f ) , V i i l : {u,x*) < ft(x,u)

Therefore the inequality " > " holds in Eq (2.45) For proving the converse inequality, let u £ X and A £ R with A < f'£(x,u) From the definition of

f'e (x, u), we have that

Vt>0:X<fiW + t u ) ; m + £ (2.47)

and

A < l i m f^ + tu)-f(x)+e = ^ f(x + tu)-m = / o o ( u ), ( 2.4 )

t—>oo t (—>oo t

Let us consider A := e p i / and B := {(x + su, f(x) + Xs — e) \ s > 0} It is obvious that A and B are nonempty closed convex sets and, from Eq (2.47),

A n B = 0, i.e (0,0) ^ A — B Moreover B is locally compact (as subset

of a finite dimensional separated locally convex space) Since Aoo = epi /oo and B00=R+- (u, A), from Eq (2.48) we obtain that 4oonBco = {(0,0)}

Then, by Corollary 1.1.8, A — B is a closed set

Applying Theorem 1.1.7, there exists (x*,a) X* x E such that

Taking x = x and letting t -> oo we obtain that a < 0; if a = 0, for s = we get the contradiction > Therefore a < and we can suppose that

a = — The above relation becomes

V x G d o m / , V s > : > (a; -x,x*) - (/(«) - /(a?) + e ) +s(A - ( u , i * » For s = we obtain that x* G dsf(x), while for s -> oo we obtain

(u,a7*) > A Therefore A < sup{(u,a;*) | a;* £ 5e/(aT)} Since A < f^(x,u)

is arbitrary, the relation (2.45) is true The equality (2.46) follows from

relation (2.45) and Theorem 2.1.14 • The subdifferentiability criterion of Theorem 2.4.9 can be extended

Theorem 2.4.12 Let f G A(X) and X0 := aff(dom/) / / f\Xo is

con-tinuous atxG domf, then df(x) ^ In particular, if dim X < oo, then df(x) ^ for every x G *(dom/)

Proof Without any loss of generality, we can suppose that x = 0; then

X0 = lin(dom/) The function g := f\x0 is convex, proper and continuous

at By Theorem 2.4.9 we have that dg{0) # Let v? G dg{0); hence

ip € XQ and

V x g d o m j : tp[x) — (p(0) < g(x) — g(0)

Using the Hahn-Banach theorem we get i* G X* such that x*\x0 =

The above inequality shows that

Vx G d o m / = dome? : {x - 0,x*) = tp{x) < g(x) - g(0) = f(x) - / ( ) , whence x* G 9/(0)

If d i m X < oo, then dimXo < oo By Theorem 2.2.21, g is continuous on int(dom/) = l( d o m / ) The conclusion follows from the first part •

Note that under the conditions of Theorem 2.4.12 df(x) is, generally, not bounded, and so it is not «;*-compact But, under the conditions of Theorem 2.4.9, in the case of normed spaces, df has supplementary properties (mentioned in the next theorem) The local boundedness of monotone operators was proved by R.T Rockafellar (see Theorem 3.11.14); the converse is true in Banach spaces for lower semicontinuous functions as shown by Corollary 3.11.16

T h e o r e m 2.4.13 Let X be a normed space, f G A(X) be continuous

i n t ( d o m / ) Moreover, if f is bounded on bounded sets then f is Lipschitz

on bounded sets and dsf is bounded on bounded sets

Proof By Theorem 2.4.9 we have that int(dom/) C dome?/ C d o m / ;

hence int(dom<9/) = int(dom/) Let xo £ int(dom/); since / is continuous at x0, from Corollary 2.2.12 we get the existence of m, p > such that

Vx,y€D(x0,2p) : \f(x) - f(y)\ < m\\x - y\\

Let us fix a; € D(xo,p) Then for y £ x + pUx we have that f(y) <

f(x) + mp, i.e Eq (2.41) holds with V := pUx and r] :— mp Using

Eq (2.44), we obtain that def(x) C (r? + e)V° = (m + e/p)Ux* for every

x £ D(x0,p)

Assume now that / is bounded on bounded sets Then d o m / = X Taking p > 0, / is bounded above on 2pUx, and so Eq (2.41) holds with

V := 2pUx and some r\ > By Corollary 2.2.12 / is Lipschitz on pUx,

and, by Eq (2.44), def(x) C (?? + e)p~1Ux* for x e D{x0,p) The proof is

complete • For other relationships between the continuity of / and the local

bound-edness of df see Corollary 3.11.16

In Theorem 2.4.4 we have seen that finding the e-subdifferential of a convex function at a point reduces to compute the subdifferential of a sub-linear function at the origin Other subdifferentials (the subdifferentials of Clarke, of Michel-Penot, etc.), for nonconvex functions are introduced through certain sublinear functions So, we consider it is worth giving some properties of sublinear functions

T h e o r e m 2.4.14 Let f,g:X->M.,h:Y->Rbe sublinear functions,

T £ &(X, Y) and B,C C X* be nonempty sets Then:

(i) / * = ^9/(0);

(ii) 5/(0) ^ o / is Isc at 0;

(hi) for every x G dom / and for every e £ R+ we have:

dm = {x*edf(p)\(x,x') = f(x)},

d£f{x) = {x* e 3/(0) I (x,x°) > f(x) - £}, def(0) = 0/(0);

(iv) if f is Isc at then

(v) Suppose that f and g are Isc Then f < g <3> 9/(0) C dg(0) (vi) The support function SB of B is sublinear, Isc, SB = (<-B)* and

0 S B ( O ) — coB, the closure being taken with respect to the weak* topology w* on X*; moreover, SB < sc if and only if B C coC

(vii) / / h is Isc then d(h o A)(0) = w*-c\ (A*{dh{0)));

(viii) if f and g are Isc, then d(f + g)(Q) = w*-c\ (3/(0) + dg(0)) Proof, (i) For every x* € X* we have

f*(x*) = s u p ^ O - f{x) I x € d o m / } > (0,a;*) - /(0) =

If a;* £ df(0) then (x,x*) < f(x) for every x £ d o m / , whence f*(x*) = If x* £ <9/(0), there exists x e X such that (x, x*) > f(x) In this situation we have

f*{x*) > sup{(tx,x*) - / ( t x ) | t > 0} = sup{i(x,a;*) - / ( x ) | t > 0} = oo

(ii) If 9/(0) ^ then, from Theorem 2.4.1(h), we have that / is Isc at Suppose that / is Isc at Then /(0) = /(0) = 0, and so, by Theorem 2.3.4, /(0) = /**(0) = Assuming that <9/(0) = we obtain that /* = £9/(0) = +00, and so the contradiction /** = — 00 Therefore 9/(0) ?

(iii) Let x £ d o m / and e > If x* £ 9/(0) and (a;,a;*) > /(a;) — e then

WxeX : {x-x,x*) = {x,x*)-(x,x*} <f(x)-f(x)+e, i.e x* £ d£f(x) Conversely, let x* £ def(x); then the above inequality

holds Taking x = we get (af, a;*) > f(x) — e; taking now x :=x + ty, t > 0, y £ X, we obtain that

*(l/,**) < / ( i + ty) - f{x) +e< fix) + tfiy) - fix) + e = tf(y) + s Dividing by t > and letting then t -> 00, we obtain that (y,x*) > fiy) for every y € X, i.e x* £ 9/(0)

For e = we have

0/(3:) = {x* € 0/(0) I (a;,a:*) > fix)} = {x* £ 0/(0) | (x,x*) = fix)},

while for x = we obtain that

(iv) Suppose that / is lsc at Then /(0) = /(0) = Using Theorem 2.3.4, we have that /** = J Therefore

J(x) = sup{(a;,a;*) - f*(x*) | x* X*} = sup{(x,x*) \ x* € / ( ) }

(v) It is obvious that 0/(0) C dg(0) if / < g The converse implication follows immediately from (iv)

(vi) At the end of Section 2.3 we noted that SB is lsc, sublinear and

sB = {LB)*'• From the obvious inclusion B C 8SB(0) we get coB c <9SB(0)

(because <9SB(0) is convex and w*-closed) Let x* fi coB Using Theorem 1.1.5 in the space (X*,w*) and taking into account that (X*,w*)* — X, there exist x E X and A s K such that

Vx* G coB D B : (x,x*) > A > (x,x*),

whence (x,x*) > ssi'x), i-e x* £ 8SB(0). The conclusion follows

(vii) By Theorem 2.4.2 we have that A*(dh(0) C d(h o A)(0) Let

x* i w*-c\ (A*(dh(0))) Using Theorem 1.1.5, there exist x € Y and A e R

such that (x,x*) > A > (x,Ay*) — {Ax,y*) for every y* e dh(Q) From (iv) we get A > h(Ax), and so x* $ d(h o A)(0)

(viii) Let H : X x X -> , H(x,y) := f(x)+g(y), a n d T : I - > X x X ,

Tx := (x,x) It is obvious that H is sublinear and lsc, and T is continuous

and linear; moreover, dH(0,0) = 5/(0) x dg{0) and T*(x*,y*) = x* + y* By (vii) we obtain that

d(f + g)(0) = d(H o T)(0) = «>*-cl (T*(dH(0,0))) = w*- cl(5/(0) + dg(0))

The proof is complete • Using the preceding result we have

C o r o l l a r y 2.4.15 Let f G \(X) and x G d o m / Then df(x) ^ if and

only if f'(x, •) is lsc at In this case

f'{x,-)(u) = sup{(u,x*) | Z* G df(x)} V u e l

Proof The conclusion follows from the formula df(x) — df'(x, -)(0) (see Theorem 2.4.4) and assertions (i) and (iv) of the preceding theorem •

Note that, even for X = E2 and / G T(X) subdifferentiable at x,

C o r o l l a r y 2.4.16 Let (X, || • ||) be a normed space, f : X -> R, f(x) :=

\\x\\, andx G X Then, denoting Ux* by U*, we have:

r = iu; df(0) = ir, df(x) = {x*eU*\(x,x*) = \\x\\}

Proof The above formulas follow immediately from Theorem 2.4.14 • Another example of sublinear function is given in the following result

Corollary 2.4.17 Let / : En ->• E, f(x) := xi V • • • V xn, and x e W1,

where n G N Then

5/(0) = An, / * = I A „ , def(x) = {y e An\xiyi + +xnyn> f(x)-e},

where

A » : = { ( A i , , An) "B I A i > , Ai + + A„ = 1}

Proof It is obvious that / is sublinear and continuous Moreover

y G <9/(0) » V x G C : xlVl + ••• + xnyn < n V • • • V xn

Taking Xj = for j ^ i and Xi = —1, we obtain that — j/» < 0, i.e yi>0 for every i Taking then X{ — t G K for every i, we obtain that £(j/H \-yn) < t

for every i, whence y±-\ \-yn = l Therefore 9/(0) C An The converse

inclusion is immediate The other relations follow directly from Theorem

2.4.14 • The following theorem furnishes a formula for the subdifferential of a

supremum of convex functions Other formulas will be given in Section 2.8

Theorem 2.4.18 (loffe-Tikhomirov) Let (A,T) be a separated compact

topological space and fa : X —> E be a convex function for every a G A

Consider the function f := supaGAfa andF(x) := {a G A | fa{x) = f(x)}

Assume that the mapping A a / „ ( i ) G E is upper semicontinuous and XQ G dom / is such that fa is continuous at XQ for every a £ A Then

df(x0)=co(\J dfa(x0)) (2.49)

Proof Since A is compact and a H-> fa{x) is upper semicontinuous we

—oo for all a £ A = F(x0), and so df(x0) = dfa(x0) — 0; the conclusion

holds We assume now that f(x0) £ R

Let us note first that in our conditions x0 £ (dom / ) ' Indeed, let x £ X

Consider a fixed 70 > f(xo) (> fa(xo))- Because fa is continuous at xo,

for every a £ A there exists ta > such that fa(xo + tax) < 70 Since

/? t-> /^(^o + tax) is upper semicontinuous, the set Aa := {/3 £ A \ fp(x0 +

£Q:r) < 70} is an open set containing a As A = UaeA-^<*> there exist

a i , , an £ A such that A = \J™=1 Aai Let t := m i n { £a i, , iQn } >

It follows that fa(x0 + tx) < 70 for every a £ A, and so f(xo + tx) < 70

Hence x0 £ ( d o m / ) \

Since fa is continuous at xo, dfa(xo) 7^ for every a £ F(xo), and so

Q ^ 0, where Q is the set on the right-hand side of Eq (2.49) Suppose that

there exists x* £ df(x0) \ Q- Using Theorem 1.1.5 in the space (X*,w*),

there exist x £ X and e > such that

V a e F ( i o ) , V i ' e /a( i0) : (x,x*) - e > (x,x*), (2.50)

or equivalently, by Theorem 2.4.9,

V a e F ( i „ ) : (x,x*)-e>{fa)'{x0;x) (2.51)

Because x0 £ (dom/)1, we may suppose that x0+x £ d o m / Let t £]0,1[;

then xo + tx £ dom / There exists at £ A such that fat (XQ + tx) —

f(x0 + tx) Because (1 - t)fa,(x0) + tfat(x0 + x) > fat(x0 + tx) and

x* £ df(x0), we obtain that

(1 - t)fat (Xo) > f(x0 + tx) - tfat (x0 + X) > f(x0 + tx) - tf(x0 + x) > f(xo) - t (x,x*) - tf(x0 + x)

It follows that limt^o fat(xo) = f(xo) The space (A,T) being compact,

there exists a convergent subnet (a^^j^j of (at)t e]0,i[ converging to ao £

A; hence fao(x0) = f(x0), i.e a0 £ F(x0)- Let s e]0,1[ be fixed (for the

moment) and take t £ ]0, s] Using Eq (2.50) we obtain that

fat(xo +SX) - fat(x0) fqt(x0 + tx) - fgt{x0) f(x0 + tx) - f (x0) s - t - t

> (x,x*)

j y js and using the upper semi-continuity hypothesis, we obtain that V a € ] >i[ : Ui^ + ^-Ui^)

s

and so (jao)'{xo;x) > (x,x*), contradicting Eq (2.51) D

Of course, for conjugates and for the e-subdifferentials it is desirable to dispose of numerous formulas There exist effectively a large set of such formulas we shall establish in Section 2.8

2.5 The General Problem of Convex Programming

By problem of convex programming we mean the problem of

mini-mizing a convex function / : X —> R, called objective function (or cost

function) on a convex set C C X called the set of admissible solutions,

or set of constraints We shall denote such a problem by

(P) f(x), xeC

Of course, to consider this problem, X must be a linear space, however most of the results will be obtained in the framework of separated locally convex spaces or even normed spaces

To problem (P) we can associate a problem (apparently) without con-straints:

(P) f(x), x G X,

where / := / +

ve-in order for the problem (P) to be nontrivial, it is natural to assume

that C n dom / ^ (<£>• dom / ^ 0) and that / does not take the value — oo on C (i.e f does not take the value — oo)

We call value of problem (P) the extended real

v(P) := v(f,C) := m£{f(x) | x G C} £ I ;

we call (optimal) solution of problem (P) an element x G C with the property that f(x) = v(P); this means that x is a (global) minimum point for the function / We denote by S(P) or S(f, C) the set of optimal solu-tions of problem (P) Therefore

S(P) = {x G C | Vx G C : f(x) < f(x)}

if C fl dom / ^ The set S(f, X) is denoted also by argmin /

Of course, an important problem is that of the existence of solutions for (P), resp (P) The most important result which assures the existence of solutions for (P) is the famous Weierstrass' theorem Because the underly-ing spaces are not compact we have to use some coercivity conditions We say that / : X —> E is coercive if limn^n^oo f(x) — oo It is obvious that / is coercive if and only if all the level sets [/ < A] are bounded (see also Exercise 1.15); when / is convex then / is coercive if and only if the level set [/ < A] is bounded for some A > inf / (see Exercise 2.41) We have the following result

Theorem 2.5.1 Let f £ T(X)

(i) If there exists A > v(f,X) such that [f < A] is w-compact, then s{f,x)^d>

(ii) If X is a reflexive Banach space and f is coercive then S(f,X) ^ ill Proof, (i) Of course, v(f,X) = v(f, [f < A]) Since / is lsc and convex, / is tu-lsc The conclusion follows using the Weierstrass theorem applied to

the function / | [ / < A ]

-(ii) Because / is coercive (see Exercise 1.15), [/ < A] is bounded for every A e E Since [/ < A] is w-closed and X is reflexive, we have that [/ < -M is w-compact for every A G E The conclusion follows from (i) •

Of course, in the preceding theorem, the condition that / is convex can be replaced by the fact that / is quasi-convex The next result shows that the reflexivity of the space X is almost necessary in Theorem 2.5.1

Theorem 2.5.2 Let (X, ||-||) be a Banach space Assume that there exists

a proper function f : X —> M satisfying the following conditions:

(i) [/ < f(x)] l'5 closed, convex and bounded for every x € dom / ;

(ii) / attains its infimum on every nonempty closed convex subset of

X;

(iii) there exist xo,xi £ d o m / and r > such that f is Lipschitz on

[f < /On)] and [f < f(x0)] + rUx C [/ < / f a ) ] , where Ux := {x € X |

IMI <

1}-Then X is reflexive

Proof Of course, f(x0) < f(xi); if f(x0) = f{xi) then the relation

f(xi)] Hence f(x0) < f(xi) Since / is Lipschitz on [/ < f(x\)], there

exists L > such that \f(x) - f(x')\ < L\\x-x'\\ for all x,x' £ [f <

f(x\)] Since /|[/</(xi)] is continuous, there exists X2 £ [a;0,a;i] such

that f(x2) = {f(x0)+f(xi))/2 The set S := [f < f(x2)\ is convex,

bounded with nonempty interior Indeed, D(XO,SQ) C S, where So := min{r, (f(Xl) - f{x0)) /(2L)}; moreover S + 80UX C [/ < /(a*)]

Consider A := — 5; A is a bounded, convex and symmetric set with nonempty interior, and so € int A Therefore there exist a, /? > such that allx C A c /SC/x- It follows that a- 11 | - | | = paux > PA > Ppux =

P~x ||-||, where p ^ is the Minkowski gauge associated to A By Theorem

1.1.1 PA is a semi-norm, and so it is a norm equivalent to ||-||; moreover, by Proposition 1.1.1, the unit closed ball with respect to PA is cl A To obtain that X is reflexive, by the famous James' theorem (see [Diestel (1975), Th 1.6]), it is sufficient to show that every x* € X* attains its infimum on A Let x* £ X* \ {0} and a* := inf{(x,x*) \ x £ S} Because S is bounded, a* € E Let H := {x £ X \ (x,x*) = a*} It is obvious that

(S+eUx)nH # for every e > Taking e ]0, S0] and xe £ (S+eUx)r\H,

we obtain that f(xe) < f{x2) + eL, and so mixen f(x) < f(x2) By (ii)

there exists xi £ H such that f(x~i) = infx €ij f(x), and so x\ £ Sf)H

Therefore (xi,x*) < (x,x*) for every x £ S Similarly, there exists x2 £ S

such that (x2,x*) > (x,x*) for every x £ S, whence there exists x :=

Xi —x2 £ A such that (x,x*) < (x,x*) for every x £ A •

Also the coercivity condition in Theorem 2.5.1(h) is essential as the next theorem will show In order to establish it we introduce some preliminary notations and results Let / T(X) and denote by 11/ the set

{g £ T(X) | domg = d o m / , supx€domf \f(x) - g(x)\ < oo, inf / = inf g}

Since for g £ Hf we have that / — < / < / + f o r some £ M+, we have that /oo = g^ and / , g are simultaneously coercive or not coercive Consider

d : Ilf x Ilf ^ M+, d{gug2) := s u px d o m / \f{x) - g{x)\

Lemma 2.5.3 The mapping d is a metric on 11/ and (11/, d) is a complete

metric space

Proof It is obvious that d is a metric Let (gn)n>i C 11/ be a Cauchy

g(x) E R for every x G d o m / Moreover, ( s u px e d o m / \gn(x) - g(x)\)n^,1 ->

0 We extend g to X setting g(x) := oo for x G X \ dom / ; hence dom g — dom / Because (gn(x)) —> g(x) for every a; G X , from Theorem 2.1.3(h) we

have that g is convex Let e > 0; then there exists nE such that <7„(a;) — s <

g(x) < 9n(x) + £ for all n > nE and a; G d o m / It follows that g is proper

(being minorized by gnc — e even on X) and inf / — e = inf gn — e < inf <? <

inf / + e = inf gn + e for every e > which shows that inf g = inf / (even

if inf/ = - c o ) We must only show that g is lsc Take first x G domg = d o m / and A < g{x) There exists e > such that A < g(x) — 2e As above, there exists n = ne such that g(y) — e < gn(y) < g(y) + £ for every

y G d o m / In particular g(x) — e < gn(x) Because gn is lsc at x, there

exists U G Mx{x) such that <?(a;) — e < gn(y) for every y E U It follows

that g(x)—e < g{y) + e for every y G E/ndom/, and so A < g(x)—2e < g(y) for y G U D dom / As <?(y) = oo for y G f/ \ dom / , we have that A < g(y) for y G U, and so # is lsc at x Consider now x G X \ d o i n g and take A G M Let n G N be such that g(y) — < ff„(?/) < g{y) + for every y G d o m / Because gn is lsc at a; and A + < gn(x) = oo, there exists C/ G Mx{x) such

that A + < gn(y) for every j/ G U It follows that A < gn(y) - < <?(y)

for all y G t/ D d o m / Since A < g(y) = oo tor y E U \ domg, we have that

A < g(y) for y E U, and so g is lsc at x Hence g Elif • L e m m a 2.5.4 i e i / G T(X) 6e bounded from below and K := {u E X \

foo(u) < 0} Consider e > andy E X Then the se£co(epi/U{(y,inf / —

e)}) is i/te epigraph of a function fyi£ E T(X) with inf fViS = inf / — e and

argmin /y>£ = y + K Moreover, if f{y) < inf / + e then f — 2e < fy<£ < f,

and so g := /y i £ + e G 11/ and d(f,g) < e

Proof Consider a := inf / — e < inf / Denote A := co (epi / U {(y, a)})

and let (x, t) E A and s > t We want to show that t > a and (a;, s) E A If these happen then epiipA = ^4 [VA being defined in Theorem 2.1.3(iv)],

fy,e := VA £ r( X ) , /y,£(y) = a = inf /j,) £ and fy>e < / Moreover, if

f(y) < inf / + e, then f(y) — 2e < a, which shows that (y, a) E epi(/ — 2e)

As / — 2e < f, we obtain that A C epi(/ - 2e), and so / - 2e < /y ) £

Indeed, there exist the nets (Ai)igj C [0,1] and ((xi,ti))i€l C e p i / such

that (a;,*) = lim,e/ (\i(xi,U) + (1 - Aj)(2/,a)), and so x - limi 6/ (XiXi +

(1 — Aj)y) and t = limjg/ (A;£; + (1 — Aj)a) Since [0,1] is a compact set, we may suppose that (A;)i e / -> A G [0,1] Of course, A;£; + (1 - Xi)a >

where (xo,to) is a fixed element of e p i / In the contrary case Aj > for

i >z_ j0 for some i0 £ I Then limi^io (Xi(xi,ti+X^1(s-t))+ (1-Xi)(y,a)) =

(x, s), and so (a;, s) £ A

Let u £ K; then (u,0) £ (epi/)oo- Fixing (xo,io) £ ep i / , we have that

(x0 + nu,io) £ e p i / for every n € N, and so ^(zo + nu, to) + ^^{y, a) £ A

for every n £ N Taking the limit we obtain that (y + u, a) £ A, whence

fy<s(y + u) < a Hence y + uG argmin/y>£ Conversely, let z € argmin/y,£

Assume that u := z - y ^ It follows that (z,a) € A, and so there exist the nets (A;)j€/ C [0,1] and ((xi,ti))ieI C e p i / such that (2,a) =

limi 6/ (Aj(x;,£;) + (1 - Xi)(y,a)) Because z ^ y, A» > for i y i0 (for

some io € / ) • As above, we may assume that (A;)j6/ ->• A € [0,1] If

A ^ then ((xj,ii))i g / ->• (A-1.z + (1 - X~1)y,a), a contradiction because

ti > inf/ > a Hence A = It follows that limjg/ A; {xi,t{) = (u,0)

Let (a;,i) £ e p i / Then Aj (xi,U) + (1 - A,)(a;,i) € e p i / , and so, taking the limit, we obtain that (x, t) + (u, 0) € epi / This shows that (u, 0) £ rec(epi/) = (epi/)oo- Therefore u £ K Hence argmin/y]E = y + K •

Theorem 2.5.5 Let X be a reflexive Banach space and f £ T(X) be such

that K n -K - {0}, where K := {u £ X | /<»(«) < 0} Then the following statements are equivalent:

(i) / is not coercive,

(ii) there exists g £Uf such that argming = 0,

(iii) {g &Uf \ argming = 0} is a dense G$ set (see page 34)-Proof It is obvious that (iii) => (ii)

(ii) =>• (i) Let g SUfbe such that argming = From Theorem 2.5.1(h) we have that g is not coercive Since g £ Uf, there exists > such that

9 — < / < + 7> which implies that / is not coercive, too

(i) =*- (iii) First of all note that for any g £ Uf, g is not coercive and /oo = Soo (because / - < P < / + 7for some € M+)

If inf/ = - 0 then Q := {g £ Uf \ argming = 0} = Uf, and so the conclusion holds

Let inf / £ R Without loss of generality we assume that inf / = For every n £ N consider the set

Gn •= {g e Uf I minx e n C/x g(x) > inf g = 0}

0 < < mixenUx g(x); then the ball B(g,8) C Qn As Q = f]n€^Gn, it

is sufficient to show that Qn is dense for any n > For this fix n > 1,

e > and /i G 11/ We may assume that /i(0) < e (otherwise replace /i by h(xo + •)> where xo is taken such that h(xo) < e)

There exists y G X such that h(y) < e and (y + if) n nUx = 0- Indeed, when K = {0} take y € X \ nUx such that h(y) < e; this is possible because the set [h < e] is not bounded (see Exercise 2.41) Assume now that

K ^ {0} Then there exist z G K and r > such that (z + if) n rUx = 0,

or equivalently z ^ r[/x — K Otherwise K C flr>o (rUx — K) C —K,

a contradiction The last inclusion is obtained as follows: consider z e Hr>o (rUx - K)\ then z = ^un — kn with un e Ux and kn € if, for every

n € N, and so (fc„) ->• —z € if

Consider g := /iy]£ +e By Lemma 2.5.4 we have that g G 11/, d(/i, g) <e

and argmin/i = y + if Since (2/ + if) D nUx = 0, we have that g G Qn

-Therefore Qn is dense in 11/ •

Note that the reflexivity of the space was used in the proof of the pre-ceding theorem only to ensure that the infimum of g on nUx is attained; so the preceding result remain valid when working on the dual of a normed space, the considered convex functions being w*-lsc Also note that the condition Xo := if fl—if = {0} in the preceding theorem is not essential; if

Xo 7^ {0} one obtains a similar result taking into account the constructions

in Exercise 2.24 For a similar result when X is not a normed space see Exercise 2.25

Another important problem in optimization theory is the uniqueness of the solution when it exists The following result gives an answer to this problem

Proposition 2.5.6 Let f G A(X) Then S(f,X) is a convex set

Fur-thermore, if f is strictly convex then S(f, X) has at most one element

Proof Let x G S(f,X); then S(f,X) = [f < f(x)], whence S(f,X) is convex

Let now / be strictly convex, and suppose that S(f,X) contains (at least) two distinct elements xi and xi\ since / is proper, S(f,X) C d o m / Then we obtain the contradiction

v(f,X) < f (\Xl + |x2) < i/On) + \f[x2) = v(f,X)

As we already know, the practical method for determining extremum points of a function is to determine the points which verify the necessary conditions then to retain those which verify the sufficient conditions In convex programming we have a very simple necessary and sufficient condi-tion for optimal solucondi-tions

Theorem 2.5.7 If f £ A(X), then x E dom / is a minimum point for f

if and only if E df(x)

Proof Indeed, f(x) < f(x) for every x E X if and only if x £ dom / and

0 < f(x) — f(x) for every x £ X, which means that € df(x) O

Therefore, in convex programming, the minimum necessary condition is also a sufficient condition

In optimization theory local optimal solutions also play an important role; if g : X —>• E, we say that x £ X is a local minimum (resp local

maximum) point if there exists V € Nx{x) such that f(x) < f(x) (resp f(x) > f(x)) for every x £ V The convex programming problems present

a particularity

Proposition 2.5.8 Let f : X —• E be a convex function

(i) Ifx(z dom f is a local minimum point for f, then x is also a global

minimum point;

(ii) if x € dom / is a local maximum point for f, then x is a global

minimum point for f

Proof, (i) By hypothesis there exists V £ Nx(x) such that f(x) < f(x) for every x € V Suppose that there exists x £ X such that f(x) < f(x); therefore f(x) £ E Since V £ J^x(x), there exists A £]0,1[ such that

y := (1 - X)x + Xx £ V Therefore

/(5?) < f(v) = / ( ( I - A)3F + Ax) < (1 - A)/(3F) + Xf(x),

whence the contradiction f(x) < f(x) Therefore x is a global minimum point for /

(ii) By hypothesis there exists U £ J^cx such that f(x + x) < f(x) for

every x £ U If f(x) = — oo, it is clear that x is a global minimum point of / Suppose that f(x) £ E (hence / is proper because x € int(dom/)) Then

Therefore f(x) = f(x) > f(x) for every x € x + U € N j ( x ) Hence a; is a

global minimum point of / D

This result explains why in a convex programming problem we look only for global minimum points

In practical problems, solved numerically on computers, frequently it is not possible to determine the exact optimal solutions (because one works with approximate values) A simple example in this sense is the problem of minimizing the function / : E —» R, f{t) := (t — 7r)2 Taking into account

this fact, the notion of approximate solution is proved to be useful More precisely, if e e R+, we say that x £ C is an e- (optimal) solution of problem (P) if f(x) < f(x) + e for every x C; we denote by SS(P) or

Ss(f,C) the set of e-solutions of problem (P) It is obvious that when

C fl dom / / we have that Se(f, C) = Se(f, X); moreover, if / is proper

and S£{f,C) ^ 0, then v(f,C) G R and Se(f,C) = {x £ C | f(x) <

v(f, C) + e} Related to e-solutions we have the following result

Proposition 2.5.9 Let f £ A(X), x e d o m / and £ € P Then Se(f,X)

is convex; the set Se(f,X) is nonempty if f is bounded from below

Fur-thermore, x € Sc(f,X) if and only if d£f(x) •

2.6 Perturbed Problems

We shall see (especially) in the following two sections that it is very useful to embed a minimization problem

(P) f(x), x X,

in a family of minimization problems

In this section X, Y are separated locally convex spaces if not stated explicitly otherwise and / : X —> R

Let us consider a function $ : I x F l having the property that

f(x) — $>(x, 0) for every x € X; $ is called a perturbation function For

every y £ Y consider the problem

(Py) $(x,y), x X

It is obvious that problem (P) coincides with problem (Po)

obtaining useful results one chooses adequate perturbation functions, as we shall see in the sequel

Let $ : I x l b e a convex function and h : Y -» E, h{y) =

v(Py), its associated marginal function (see page 43); h is also called the

value or performance function associated to problems (Py) As noted in

Theorem 2.1.3(v), h is convex, while from Eq (2.8) we have that dom/i = P r y ( d o m $ )

The problem

(P) $ ( z , ) , x £ X,

is called the primal problem; we associate to it, in a natural way, the following dual problem

{D) max (-$*(0,2/*)), y* € Y*

It is obvious that (£>) is equivalent to the convex programming problem (!?') $*(0,2/*), y* eY*

The equivalence has to be understood in the sense that the problems (D) and (£>') have the same (e-)solutions; moreover v(D') = —v(D) (of course, for a maximization problem the notions of (e-)solution, local solution and value are defined dually to those for minimization problems) It is nice to observe that (£>') and (P) are of the same type In the following results we establish some properties which connect the problems (P), (D) and the function h

Theorem 2.6.1 Let $ : X x Y -> E and h : Y ->• E be the marginal

function associated to $ Then:

(i) h* (y*) = $* (0, y*) for every y* eY*

(ii) Let (x,y) € X x Y be such that ${x,y) R Then (0,j/*) € d$(x,y) O h(y) = $(x,y) and y* £ dh{y)

(hi) v(P) = h(0) and v(D) = /i**(0) Therefore v(P) > v(D); in this

case we say that one has weak duality

(iv) Suppose that $ is proper, x € X and y* € Y* Then (0,y*) €

d$(x~, 0) «/ and only if x is a solution of problem (P), y* is a solution of (D) and v(P) = v{D) G E

Assume, moreover, that $ is convex

(v) /i(0) £ E and /i is Isc at <£> u(P) = u(D) £ E; in this case one

(vi) h(0) G E and dh(0) / & v(P) = v(D) G E and (D) has optimal

solutions In this situation S(D) = dh(0)

(vii) Suppose that $ is proper Then [ h is proper] •£> [ h* is proper] •£> [ h is minorized by an affine continuous functional] •&

3y* ' , a GE, V(x,y)eXxY : $(x,y) > (y,y*) + a (2.52) Proof, (i) We have

h*(y*) = sup({y,y*) - h{y)) = sup ((y,y*) - inf $(x,y))

y€Y y€Y \ ^ ^ /

= sup sup((y,y*) -$(x,y)) = sup ((x,0) + (y,y*) - ${x,y))

y€YxeX (x,y)€XxY

= **(<), y*)

(ii) Assume that $(x,y) G E Let (0,j/*) G d$(x,y) Then by (i) and Theorem 2.4.2 (iii),

h(y) < *(x,y) = (x,0) + (y,y*) - $*(0,i/*) = (y,y*) - h*(y*) < h(y),

and so h(y) = $(x,y) and y* G dh(y) Conversely, if these two conditions hold then

*{x,y) = h(y) = {y,y*) - h*{y*) = (x,0) + (y,y*) - * * ( , y ' ) ,

whence, again by Theorem 2.4.2 (iii), (0,2/*) G d$(x,y) (iii) It is obvious that v(P) — /i(0); moreover,

v(D) = sup (-$(0,r/*)) = sup ((0,2/*) -h*(y*)) = h**{0)

y*EY* y*&Y*

Therefore v(P) >v(D) (iv) If (0,|7*) G<9$(x,0) then

v(P) = h(Q) < $(x,0) = -$*(0,2/*) < v(D) = x**(0) < h(0),

whence x is a solution of (P), y* is a solution of (£>) and v(P) = v(D) G E Conversely, if these last assertions are true, we have

-$*(0,j7*) = h**(0) = h(Q) = S(z,0) G E, whence

which shows that (0,y*) G d$(x,0) Assume now that $ is convex

(v) If h is lsc at and h(0) G R we have that h(0) G E, whence h** = ~h Therefore v(D) = h**(0) = ft(0) = v{P) G E Conversely, if this last relation is true, then h(0) = h(0), i.e h is lsc at The conclusion holds

(vi) Suppose that /i(0) G E and dh(0) ^ 0; then /i is lsc at 0, whence, by (iv), we have that v(P) = v(D) G E Let y* G dh(0) Then h(0) + h*(y*) = 0, whence

Vy* G Y* : v(D) = v(P) = h(Q) = -h*(y*) = -**(<), IT) > -*'{0,y*)

Therefore ^ dh(0) C S(D) Conversely, suppose that v(P) = v(D) G R and that (D) has solutions Let y* G S(D) Then ft(0) = h**(0) =

-h*(y*) G E, whence y* G 9ft(0) Thus we have that # S(D) C 0ft(O)

(vii) The mentioned equivalences are obvious since h is convex, and

d o m / i ^ •

We say that the problem (P) is normal if v(P) = v(D) G E; (P) is

stable if w(P) = v(D) G R and (D) has optimal solutions Theorem 2.6.1

above gives characterizations of these notions

In the following result we establish formulas for deh(y) and deh*(y*)

when h is proper

Theorem 2.6.2 Let $ G A(X x Y) satisfy condition Eq (2.52) and

e G E+ Then:

(i) /or every y G dom h

deh(y) = f l ,> 0 LUjcfo* e y* I (°'^) G ^ + ^ ^ ) }

7)>0 * ( x , y ) < f t ( y ) + J

(ii) /or every y* G dom/i*;

Proof, (i) It is obvious that

n n {»*i(o,!/,)6flt+,^»)}

V>0 $(x,y)<h(y)+T]

c n „ U( => * l ( o , i / * ) e ae + ) J$ (a: , i / ) }

1 lri>0 ^^x€X

C ae/i(y)

Let y* G deh(y), i.e h(y) + h*{y*) < (y,y*) +e, and let rj > and a; € X

be such that $(x,y) < h(y) + rj Then

$(x, 2/) + $*(0, y*) < ft(y) + r, + h*{y*) < (y, y") + e + r,,

i.e (0,2/*) € d£+n$(x,y) Therefore

dsh(y) C f | f | {y*\ (0,y*) G &+„*(*, y)},

V>0 <b(x,y)<h{y)+ri

and so the desired equalities hold (ii) Let y € d£h*(y*) and r\ > Then

/>**(</) + W ) = % ) + h*(y*) < (y,y*) + e

It follows that for every V G N(j/) there exists yv G V such that

M l / v ) + &*(!/*) < <yv,l/*> + £ + ??•

From the definition of h we get xy E X such that

$(xv,yv) + h*(y*) = $(xv,yv) + $*(0,y*) < (yv,y*) +e + rj,

i.e (0,y*) G de+n<b(xv,yv)- Therefore

j / G c l { t / G y | a ; G X : (0,y*) G 3£ + ?$(z,y)}

Taking into account that 77 > is arbitrary, we obtain that

d£h*(y*)cf) d{y\3xGX : (0,y') G de+r,*(x,y)}

(xv,Vv) € X x Y such that yv GVTlVo and (0,y*) G de+r,/2$(xv,yv); hence

Kw) + h*{y*) < ${xv,yv) + $*(0,y*) < (yv,y*)+e + i]/2 < (y,v*)+e+T]

Therefore

V V e N f e ) : inf % ) + A*(y*)< (J/,!/*>+£ + »/, yev

i.e h(y) + h*(y*) < (y,y*) +E + TJ Since TJ > is arbitrary, we obtain that h"(y) + h*(y*) = h(y) + h*(y*) < (y,y*) + e,

i.e.ytd£h*(y*) •

The preceding result is useful for deriving formulas for e-subdifferentials for other functions

Theorem 2.6.3 Let $ E T(X x Y) and cp : X -» E, ip(x) := F(x,0)

Then for every e E E+ and every x E dom ip we have

dMx)=C\ w*-c\{x*eX*\3y*eY* : (x*,y*) E d£+v$(x,0)}

= f | w*-d{x*eX*\3y*eY* : (x,0) E de+v^(x*,y*)}

Proof Let us consider the spaces X*, Y* endowed with their weak* topologies and the function

k:X*^W, k(x*):= inf $*(x*,y*)

y*€Y*

Then k* : X ->• I , A;*(a;) = $**(x,0) = $(x,0) = ip(x) Using Theorem 2.6.2, we have

dMx) = dEk*(x) = f l w*-c\{x* | 3j/* : (a;,0) E &+-,**(**>!/*)}

= f | W * - C { Z * | T / * : ( x ' y ' J e f t + ^ O ) }

1 '77 X ) >J1>0

The proof is complete D Let us apply the results of Theorems 2.6.2 and 2.6.3 in some particular

C o r o l l a r y 2.6.4 Let A G H{X,Y) and f : X —> R be a convex function

for which

3y* £Y*, 3a G R, Vz e X : f(x)> (x,A*y*)+a If (Af)(y) G (O y G A(dom/) = d o m A / ) , j / * G dom(/* o A*) and e > 0, then

de{Af){y) = fl [J A-\de+rif{x)) n>0 Ax=y

= n n A*-\d£+rif(x)),

»7>0 Ax=y,f(x)<(Af)(y)+r,

and

ds(f*oA*)(y*) = f]d{yeY\3xeX : Ax = y, A*y* G de+nf(x)}

Proof Let us consider

*,y) •= |

K ,in ' oo if Axjty

Then $*(x*,t/*) = /*(x* + A V ) and

(O.y*) G 0„$(a;,y) <S> Ax = y, A*j/* G fy/fr)

The result follows immediately from Theorem 2.6.2 D C o r o l l a r y 2.6.5 Let A G L(X, Y) and f G T(Y) Then

d£(foA)(x) = O^lv'-dA^de+rffiAx))

for every x G A_ 1( d o m / ) = dom(/ o A) and every e >

Proof Let us consider $ : I x y ^ l , $(x,y) := f(Ax+y), and ip(x) :— $(x,0) = f(Ax) Then$*(x*,y*) = f(y*) ii A*y* =x*, $*(x*,y*) = oo

otherwise Moreover

(x*,y*)edv*(x,0)&A*y*=x' and f(Ax) + f*(y*) < (Ax,y*) + rj

& A*y* = x* and y* G dvf(Ax)

Corollary 2.6.6 Let / i , /2 € A(X) for which

3x* eX*, 3a e l , \/xeX, V i e {1,2} : fi{x) > {x,x*) + a If (hnf2)(x) e E and e > 0, then:

de(fiDf2){x) = f ) 1J (0e i/i(a; - y)ndeJ2(y))

n>0 yeX,Ei>0,e+n=ei+£2

= n n u (^x/i^-^n^/ad/)),

ri>0yeSrl(x) £i>0,e+r)=£i+£2

d{huf2){x) = r\v>0Uy€X (Wx - y)n ^/a(i/))

where Sv(x) := {y e X \ Mx - y) + f2{y) < {fiDf2)(x) + r?}

Proof Let us consider / : X x X -)• E, f{x\,x2) := fi(xi) + f2(x2)

and A e £ ( X x X , X ) , A(xi,x2) := £i + £2- The conclusion follows from

Corollary 2.6.4 •

Corollary 2.6.7 Let / i , /2 € T(X) / / a; e d o m / i n d o m /2 and e >

then:

ds{h+mi)=n ™*-d ( u (^Aw+aeaMx))

7)>0 y£i>0,£+77=ei+e2

5 ( / i + f2)(x) = f l t i T - d ^ / i C * ) + d„f2(x))

Proof Let us consider / : I x I - > I , / ( ^ i i x2) := / i ( x i ) + 72(^2) and

A e £ ( X , X x X), Ax := (x,x) The conclusion follows from Corollary

2.6.5 •

2.7 The Fundamental Duality Formula

The following theorem is very useful for obtaining important results in convex programming; this is the reason for calling formula (2.53) the

fun-damental duality formula of convex analysis

Theorem 2.7.1 Let $ e A(X x Y) be such that P r y ( d o m $ )

(i) there exists Ao £ E such that VQ := {y £ Y \ 3x £ X, $(x,y) <

AO}S:NVO(O);

(ii) there exist Ao G E and x0 £ X such that

VUeNx : {y£Y\3x£x0 + U, $(x,y) < Ao} e Kyo(0);

(iii) there exists XQ £ X such that (a;o;0) £ d o m $ and $(xo,-) is

con-tinuous at 0;

(iv) X and Y are metrizable, epi $ satisfies condition (Hwz) on page 14

and 0£ i 6( P ry( d o m $ ) ) ;

(v) X is a Frechet space, Y is metrizable, $ is a li-convex function and e i 6( P ry( d o m $ ) ) ;

(vi) X is a Frechet space, $ is Isc and £ j 6( P r y ( d o m $ ) ) ;

(vii) X, Y are Frechet spaces, $ is Isc and € l c( P r y ( d o m $ ) ) ;

(viii) dimlo < 00 and £ *(Pry(dom$));

(ix) there exists Xo £ X such that $(XQ,-) is quasi-continuous and the sets {0}, P r y ( d o m $ ) are united

Then either h(0) = —00 or h(0) £ E and h\y0 is continuous at In

both cases we have

inf $ ( x , ) = max ( - **(0,i/*)) (2.53)

x € X ' y*€Y' v " '

Furthermore, x £ X is a minimum point for $(-,0) if and only if there exists y* £ Y* such that (0,y*) £ d$(x,0)

Proof Since £ P r y ( d o m $ ) , h(0) < 00 If h(0) = —00 we have h*(y*) — °° f°r every y* Y*, whence —$*(0, j/*) = - 0 = h(0) for every

y* £ Y* Therefore the conclusion is true in this case Let us consider now

the case h(0) £ K

Suppose that condition (i) is verified Then h\y0 is bounded above by

Ao on VQ Since h is convex, h\y0 is continuous at By Theorem 2.4.12 we

have that dh(0) f Then relation (2.53) follows Theorem 2.6.1 (vi) (ii) =£- (i) This implication is obvious (just take U — X)

(iii) => (ii) Suppose that (iii) holds Since $(xo,-) is continuous at 0, the set V0 := {y | $(xo,y) < $(x0,0) + 1} is a neighborhood of

(in particular YQ = Y) Taking Ao := $(a;o,0) + 1, it is obvious that

V0 C {y \3x £ x0 + U : $(x,y) < A0} for every U £ Nx- Therefore (ii)

(iv) => (ii) Suppose that (iv) holds Let us consider the relation 31 :

X xR=$Y whose graph is given by

grtt := {(x,t,y) \ (x,y,t) € epi$}

From the hypothesis 51 satisfies condition (Hwi), whence, by Proposition 1.2.6(i), % satisfies condition (Hw(z,i)), too Moreover € i6(Im3?) Let

(xo,io) e X x R be such that Jl(xo,to)- Applying Theorem 1.3.5 we

obtain that "R{{x0 + U)x ] - oo,t0 + 1]) £ ^y0(0) for every U eNX- This

shows that (ii) holds with Ao := t0 +

(v) => (ii) By Proposition 2.2.18, there exist a Frechet space Z and a cs-closed function F : Z xX xY -^R such that $(x, y) = infz ez F(z, x, y)

for all (x, y) € X xY The conclusion follows like in the preceding case by replacing X by Z x X, x by (z,x) and U by Z x U

(vi) => (ii) The proof is the same as for (iv) =$• (ii) with the excep-tion that one uses Ursescu's theorem (Theorem 1.3.7) instead of Simons' theorem

It is obvious that (vii) implies conditions (iv), (v) and (vi) If (viii) is verified, the conclusion follows from Theorem 2.4.12

(ix) => (i) It is obvious that $(a;0, •) > h By Proposition 2.2.15 we have

that h is quasi-continuous It follows that rint(dom/i) ^ Using Proposi-tion 1.2.8 we obtain that £ rint(dom/i) Therefore h\y0 is continuous at

0, and so (i) holds

Of course, if x is a minimum point for $(-,0) then x is a solution of (P) (from p 107); it follows that $(x,0) = v(P) = v{D) G R Let y* be a solution of (D) (in our conditions a solution exists) Then, from Theorem 2.6.1(iv), (0,2/*) £ d$(x,0) The converse implication follows from the

same result •

When the properness condition in the preceding theorem is violated relation (2.53) is automatically verified Indeed, in this case there exists

(xi,yi) £ X x Y such that $(xi,j/i) = - o o , and so h(yi) = —oo Because

0 € '(dornh), by Proposition 2.2.5 we have that h(0) — - o o

Other conditions of this type are:

3A0 G R, 3B G %x : {y G Y | 3x G B, $(x,y) < A0} G >JVo(0), (2.54)

V?7 G Kx, 3A > : {yeY\3x£XU, $(x,y) < A} G Nyo(0), (2.55)

where "Bx is the class of bounded subsets of X and, as in Theorem 2.7.1,

Y0 = l i n ( P ry( d o m $ ) )

Proposition 2.7.2 Let $ G A(X x F ) Conditions (i) — (viii) foez'ng

those from Theorem 2.7.1, we have: (iii) =>ã (2.54) =S> (2.55) ôã (ii) => (i),

(2.55) => (2.54) «/ X is a normed vector space, (vii) =>• (iv) A (v) A (vi), (iv) V (v) V (vi) =* (ii) and (viii) =* (2.54)

Moreover, taking D = P r y ( d o m $ ) , one has: if dim(linZ)) < oo t/ien

*D = rint£>; i/ X, Y are metrizable, e p i $ satisfies H(x) and lbD ^

then thD = rintZ?; similarly for the situations corresponding to conditions

( v ) - ( v i i )

Proof The implications (vii) =>• (iv) A (v) A (vi), (iv) V (v) V (vi) =>• (ii) =*• (i) were already observed (or proved) during the proof of the preceding theorem

The implication (iii) =$• (2.54) is obvious; just take B — {XQ}

(2.54) =>• (2.55) Let U G Kx; there exists /x > such that B C fill

Taking A = max{Ao,/i} we have that

{yeY\3x£B, $(x,y) < A0} C {y G Y \ 3x G ^?7, $(a;,j/) < A0}

C f e e Y | x G XU, 9(x,y) < A} The conclusion follows

(ii) =>• (2.55) Consider A0 G M and zo G X given by (ii) Let U G N *

There exists fi > such that a;0 € /if/ Let V = {y€Y\3x£x0 +

U, $(x,y) < Ao} G Ny0 Taking A = max{Ao,/« + 1} and y € V, there

exists x G x0 + U such that $(x,y) < A0 As x G x0 + U C /J,U + U =

(fi + 1)U C At/, the conclusion follows

(2.55) =>• (ii) It is obvious that there exists x0 G X such that $(xo, 0) <

oo Consider Ao = max{$(a;o,0),0} + and let U G Nx- There exists

Uo G Nx such that UQ + Uo C U There exists also Ai > such that XQ G Aif/0 By hypothesis, there exists A > Ao + Ai such that Vb = {y G

y | 3a; G AE/0, #(a;,i/) < A} G Nyo(0) Let V = X^VQ and take yeV As

Xy G Vo> there exists x' G Af/o such that $(x',Xy) < X It follows that

whence $ (x,y) < Ao, where x := xo + X~1(xl — Xo) G Xo + Uo + Uo C Xo + U

(2.55) => (2.54) when X is a normed vector space Take U = Ux = {x G X | ||x|| < 1} There exists A0 > such that {y G Y \ 3x G X0U, ${x,y) <

Ao} € Nyo(0) As XoU is bounded, the conclusion follows

(viii) =$• (2.54) Suppose that dim Yb < oo and £ i( P ry( d o m $ ) ) It

follows that there exist j / i , ,ym G P r y ( d o m $ ) such that Vo = co{y\, ,

2/m} £ !Nyo(0) For every i € l , m there exists Zj G X such that (xi,yi) G

d o m $ Let Ao = max{$(a;j,y;) | < i < m} and B — co{xi, , xm}

It is obvious that B is bounded and for y G Vo there exist A j , , Am >

with Y!T=i ^i = 1 s uch that j / = YlT=i ^'Vi- Then a; = YllLi ^ixi € -^ anc^

*(z,2/) < A0

The fact that lD — v'mtD if dim(lin£>) < oo is obvious Let X, Y be

metrizable, e p i $ satisfy (Hx), and consider y0 G lbD Taking <J?0 defined

by <b0(x,y) = $(x,y + y0), we have that G j 6Pry(dom$o)- It is easy to

show that $o verifies condition (iv) of Theorem 2.7.1 As remarked above, condition (ii) holds for $o, which implies that G rint (Pry(dom<i?o)), »-e

2/o € rintZ) Similarly one obtains the other relations •

Corollary 2.7.3 Let $ G T(X xY) If one of the conditions (ii)—(ix) of

Theorem 2.7.1, (2.54) or (2.55) holds, then

($(•,0))* (&") = $*(£*,</*) =:i/<(z*)

y*£Y'

for every x* G X* In particular ip G T*(X*)

Proof It is obvious that $(-,0) G r ( X ) in our conditions Let x* G X* and consider $ : I x F - > I denned by $(x,y) := $(x,y) — (x,x*) As observed above, the function $ satisfies the same condition as $ among those mentioned in the statement of the corollary Applying Theorem 2.7.1

(and eventually the preceding proposition), we have that inix&x $(x,Q) =

m a xy.ey ( - $ * ( , ?/*)) But $*(0,2/*) = $*(x*,y*), and so the conclusion

follows • We state another duality formula which will be useful in the sequel

Theorem 2.7.4 Let F G A(X x Y), : X =t Y be a convex

multi-function, and D = U{G(x) — y \ (x,y) G d o m F } Assume that G D and let Y0 = linD If one of the following conditions holds:

(i) for every U G Nx there exist A > and V G Ny0 such that

(ii) there exist Ao G K, B G T>x and VQ G Ny0 such that

{0} x V0 C grC n (B x Y) - [F < A0];

(iii) there exists (xo,yo) G grC (~l d o m F such that F(xo, •) is continuous

at y0;

(iv) X, Y are metrizable, lbD and either F is cs-complete and C is

cs-closed, or F is cs-closed and C is cs-complete;

(v) X, Y are Frechet spaces, F and C are li-convex, and G lbD;

(vi) dim YQ < oo and € *£>,

£/ien £/iere exists z* G Y* such that

inf{F(x,y) \ (x, y) G gr 6} = inf{F(x,y) + (z, z*) \ (x,y + z) e gr e }

Proof Let Z := Y and

* : ( I x Z ) x l , $ ( x , z ; y ) := F(x,z) + tg r e( x , y + z)

It follows easily that $ is convex and P r y ( d o m $ ) = D It is obvious that the conclusion of the theorem is equivalent to inf(X ] 2)exxZ^(a ;,2 :iO) =

max^.gy — $*(0,0;2/*) So we have to show that if one of the conditions of the theorem is verified then a condition of Theorem 2.7.1 holds

If (i) holds it is immediate that $ verifies condition (i) of Theorem 2.7.1 The implication (ii) => (i) is obvious We also have that (iii) => (ii); just take B := {x0}, A0 := F(x0,yo) + and V0 = {y G Y \ F(x0,y0+y) < A0}

(in this case Y0 — Y) Similar to the proof in Proposition 2.7.2 we have

that (vi) =>• (ii)

(iv) => (i) Let Y^n>i ^n(xn, zn, yn, tn) be a convex series with elements of

epi $ such that J2n>i ^nXn and J2„>i ^nzn are Cauchy, J2n>i Kyn = y G

Y and J2n>i ^tn = t G K Then (xn,zn,tn) G e p i F and (xn,zn + yn) G

grC for every n > If F is cs-complete it follows that J2n>1 Xnxn and

S n > i ^nZn are convergent with sums x G X and z G Z, respectively;

moreover, (x,z,t) G epi.F, whence (x,z + y) G grC since grC is cs-closed The same conclusion holds in the other case Therefore $ verifies condi-tion (H(x,z)) in our hypotheses Hence condicondi-tion (iv) of Theorem 2.7.1 is verified So, by Proposition 2.7.2, relation (2.55) holds Therefore for

U x Y G Nxxz there exists A > such that {y G Y | (x, z) G XU x Y, $(x, z; y) < A}

and so (i) holds

(v) => (i) Consider the set

A:={(x,z,y) eX x Z xY \y + z£ G(x)} = {{x,y' -y,y)\{x,y')egre, yeY}

= grC x {0} + {0} x {(-y,y) \y eY}

Using Propositions 1.2.4 (ii) and 1.2.5 (ii) we obtain that A is li-convex (as sum of two li-convex subsets of a Frechet space) Since $(x,z;y) —

F(x, z) + LA{X, Z, y) and the functions F and LA a r e li-convex, $ is li-convex,

too Thus $ satisfies condition (v) of Theorem 2.7.1; the other conditions being obviously satisfied, as in (iv) =4- (i), we get that (i) holds, too •

Note that every condition of the preceding theorem is verified by F,

F(x,y) — F(x,y) — (x,x*), where x* € X*, when the same condition is

verified by F

Taking gr = X x {0}, the conclusion of the preceding theorem is just the conclusion of Theorem 2.7.1 Conditions (i), (ii), (iii) and (vi) become conditions (2.55), (2.54), (iii) and (viii) of Theorem 2.7.1, respectively; to conditions (iv) and (v) correspond slightly stronger forms of conditions (iv) and (v) of Theorem 2.7.1, respectively

Remark 2.7.1 If F(x,y) = f{x) + g(y) with / € A(X), g € A(Y), for

condition (i) of Theorem 2.7.4 it is sufficient (and necessary if f,g have proper conjugates) to have

VUelSx, A > , 3V € X y0 : V C [g < A] - e(\U n [/ < A]),

while for condition (ii) of Theorem 2.7.4 it is sufficient (and necessary if / , g have proper conjugates) to have

3 A0 S 1, Be Sjr, V0 £ NYo : V0 C [g < A0] - G{B n [/ < A0])

Of course, these two conditions are equivalent if X is a normed space

Corollary 2.7.5 Let f € A(X) and A &(X,Y) Suppose that one of

the following conditions is verified:

(i) / is continuous on int(dom/), assumed to be nonempty, and A is relatively open, i.e A(D) is open in ImA for every open subset D C X;

(ii) X and Y are metrizable, either (a) / is cs-complete or (b) / is

(iii) X is a Frechet space, Y is metrizable, f is a li-convex function and

ibA(domf)jt<fr;

(iv) X is a Frechet space, f is lower semicontinuous and %bA(dom f) ^ 0;

(v) dim (lin A(dom/)) < oo

Then, either Af is —oo on *(^4(dom/)) or Af is proper and {Af)\y0 is

continuous on ' (A(dom f)) Moreover

V j / G ^ d o m / ) ) : (Af)(y) =max{{y,y*) - r(A*y*)\y* eY*}

Proof Let us consider y0 l( A ( d o m / ) ) and

$ : X x Y -*• I , $(ar,y) := f(x) + tgTA(x, 2/o + y)

We have that Pry (dom $) = yl(dom/) - y0- If condition (ii), (iii), (iv) or

(v) is verified, then $ satisfies condition (iv), (v), (vi) or (viii) of Theorem 2.7.1, respectively Suppose that condition (i) is satisfied (Obviously, it is impossible that condition (iii) of Theorem 2.7.1 be verified in this case:

F(xo, •) = i{Ax0}-) Since A is relatively open we have that *(A(dom/)) =

intim/i (A(dom/)) = A(int(dom/)) (Exercise!), whence j/o = AXQ for some

Xo € int(dom/) It follows that / is bounded above on a neighborhood

Vo of Xo, and so h (the marginal function associated to $ ) is bounded above by the same constant on J4(VO) — yo, which is a neighborhood of Therefore condition (i) of Theorem 2.7.1 is verified So, under each of the five conditions, we have that

(Af)(y0) = inf $(a:,0) = max (-$*(0,y*)) = max « y0, y * > - / * ( > l V ) ) ,

xex y*eY* yer*

doing similar calculations to those from Theorem 2.3.1(ix) •

Corollary 2.7.6 Let f1} , fn G A(X) and take f :- / i D • • • • / „

Sup-pose that one of the following conditions is verified:

(i) / i is continuous at some point in dom / i ,

(ii) X is a Frechet space, the functions / i , , /n are li-convex and

i 6( d o m / ) ^ ,

(iii) d i m X < oo

Then either f is —oo on !( d o m / ) or /|aff(dom/) is finite and continuous on t(domf), in which case f is sub differentiable on this set Furthermore, for

every x € ' ( d o m / ) we have that

Proof Recall that dom / = dom / i -\ + dom / „ In the cases (ii) and (iii) the result is an immediate consequence of Corollary 2.7.5 taking $ and

A as in Corollary 2.4.7, while for (i) one does the proof by induction •

2.8 Formulas for Conjugates and e-Subdifferentials, Duality Relations and Optimality Conditions

In the preceding sections we have considered only situations in which it

was simple to compute conjugates and e-subdifferentials: sum of functions with separated variables, convolution of convex functions, and functions of type Af For the other types of functions, generally, it is more difficult to compute the conjugate functions or the e-subdifferentials In this sec-tion, we intend to establish sufficient conditions, as general as possible, in order to ensure the validity of such formulas In this section the spaces

X,Xi, , Xn and Y are separated locally convex spaces if not stated

ex-plicitly otherwise

We begin with the following result

T h e o r e m 2.8.1 Let F £ A(X x Y), A £ H{X,Y) and <p : X -> E, <p(x) := F(x,Ax) Assume that € D := {Ax - y \ (x,y) £ d o m F }

and take Y0 := linD Assume that one of the following conditions holds:

(i) there exist Ao £ K, Vo € Ny0 and B G "S>x such that

{0} x Vo C {{x,Ax) \xeB}-[F< Ao];

(ii) for every U £ Nx there exist A > and V £ Ny0 such that

{0} x V C {(x, Ax)\xe At/} ~[F< A];

(iii) there exists xo £ X such that (:ro,j4a;o) £ d o m F and F(xo,-) is

continuous at AXQ;

(iv) X and Y are metrizable, E tbD and either F is cs-closed and gr A

is cs-complete or F is cs-complete;

(v) X is a Frechet space, Y is metrizable, F is li-convex and £ lhD;

(vi) X is a Frechet space, F is Isc and £ lbD;

(vii) X and Y are Frechet spaces, F is Isc and £ %CD;

(ix) there exists XQ £ X such that F(xo, •) is quasi-continuous and {0}

and D are

united-Then for x* £ X*, x £ domip and e > we have:

tp*(x*) = mm{F*(x* - A*y\y*) \ y* £ Y*}, (2.56) dM*) = {A*y*+x* | (x*,y*) € dEF(x, Ax)} (2.57)

Proof It is easy to verify that

Vx* £ X* : <p*{x*) < M{F*{x* - A*y*,y*) \ y* £ Y*}

for every function F and every operator A £ &(X, Y)

Consider $ : X x Y -» E, $(x,y) = F(x,Ax — y) Then $ satisfies one of the conditions (2.54) or (ii) - (ix) of Theorem 2.7.1 when F satisfies one of the conditions (i) — (ix), respectively It follows that the function $ , defined by <t(x,y) := $(x,y) — (a;, a;*), where x* £ X*, satisfies one of the conditions of Theorem 2.7.1, too But

-<p*(x*) = mf{F(x,Ax) - (x,x*) \ x £ X} = inf {$(z,0) | x £ X)

Applying Theorem 2.7.1, we obtain that

- V ( a ; * ) = i n f { S ( x , ) | X e X} = max { - * ' ( , -y") \y* £Y*} (2.58)

But

**(0, -V*) = sup{(ar, a;*) + (y, -y*) - F(x, Ax - y) \ (x,y) £ X x Y}

= sup{(a;,a;*) + (z - Ax,y*) - F(x,z) \ (x,z) £ X x Y} = sup{(a;, x* - A*y*) + (z, y*) - F(x, z) \(x,z)£Xx Y}

= F*(x*-A*y*,y*)

Prom Eq (2.58) we get immediately Eq (2.56)

Note that the inclusion "D" in Eq (2.57) is true for every function F and every operator A £ L(X,Y) Let x £ domip and x* £ detp(x) Then

(see Theorem 2.4.2)

cp(x) + ip*(x*) < (x,x*)+e

By Eq (2.56) there exists y* £ Y* such that <p*(x*) = F*(x* - A*y*,y*), whence

i.e (x* - A*y*,y*) =: (x*,y*) £ deF(x,Ax) Therefore x* = x* + A*y*,

which proves that the inclusion "C" in Eq (2.57) holds, too • Note that (viii) V (hi) => (i) => (ii), (iv) V (v) V (vi) =>• (ii), (vii) =>

(iv) A (v) A (vi), and (ii) =$• (i) if X is a normed space

Note also that similar properties to those stated in the second part of Proposition 2.7.2 can be given for the situations of Theorem 2.8.1

Corollary 2.8.2 Under the conditions of Theorem 2.8.1 we have that

inf F(x,Ax) = max ( - F*(-A*y*,y*)) (2.59)

xeX y*€Y*

Furthermore, x~ is minimum point for <p if and only if there exists y* £ Y* such that (-A*y*,y*) £ dF{x,Ax)

Proof The relation (2.59) follows from relation (2.56) taking x* = Moreover we have that x is minimum point for ip if and only if € dtp(x),

i.e., using Eq (2.57), if and only if there exists (x*,y*) £ dF(x,Ax) such

that = x* + A*y* Therefore the conclusion holds • We note that the result of Corollary 2.8.2 can be used to obtain the

relation (2.56), and so obtain Theorem 2.8.1

When the function F has separated variables, to Theorem 2.8.1 corre-sponds the next result

Theorem 2.8.3 Let f £ A(X), g £ A(Y) and A £ L(X,Y) Assume

that dom / D A- 1 (domg) ^ and let Ya := lin (il(dom / ) — dom g)

Con-sider ip £ MX), ip(x) :— f(x) + g{Ax) Assume that one of the following conditions holds:

(i) there exist Xo £ R, B £ "Bx and VQ £ Ny0 such that

V0CA{[f<\0\nB)-[g<\0\;

(ii) for every U £ J^x there exist A > and V £ Ny0 such that

VcA([f<X]nXU)-[g<X\;

(hi) there exists xo £ d o m / n A~l(domg) such that g is continuous at

AxQ;

(iv) X,Y are metrizable, £ lb(A(dom f) — domg), f and g have proper

(v) X is a Frechet space, Y is metrizable, f, g are li-convex functions

and Oeib (A(dom / ) - dom g);

(vi) X is a Frechet space, f,g are Isc and £ lb(A(dom f) — domg);

(vii) X, Y are Frechet spaces, f,g are Isc and £ tc(^4(dom/) — d o m g ) ;

(viii) d i m l o < oo and € *(A(domf) — domg);

(ix) g is quasi-continuous and A(dom/) and domg are united; (x) Y = Wl, qri(dom/) ^ and ( q r i ( d o m / ) ) l~l Momg ^

Then for every x* X*, x domy? and e > we have:

p ' O O = min{/*(** - A*y*) + g*(y*) | y* € F * } ,

de<p(x) = [j{d£J(x) + d£2g(Ax) \ si,e2 > 0, ei + e2 = e},

^ ( x ) = fl/(a:) + A*(dg(Ax)) (2.60) P r o o / We apply Theorem 2.8.1 to A and F : X x Y -> defined by

F(x,y) := f(x) + g(y) It is easy to see that if one of the conditions

(i)-(ix) holds, then the corresponding condition of Theorem 2.8.1 is verified If (x) holds, using the properties of the algebraic relative interior in finite dimensional spaces (p 3) and Proposition 1.2.7, we have

*(i4(dom/)-domff) = i( A ( d o m / ) ) -i( d o m g ) = ^ ( q r i ( d o m / ) ) -i( d o m g ) ,

and so (viii) holds, too The conclusion follows then applying Theorems

2.8.1, 2.3.1 (viii) and Corollary 2.4.5 • Note that, as in Theorem 2.8.1, we have that (viii) V (iii) => (i) => (ii),

(iv) V (v) V (vi) =» (ii), (vii) =>• (iv) A (v) A (vi), (ii) => (i) if X is a normed space, and of course, as mentioned in the proof, (ix) =>• (viii)

Applying the preceding result we obtain a formula for normal cones

Corollary 2.8.4 Let A Z(X,Y) and L C X, M C Y be convex sets

Suppose that one of the following conditions is verified:

(i) there exists XQ S L such that Ax0 int M,

(ii) X,Y are of Frechet spaces, L,M are li-convex and € lb(A(L) —

M),

(iii) d i m F < o o and £ 1{A(L) - M)

Then for every x € LC\ A~1(M)

Proof Using the preceding theorem for f := IL, •= IM and A, formula

(2.61) follows from formula (2.60) • Corollary 2.8.5 Under the conditions of Theorem 2.8.3 we have the

fol-lowing relation, called the Fenchel-Rockafellar duality formula,

inf {f(x)+g(Ax)) = ma* ( - / ' ( - A Y ) - g*(y*))

i g A y €'

Furthermore, x is a minimum point for f + go A if and only if there exists y* G Y* such that -A*y* £ df(x) andy* £ dg(Ax)

Proof We proceed as in Corollary 2.8.2 (or apply this corollary) •

Two particular cases of Theorem 2.8.3 are important in applications: / = and A = Idx- The next theorem is stated even for A replaced by a convex process C

T h e o r e m 2.8.6 Let g £ A(F) and C : X =4 Y be a convex process

Assume that G D, where D := ImC — domg Consider YQ := linD and the function ip : X —> R, <p(x) = ini{g(y) \ y £(x)} Assume that one of the following conditions holds:

(i) for every U G N x there exist A > and V € Ky0 such that V C

[g < A] - e(XU);

(ii) there exist XQ € R, B € 3x and V0 € Ny0 such that V0 C [g <

X] - G(B);

(iii) there exists j/o € dorngfllmC such that g is continuous at yo; (iv) X, Y are metrizable, g has proper conjugate, either g is cs-closed

and C is cs-complete, or g is cs-complete and C verifies (Hz), and £ lbD;

(v) X, Y are Frechet spaces, g, C are li-convex and lbD,

(vi) dim Y0 < oo and ' D

Then

V i ' e l * : ip*{x*)=mm{g*{y*) \ x* G C*(</*)}

Moreover, ifxE domtp = C- 1(domg) is such that <p(x) — g(y) with y £

A(x), and e > then de(p(x) C C* {deg(y)) (with equality i / g r C is a linear

sub space)

Proof Let x* £ X* Consider F : X x Y -)• I , F(x,y) := g(y) - (x,x*)

Then

If one of the conditions (i)-(vi) holds then the corresponding condition of Theorem 2.7.4 holds (In fact, in case (iv), if g is cs-complete and C satisfies (Ha;) one verifies directly that the function $ from the proof of Theorem 2.7.4 satisfies (E(x,z)).) So, by Theorem 2.7.4, there exists y* £ Y* such that

<p*(x') = -M{F(x,y) + (z,y*) | (x,y + z) £ grC}

= sup{(z, a;*) - g(y) - (y1 - y, y*) | (x, y') £ gr C, y £ Y)

= sup{(y,y*) -g{y) \ y £ Y}

+ sup{(a;,a;*) + (z,-y*) - igre(x,z) \ x € X,z € Y}

= 9*(v*) + t-(gr e)+ (x*, -y") = g*{y") + tgr e* (y*,x*)

Since

V / e T : V*(x*)<g*(z*) + Lgve.(z*,x*), (2.62)

the conclusion follows Let x £ d o m y = A~1(domg) be such that <p(x) =

g(y) with y £ Q(x) and e > Let x* € dE(p(x) Since <p(x) + <p*(x*) <

{x,x*) + e , there exists y* £ G*~1(x*) such that (f*(x*) — g*(y*) It follows

that g(y)+g*(y*) < (x,x*)+e < (y,y*)+e, whence?/* £ deg{y) Therefore

de(fi(x) C G*(deg(y)) Conversely, suppose that gr C is a linear subspace and

take x* £ Q*{y*) with y* £ deg{y) Then (x,x*) = (y,y*); using Eq (2.62)

we obtain that (p(x) + ip*{x*) < (x, x*) + e, i.e x* £ deip(x) •

Note that (vi) V (hi) =>• (ii) =» (i) and (iv) V (v) => (i)

Important situations when gr C is a linear subspace are: C = A"1 with

A £ &(X, Y) and = A with A a densely defined closed operator {i.e A : D(A) -> Y, D(A) being a dense linear subspace of X, A being a linear

operator and gr A a closed subset of X x Y; see also Exercise 1.11) T h e o r e m 2.8.7 Let / , </ € A(X) Assume that d o m / D domg ^ and

let Xo := lin ( d o m / — doing) Assume that one of the following conditions holds:

(i) there exist Ao £ K, B £ S ^ and VQ £ 3sfx0 SMc/t ^f l£

V o C [ / < A o ] n - B - [ < / < A o ] ;

(iii) there exists XQ G dom / n dom g such that g is continuous at XQ ; (iv) X is metrizable, f,g have proper conjugates, f is cs-closed, g is

cs-complete and G l 6( d o m / - domg);

(v) X is a Frechet space, f,g are li-convex and G l i >(dom/ — domg);

(vi) X is a Frechet space, f,g are Isc and € l 6( d o m / — domg);

(vii) X is a Frechet space, f,g are Isc and G I C(dom/ — domg);

(viii) dimX0 < oo and G *(dom/ — domg);

(ix) g is quasi-continuous and d o m / and domg are united;

(x) X is a Frechet space, f,g are li-convex and (0,0) G lb[{(x,x) \ x G

X} — dom / x dom g)

Then for x* € X*, x G dom / Pi domg and e > we have:

( / + <?)*(**) = min{/*(** - y*) + g*(y*) | v* G X*} = (/*Og*)(x*),

(2.63)

de(f + 9)(x) = {J{deif(x) + dE2g(x) \ei,e2 > 0, ex + e2 =e),

d(f + g)(x) = df(x) + dg(x)

Proof Taking A = I d x , the conclusion follows from Theorem 2.8.3 under conditions (i) - (iii), (v) - (ix) If (iv) holds, taking Y = X and $(x,y) :=

f(x) + g(x — y), condition (iv) of Theorem 2.7.1 is verified If (x) holds

consider F : X x X -)• I , F{x,y) := f{x) + g{y) and A : X -> X x X,

A(x) :— (a;, a;) Then condition (v) of Theorem 2.8.6 holds; applying it we

obtain the conclusion, taking into account that A*{x*,y*) = x* + y* •

The same implications as in Theorem 2.7.1 (mentioned in Proposition 2.7.2) hold

Corollary 2.8.8 Let X be a Frechet space and f, g G A(X) / / / and g

are li-convex then for every x G *6(dom/ + domg) we have

(/Dg)(x) = ( / * + $ * ) • ( * ) = m&x{(x,x*) - f*(x*) - g*(x*) \ x* € X*}

(2.64)

Proof Let a;o G s 6( d o m / + domg) and consider h G A(X), h(x) :=

Theorem 2.8.7 is satisfied From formula (2.63) applied for x* = we get

(fn9)(x0) = inf (/Or) + h{x)) = - ( / + h)*(0)

= - ( / • ( * * ) + >»*(-**))•

X*£X *

But h*{—x*) = s u p { - (x,x*) — g(xo — x) \ x € X} = g*(x*) — (x0,x*), and

so (2.64) holds for x = x0 O

Of course, one can obtain the conclusion of the preceding corollary also for other situations corresponding to conditions of Theorem 2.8.7 For example, if there exist A0 £ M, B € "S>x, VQ € Nx0 (x) s u c n that Vo C [/ <

A0] n B + [g < A0], where X0 - aff(dom/ + dom^), then Eq (2.64) holds

Proposition 2.8.9 Let f,g € A(X) and take D = d o m / - domg If

dim(lin£>) < oo then {D = rintZ?, while if X is metrizahle, f,g have proper

conjugates, f is cs-closed, g is cs-complete and tbD ^ then lbD = rint D;

similarly for the situations corresponding to conditions (v)-(vii) of Theorem 2.8.7

Suppose that X is a Banach space and f, g are li-convex Then for every x € lbD there exist n, A > such that

(x + nUx) n aff D C [f < A] n \UX - [g < A] D \UX- (2.65)

Proof The first part follows from Proposition 2.7.2 taking Y = X and $(x,y) =f(x)+g(x-y)

Suppose that X is a Banach space and / , g are li-convex; consider x £

lbD Replacing / by / , f(u) — f(u + x), we may suppose that x = It

follows that condition (v) of Theorem 2.8.7 holds, and, as noted after its proof, condition (i) is verified Therefore there exist 77 > 0, Ao G B and

B £"Bx such that

rjUx n aff D C [/ < Ao] n B - [g < A0]

Taking A' > max{Ao, 0} such that B C X'Ux and A = A' + n we obtain that

rjUx n aff D C [/ < A0] B - [g < A0] D (B + nUx)

c [/ < A]n xux - [f < A]n \ux

Theorem 2.8.10 Let Y be ordered by the convex cone Q, f £ A(X),

H : X —> Y' be convex and g £ A(y) be Q-increasing on H(domH) + Q Then :— f + goH is convex Assume that £ D, where D :— H(domHC\

d o m / ) — domg + Q, and consider Yo := linD Assume that one of the

following conditions holds:

(i) for every U £ J^x there exist A > and V € Ky0 such that

V C H (XU n [/ < A] n d o m F ) - [g < A] + Q;

(ii) there exist XQ R, B £ 1$x and VQ £ Ny0 such that

V0 C H (B n [/ < A0] n domH) -[g< A0] + Q;

(iii) there exists XQ £ d o m / fl i f- 1 (domg) such that g is continuous at

H(x0);

(iv) X, Y are metrizable, f, g have proper conjugates, either f, g are

cs-closed and epiif is cs-complete, or f,g are cs-complete and epiH is cs-closed, and G lbD;

(v) X, Y are Frechet spaces, f,g, epiH are li-convex and £ tbD;

(vi) dim Y0 < oo and 0£iD;

(vii) g is quasi-continuous, and domg and H(domH f~l d o m / ) + Q are

united

Then for x* £ X*, x £ dormp = d o m / n H~1(domg) and e >Q, we

have:

<p*(x*) = min{(/ + y* o H)*(x*) + g*(y*) \ y* £ Q+}, (2.66)

deV{x) = | J {0Er (f + y*° H){x) \y*£Q+n dE2g(H(x)), El + e2 = e}

(2.67)

Proof First observe that for all x* £ X*, y* £ Q+ and x £ dom<£ we

have

V ' O O < (/ + V* o H)*(x*) + g*(y*), (2-68)

de<p(x) D \J{dei(f + y*o H)(x) \y*£Q+n de2g(H(x)), El + e2 = e],

without any supplementary condition on / , g and H Indeed, let x € dom<^ = d o m / n H-^domg), x* € X* and y* £ Q+ Then

f{x) + (y* o H)(x) + (f + y*o H)* (x*) > (x, x*), g(H(x))+g*(y*)>(H(x),y*),

whence, adding them side by side, we get g*(y*) + ( / + y* ° H)* (x*) > (a;, a;*) - <p{x) Thus Eq (2.68) holds

Let now x £ domtp, y* Q+ndS2g(H(x)), and x* £ dei(f + y*oH)(x),

where e1,e2 > 0, £ = £x + e2 Then

f{x) + (y* o H){x) + (f + y*o H)* {x*) < (x, x*) + ex,

g(H(x))+g*(y*)<(H(x),y*)+e2

Adding them side by side we get ip(x) + g*(y*) + (/ + y* ° H)* (x*) <

{x,x*) + e Using Eq (2.68) we obtain that x* € deip(x) Therefore Eq

(2.69) holds

Let F(x, y) := f(x) + g(y) and G : X =t Y with gr C := epi H; then F £

A(X x Y) and is convex Since g is increasing on H(dom H) + Q, it follows

that (p(x) = int{F(x,y) + iep\H{x-,y) \ y Y] for every x € X Hence ip is

the marginal function associated to the convex function F + iepi H ', hence

ip is convex

If one of the conditions (i) — (vi) is verified, then F and satisfy the corresponding condition of Theorem 2.7.4 (When / and g are cs-complete and have proper conjugates, similarly to the proof of Proposition 2.2.17, we get that F is cs-complete.) As noticed after the proof of that theorem, also the perturbed function F, F(x,y) = F(x,y) — {x,x*), satisfies the same condition Therefore there exists z* £ Y* such that

a : =, jn f 1TUix)+g{y)-{x,x''))

= inf (f(x)+g(y)-(x,x*) + (z,z*))=:(3

{x,y+z)EepiH

Using again the fact that g is increasing on H(domH) + Q, we have

a = Jn f w j?^ n VW + 9{y) ~ (x,x*))

xedom H yeH(x)+Q

= inf (f(x)+g(H(x)) - {x,x*)) = ~<P*(x*),

and

P = inf inf (f(x) + g(y) - (x, x*) + (H(x) +q-y, z"))

iGdomif q€Q, y€Y

It follows that P = - o o if z* $ Q+ If z* £ Q+ then

-0= sup sup((x,x*)-f(x)-(H(x),z*) + (y,z*)-g(y))

x€dom Hy£Y

= (/ + Z*o #)*(*•) + </•(**)•

Taking into account Eq (2.68), it is clear that Eq (2.66) holds

In the case (vii) consider F : I x Y ->• K, F(x,y) :=

f(x)+g(H(x)+y)-(x,x*) Because € D and g is Q-increasing on H(domH) + Q, there exists xo € d o m / n H~1(domg) It follows that F(x0, •) is quasi-continuous and

{0} and D — Pry (dom F) are united Applying Theorem 2.7.1(ix) we have that i n fx ex F(x,0) = maxy.ey.(—F*(0,y*)) With a similar calculation

as above we obtain that Eq (2.66) holds

Let x donup and x* d£ip(x) Using Eq (2.66), there exists y* € Q+

such that ip*(x*) = (f + y* o H)*(x*) + g*{y*) It follows that

{f+y*oH){x)+{f+y*oHY{x*)-(x,x*)+9{H{x))+g*{y*)-{H{x),y*) < e

Taking £ l := ( / + y* o tf)(x) + ( / + y* o #)*(x*) - (x,x*) (> by the

Young-Fenchel inequality) and e2 = e - ei, we have that y* € d£2g(H(x))

and Z* € d£l(f + y* oH)(x) Therefore the inclusion C holds in Eq (2.67);

taking into account Eq (2.69), we have the desired equality D

We use the preceding theorem to obtain formulas for conjugates and subdifferentials of functions of "max" type

Corollary 2.8.11 Let / i , , / „ E \(X) and

y.X^W, ¥ > ( * ) : = / i ( z ) V - - - V / „ ( z )

Suppose that dom ip = f}™=1 dom /i ^ and e € IR+ For x € dom ip denote

while for every x £ dom <p,

n fn){x) (Ai, , A„) G A„,

77 e [0,e], Yli=1Xi^x^ - ¥>(x)+ri-£o} ,

dV{x) = U {5 (E"= A J i) {x) I (Al!' • •'An) € A"'

Vifl{x) : Ai = o}

Proof Let us consider the functions

H:X->(W\Rl), H(x):=( (A(*)>ãã"'/ô(*)) if x € fX=i d o m / , ,

"ã" [ 0 otherwise, 3 : ô " - ã ô , g(y):=yiV. -Vyn

It is clear that condition (iii) of the preceding theorem is verified Taking into account the expressions of g* and deg(y) given in Corollary 2.4.17 we

obtain immediately the relations from our statement D

An application of the previous result is given in the following example

Example 2.8.1 Let / € A(X) and consider / + := / V Then f df(x) if f(x) > 0,

df+(x) = l U { W ) ( x ) | A e [ , l ] } if f(x) = 0, { d(0f)(x) = didomf(x) if f(x)<0

A useful particular case of the result in Corollary 2.8.11, in which we can give explicit formulas for ip* and detp without supplementary conditions, is

presented in the following corollary

Corollary 2.8.12 Let f{ A(X,) for i € T~n and

<p:X :=Tf Xi-+W, tp(x1, ,xn):=f1(xi)V -Vfn(xn) J- -1J = I

For x = (xi, ,xn) S domtp = Yl7=i dom/j consider I(x) := {i G l , n |

fi(Xi) = <p(x)} Let x* = (x*1, ,x*n)£ U7=iX* = X* Then

For x — (xi, , xn) E dom <p and e G P we have:

dMx) = {j{~[[n.=1dei(^fiKxi)\ ( A i , , A „ ) G An, et > 0,

En V~^n

i=o£i = £' Z^i=1Xi^Xi^ - ^ - £o ) >

S ^ ^ - l J l n ^ ^ A i / O C ^ ) ! (A1, ,An)€A„, VigJ(a:) : A; = o}

Proof We apply the preceding corollary to the functions fa : X ->• R, /i(a;) := fi(xi), then we use Theorem 2.3.1 (viii) for arbitrary n and

Corol-lary 2.4.5 • Other situations when one has explicit formulas for tp* and d<p{x) in

Corollary 2.8.11 are specified in the next result

Corollary 2.8.13 Let f,g£ A(X) satisfy one of the conditions (i)-(iii)

or (v)-(x) of Theorem 2.8.7 and consider <p := / V g Then for all x* G X* and x G dom (f = dom / n dom g,

<p'(x') = {(A/)*(iO + («/)*(«*) | (A,/i) G A2, u*,v* G X\

U + V = X j ,

3<p(x) = {J{d(\f)(x) + d(»g)(x) I (A,M) G A2, A / ( X ) + ng(x) = <p{x)}

Proof When / and g verify one of the conditions (i)-(iii), (v), (viii)— (x) of Theorem 2.8.7 and A,/x > 0, then the functions A/ and fig also verify the same condition If condition (vi) or (vii) holds then, by the relations among the classes of convex functions on page 68, condition (v) holds So, (Xf + ng)*{x*)=Toia{(Xf)*{u*) + (jjig)*{v*) \u*+v* = x * } a n d

d(\f + ng)(x) = d(Xf)(x) + d(fig)(x) Using Corollary 2.8.11 one obtains

the conclusion • Taking into account that for x G dom / , one has df(x) + N(dom f, x) =

df(x) (see also Exercise 2.23), d(Xf)(x) = \df(x) for A > and d{0f)(x) = N(domf,x), we get for f,g G A(X) and x G X with f(x) = g(x) G R,

U{W)W+8WWI(A,|f)6A2}

-co(df(x)Udg{x)) + N(domf,x) +N(domg,x)

Corollary 2.8.14 Let / i , /2 € A{X) and e £ 1+ For every x* £ X* we

have that

(hQf2y(x*)=rnin{(\1f1y(x*) + (\2f2y(x*) | (A1;A2) £ A2} (2.71)

Suppose that (/i0/2)(a:) = max{/i(2;i),/2(a;2)}, where xi £ d o m / i , x~2 £

dom /2 and x = x~i + x2 Then

de{hOh){x) = \J{dEl(^h)(xi)ndE2(\2f2)(x2) I (Ai,A2) e A2,

e i , e2> , £i+e2<e + X1f1(x1) + X2f2(x2)-(f10f2)(x)} (2.72)

Furthermore, if fi(x\) = f2(x2) *^en Eq (2.73) is verified, while if fi(xi) >

72(^2) *^en i?g (2.74) is verified, where

d(fi0f2)(x)=[j{d(X1f1)(x1)nd(X2f2)(x2) I (A1;A2) G A2} , (2.73)

d(fi0f2)(x) = dMxx) nN(domf2-:x2) (2.74)

Proof Let x* £ X*; then

{fi0f2)*(x*) = s u pi e X( ( x , x * ) - i n f f / ^ ) V /2( z2) I *! + x2 = x})

= sup{(ii,a;*) + (a;2,a;*) - fi(xi) V f2(x2) \x1} x2 £ X)

= min{(A1/1)*(x*) + (A2/2)*(s*) | (Ai,A2) £ A2} ,

the last equality being obtained by using Eq (2.70)

The inclusion "D" of Eq (2.72) can be verified easily Consider x* £

de(fi0f2)(x) By Eq (2.71), there exist (Ai, A2) £ A2 such that

(/i0/2)*(x*) = ( A i / i ) * O + (A2/2)*(x') (2.75)

Using the preceding relation we obtain that

0 < [ ( A1/1) ( ^1) + (A1/1)*(2;*)-(5;1,a;*)]

+ [(A2/2)(i2) + (A2/2)*(:c')-<S2 ) a:*)]

< (fiOh)(x) + ( / i O /2) * 0 - (x,x*) < e

Taking e, := (Aj/i)(zj) + {Xifi)*(x*) - (x~i,x*) > 0, i £ {1,2}, using the preceding relation and Eq (2.75), we get

( / i /2) ( z ) + £ - Ai/i(3fi) + e2 - A2/2(x2) < e,

Let us prove now the equalities (2.73) and (2.74) The inclusions "D" follow directly from Eq (2.72) Let us prove the converse inclusions Let

x* G d{faOfa)(x) = d0(faOfa)(x) By Eq (2.72), there exist (Ai, A2) G A2

and £i, £2 > such that

x* £dei(Xifi)(x1)ndea(Xif2)(x2),

£1 + e2 < Xifi(xi) + A2/2(x2) - (faOfa){x) <

Therefore £1 = £2 = and Ai/i(xi) + A2/2(x2) = {fa(>fa){x); hence x*

belongs to the set on the right-hand side of Eq (2.73) when / i ( x i ) = fa{x2

)-If fi{x~i) > /2( ^2) , the preceding relation shows that A2 = , Ai = 1, and

so

x* G 9/i(11), x* G 0(0 • /2) ( z2) = <9idom/2(^2) = -/V(dom/2;x2)

This shows that x* belongs to the set on the right-hand side of relation

(2.74) • Note that in Eq (2.72) we can take E\ + £2 = £ + ••• Moreover, if

/1 is continuous at x\ and fa is continuous at a;2, then in Eq (2.74) we

have d(fiOfa)(x) = {0} Indeed, in this situation, N(domfa;x~2) = {0} (since x~2 G int(dom/2)); since /1O/2 is continuous at x, d(fiQfa)(x) ^

In particular, it follows that X\ is a minimum point of fa Furthermore, in this situation, /1O/2 is even Gateaux differentiate

Introducing the convention that • df(x) := N(domf;x) for / G A(X) and x G d o m / , formula (2.75) may be written in the form

d(fa0fa)(x) = dfa(x1)odfa(x2),

where A o B represents the harmonic sum of the sets A and B, which generalizes the inverse sum of A and B

In order to have more explicit formulas in Corollary 2.8.11 it is necessary to impose some supplementary conditions Let us give such conditions and the corresponding formulas in three situations for n =

Corollary 2.8.15 Let fa, fa G A(X), e G 1+ and <p : X -> E, >p :=

m a x { / i , /2} Suppose that one of following conditions is verified:

(i) there exists XQ G dorafa n d o m /2 such that fa is continuous at XQ;

(ii) X is a Frechet space, fa, fa are li-convex and G l 6( d o m / i —

d o m /2) ;

Then, for all x* € X* and x £ d o m / i D dom/2 we have:

Ơ>*(*ã) = min{(Ai/i)*(a;I) + (A2/2)*(^) | (Ai, A2) e A2, x\ + x*2 = x*},

dMx) = \J{dEl(Xifi)(x)+dE2(X2f2){x) I (Ai,A2) G A2, e0 >ei,e2 > 0,

e0+ei + e2 = e, Xifi(x) + X2f2{x) > <p(x) -e0},

dtp(x) = \J{d(X1h)(x)+d(X2f2)(x) I (Ai,A2) G A2>

Xih(x) + x2f2{x) = tp(x)} a

Let * £ X* and consider the functions / i , /2 : X —>• K

defined by

/i(a;) : = m a x { ( i , a ; ; ) , , ( j ; , < ) } , /2(x) := |{a;,a;*)|

Applying the preceding corollary, we get the following formulas for every

xeX:

dh(x) = { ]C"= A^i I (Ai, • • •, A„) € A„, ^ = if (x, x*) < /!(a;)} ,

r {x*} if (x,x*) > , 0/2 (*) = < {Az* I A € [ - , +1]} if (x, x*) = 0,

1 {-a;*} if (x,x*) <

2.9 Convex Optimization with Constraints

Let us come back to the general problem of convex programming as con-sidered in Section 2.5,

(P) f(x), x £ C,

where / € A(X), C C X is a convex set and C (~l d o m / ^ 0; the spaces considered in the present section are separated locally convex spaces if not stated explicitly otherwise Since x is solution of (P) exactly when it minimizes f + ic, x is solution of (P) if and only if € d(f + i<c)(x)- Taking into account Theorem 2.8.7, we have

Theorem 2.9.1 (Pshenichnyi-Rockafellar) Let f € A(X) and C C X

Proof In the conditions of our statement we have that

V x e C n d o m / : d(f + ic){x) = df(x) + duc{x) = df(x) + N{C;x),

whence the conclusion is obvious •

Very often the set C from problem (P) is introduced as the set of solu-tions of a system of equalities and/or inequalities

Let Y be ordered by a closed convex cone Q CY and H : X -» Y' be a Q-convex operator; the set C := {x € X \ H(x) <Q 0} is a convex set In this case the problem (P) takes the form:

(P0) f{x), H{x) <Q

An element x € X for which H(x) <Q is called an admissible solution of problem {PQ) Of course, we assume that d o m / f~l {x € X \ H(x) <Q 0} ^ 0; in particular d o m / fl domH ^

The problem (PQ) may be embedded in a natural way into a family of minimization problems {Py), y &Y:

(Py) f(x), H(x) <Q y

Let us consider the function

(x, y) := |

$ : I X F l , *(*,y):=«{ / ( l ) i f ^ ^ ^ (2.76) v 'w y ' oo otherwise v '

The problem (Py) becomes now

(Py) $(x,y), i £ l

We obtain, without difficulty, that $ is convex Moreover

$*(z*,-y*) = sup{(x,a;*) + (y, -y*) - *(x,y) \x€X,yeY} = sup{(x,x*) - (y,y*> - / ( z ) | x € X , y £ Y,#(aO <Q y}

= sup{(a;,a;*) -(H(x)+q,y*) - f(x) \ x € d o m / / , g € Q} = sup{(z,:r*) - (H(x),y*) - f{x) \ x £ domH}

+ sup{-(g,y*) | € Q }

Hence

$*(x*,-y*) = sup{(x,x*) - f(x) - (H(x),y*) \x£domH}

The function

L:XxQ+->R, L(x,y*):= { ^x

^ oo

) + (H(x),y*) i f x G d o m t f ,

if x $ dom H, is called the Lagrange function (or Lagrangian) associated to problem

(Po)- The above definition of L shows that L(x,y*) — f(x) + (y* o H){x)

with the convention that y*(oo) = oo for y* G Q+, convention which is

used in the sequel By what was shown above, we have that

V y * e Q + : &(0,-y*) = 8up(-L(x,yt)) = -hdL(x,y*)

xex x^x

Therefore, the dual problem of problem (Po) (see Section 2.6) is (£>o) max ( - $*(0, </*)), y*£Y*,

or equivalently,

(Do) max (mfxeX L(x,y*)), y* £ Q+•

We have the following result

Theorem 2.9.2 Let f G A(X), x G domf and H : X -> (Y',Q) be a

Q-convex operator, where Q <ZY is a closed convex cone Suppose that the following Slater's condition holds:

(S) 3x0edomf : -H{x0) € intQ

Then the problem (D0) has optimal solutions andv(Po) = v(Do), i.e there

exists y* G Q+ such that

inf{/(x) | H(x) <Q 0} = inf{L{x,y*) \ x G X}

Furthermore, the following statements are equivalent:

(i) x is a solution of (Po);

(ii) H(x) <Q and there exists y* G Q+ such that

O£d{f + y*o H)(x) and (H(x),y*) = 0;

(hi) there exists y* G Q+ such that (x, y*) is a saddle point for L, i.e

Vx E X, Vy* G Q+ : L(x,y*) < L{x,y*) < L{x,y*)- (2.77)

Proof The Slater condition (S) ensures that (XQ, 0) G dom $ and $(x0, •)

is continuous at 0, where $ is the function denned by Eq (2.76) Applying Theorem 2.7.1, there exists y* G Y* such that v(P0) = - $ * ( , - y * ) If

take y* = If f(Po) > —oo, using the expression of $*, we have that

y* e Q+ Therefore

v(P0) = inf{/(x) | H(x) <Q 0} = - $ * ( , - | T ) = i n f { i ( x , r ) I a; € X }

= v(D0)

(i) =S- (ii) Of course, x being a solution for (P0), we have that H(x) <Q

By what was proved above, there exists y* £ Q+ such that

f(x)=v(P0)=in{{L(x,y*)\xeX},

whence

Vx € X : f(x) + (H(x),?) < f(x) < f(x) + (H(x),y*)- (2.78) Relation (2.78), without the term from its middle, says that x is a minimum point for f + y* o H, and so € d(f + y* o H)(x); taking x = x in relation (2.78) we obtain that (H{x),y*) -

(ii) =>• (hi) Since € d(f + y* o H)(x) we have

Vx G X : L ( x , r ) = / ( x ) + (^(x),y*) < f{x) + (H(x),y*) = L f o i T ) Furthermore, for y* £ Q+ we have that

L(x,y*) = f{x) + (H(x),y*) < f(x) = f{x) + (H{x),y*) = L(x,?) Therefore Eq (2.77) holds, i.e (x,y*) is a saddle point of L

(iii) => (i) Taking successively y* = and y* = 2y* on the left-hand side of Eq (2.77) we obtain that (Hix),y*) = Using again the left-hand side of Eq (2.77), we obtain that (i?(x), y*) < for every y* e Q+; thus, using

the bipolar theorem (Theorem 1.1.9), we have that —i?(x) € Q++ = Q,

i.e x is an admissible solution of (P0) From the right-hand side of relation

(2.77) we get

Vx e X, H{x) <Q : fix) = f(x) + (H(x),r) < f(x) + (H{x),F) < fix)

Therefore x is solution of problem (Po)- D The element y* € Q+ obtained in Theorem 2.9.2 is called a L a g r a n g e

m u l t i p l i e r of problem (P)

Note that when H is finite-valued (i.e d o m # = X ) and continuous,

the fact that Q is closed was used only for the implication (iii) =4- (i) of the above theorem, to prove that x is an admissible solution

An important particular case is when there are a finite number of con-straints Let / , g i , ,gn A(X) and consider the problem

(Pi) f(x), gi{x) < 0,1 < i < n The dual problem of (Pi) is

(Di) max mix€X(f(x) + \igi(x)-] hAn(/n(i)), Ai > , , A„ >

Then the following result holds

Theorem 2.9.3 Let f,gi, ,gn € A(X) Suppose that the Slater

con-dition holds, i.e

3xo € d o m / , \/i£l,n : gi(x0) <

Then:

(i) the dual problem (Di) has optimal solutions and u(Pi) = v(D{),

i.e there exist (Lagrange multipliers) A i , , A„ £ M+ such that mf{f(x)\gi(x)<0, ,gn(x)<0}

-ini {f(x)+J1g1(x)-\ + Xn9n(x) \ x X)

(ii) Let x dom / ; x is a solution of problem (Pi) if and only if gi(x) < 0 for every i l , n and there exist A i , , An € K+ such that Xigi(x) =

for i £ l , n and

0 € d ( / + Aiffi + +

*ngn) \X

J-If the functions g\, , gn are continuous at x, the last condition is

equiv-alent to

0 € df(x) + A-!0ffi(2c) + + \ndgn(x)

Proof Let us consider H : X -> (Mn*,E!J:), H(x) := (gi(x), ,gn(x))

for x G n r = i d o m j , i?(x) = oo otherwise; H is R™-convex The result stated in the theorem is an immediate consequence of the preceding theo-rem When gi are continuous at x one has, by Theorem 2.8.7 (iii), that

d(f + Aiffi + + \n9n)(x) = df(x) + d(Xigi)(x) + + d(\ngn)(x),

Corollary 2.9.4 Let f,gi, .,gn '• X —> R be Gateaux differentiable

continuous convex functions Suppose that there exists x$ £ X such that 9i{xo) < for every i l,n Then x £ X is solution of problem (Pi) if and only if gi(x) < for every i £ l , n and there exist A i , , An £ R+

such that

—\7f(x)=\iVgi(x) + + \nVgn(x) and \igi(x) = for i £ l , n •

Corollary 2.9.5 Let g A(X), £ ] inf g, 00 [ and S £ [5 < 7] Tften

N([g < l\,x) = \J {d(Xg)(x) | A > 0, X(g(x) - 7) = 0} (2.79)

Proof The inclusion "D" in Eq (2.79) holds without any condition on g and Indeed, let x* £ d(Xg)(x) for some A > with X(g(x)— 7) = Then for x £ [g < 7] we have that (a: — x, x*) < Xg(x) — Xgix) = X(g(x) — 7) < 0, and so x* £ N([g < ^],x)

Let x* £ N([g < j];x) Then x is a solution of the problem (Pi) {x, —x*), h(x) := g(x) — < 0,

whence, by Theorem 2.9.3, there exists A > such that Xh(x) = X(g(x) — 7) = and G d(-x* + Xh){x), i.e x* £ d{Xh){x) = d(Xg)(x) Therefore

the inclusion " c " of Eq (2.79) holds, too • Note that d(Xg)(x) = Xdg(x) if A > and d(0g)(x) = didomg(x) =

N(domg;x)

In the case of normed vector spaces, taking A = X in Proposition 3.10.16

from Section 3.10, we have another sufficient condition for the validity of formula (2.79)

Note that we can obtain the characterization of optimal solutions in Theorem 2.9.3, when the functions gi are continuous, using Corollary 2.9.5 and the formula for the subdifferential of a sum Indeed, x £ dom / is a solution of (Pi) if and only if x is minimum point of the function / + icx +

htc„, where Cj := {x | gi(x) < 0} By hypothesis d o m / f l f)"= 1 int d ^

0, and so, for every x £ d o m / (~l H i l i ^ i w e have

d(f + tCl+ + icn) (x) = df{x) + N(Cf,x) + + N(Cn;x)

Using formula (2.79) we obtain the desired characterization

Theorem 2.9.2 can be extended further to the case when there are also linear constraints Let us consider the problem

where T G £ ( X , Z) Consider the Lagrange function associated to problem

L2 : X x (Q+ x Z*) - • , L2( a M , V ) := /(a:) + (H(x),y*) + (Tx,z*)

The dual problem associated to (P2) is

(£>2) max (mixeXL2(x,y*,z*)), y* G Q+, z* G Z* Then the following result holds

Theorem 2.9.6 Let f G A{X), H : X -> (Y,Q) be a Q-convex

oper-ator, where Q C Y is a closed convex cone, T G L(X,Z) be a relatively open operator and x G d o m / Suppose that there exists XQ € d o m / such that f,H are continuous at xo, —H{XQ) G int<5 and TXQ = Then the problem (D2) has optimal solutions and V{PQ,) = v{D2), i.e there exists

y* G Q+, z* G Z* such that

inf{/(x) I H(x) <Q 0, Tx = 0} = mi{L2(x,y*,z*) \ x G X)

Furthermore, the following statements are equivalent:

(i) x is solution of problem (P2);

(ii) H(x) <Q 0, T{x) = and there exists y* G Q+, ~z* G Z* such that

T*z* G df(x) + d(y* o H)(x) and (H(x),y*) = 0;

(iii) there exists (y*,z*) G Q+ x Z* such that (x,(y*,z*)) is a saddle

point of L2, i.e

L2(x,y*,z*) < L2{x,y*X) < L2{x,y*,T)

for all x G X and all y* G Q+, z* G Z*

Proof Let us consider the perturbation function

f(x) if H(x) <Q y, T(x) = z, * : I x ( x Z ) - > I , $ ( , y , z ) := ,

v ; ' ^ 'y' y ' 00 otherwise

We intend to prove that condition (i) of Theorem 2.7.1 is verified Note first that

Taking into account that / is continuous at xo G d o m / , there exists Uo G N x ( i o ) such that

V x e % : f{x)<M:=f(x0) + l

Since —H{XQ) G intQ, there exists Vo G Ny such that —H(XQ) +VQCQ

There exists V € Ny such that V + V CV0 Since if is continuous at x0,

there exists {/ € Nx(zo)) [/ C t/o, such that H(x) € H(x0) — V for every

x € U Since T is relatively open, there exists W G NimT such that W C

T(LT) Of course, V x W G N yxim r( , ) ; let (j/,z) G V x W There exists

x £U such that Tx = z Since x G f/, we have that i ? ( i ) = H(xo) — y' for

some j / ' G V Then

y - H(x) =y + y'- H(x0) G - # ( x0) + V + V C - # ( x0) + V0 C Q;

hence -ff(x) < Q y Since x £ U C Uo, we have that $(x,2/,z) < M , i.e condition (i) of Theorem 2.7.1 is verified Therefore there exists (y*,z*) G

Y* x Z* such that v(P2) = -4>*(0, -y*, -z*) But

$*(0, -y*, -z") = s u p( w ) X xyx Z( ( i / , - y * ) + (z, -z*) - $ ( x , y , z))

= sup{-(y,y*) - (z,z*) - f(x) | H ( i ) <Q y,Tx = z)

= s u p { - ( # (x) + q, y*) - (Tx, z*) - f(x) \xeX,qeQ} = - inf{f(x) + (H(x),y*) + (Tx, z") \ x G X }

+ sup{-(q,j/*) | q G Q}

Thus

**cn _ „ * —?*\ - J - i n f i e x ^ C i y * , ^ * ) if 2/* e Q+,

* l U' y ' z ) ~ \ oo if j / * £ Q+

Therefore y* G Q+ if u(P2) G K and we can take j7* = if v(P2) = - o o

The proof of the second part is completely similar to that of Theorem 2.9.2 We note only that, / and H being continuous at xo, we have d(f +

y* o H)(x) = df(x) + d(y* o H)(x) for every x G d o m / •

2.10 A Minimax Theorem

existence of saddle points as well as a minimax theorem Let A and B be two nonempty sets and / : A x B -> JR It is obvious that

sup inf f(x,y) < inf sup f(x,y) (2.80)

xeAvtB y£BxeA

The results which ensure equality in the preceding inequality are called "minimax theorems," the common value being called saddle value Note that if / has a saddle point, i.e there exists (x,y) € A x B such that

VxeA,Vy&B : f(x,y) < f(x,y) < f(x,y), (2.81)

then

max inf f(x,y) = minsup f{x,y), (2.82)

x£A yeB yeB xeA

where max (min) means, as usual, an attained supremum (infimum) Indeed, let (x,y) € Ax B satisfy Eq (2.81) It follows that

ini: sup f(x,y) < sup f(x,y) < f(x,y) < inf:(x,y) < sup inf f{x,y)

v£Bx€A xeA y£B xeAv&B

Using Eq (2.80) we obtain that all the terms are equal in the preceding relation, and so (2.82) holds (the maximum being attained at x and the minimum at y) Conversely, if Eq (2.82) holds then / has saddle points Indeed, let x G A and y e B be such that

inf (x,y) = sup ini' f(x,y) = irtf swp f(x,y) = sup f(x,y)

y£B xeAVZB y€Bx€A x£A

Since infy 6s(x,y) < f(x,y) < supx€Af(x,y), from the above relation it

follows that all the terms are equal in these inequalities; this shows that

(x, y) is a saddle point of / So we have obtained the following result

Theorem 2.10.1 Let A and B be two nonempty sets and f : A x B ->• R

Then f has saddle points if and only if condition (2.82) holds •

The following result is an enough general minimax theorem (frequently utilized) which gives also a sufficient condition for the existence of saddle points

Theorem 2.10.2 Let X be a locally convex space, Y be a linear space,

is concave and upper semicontinuous for every y G B and f(x, •) is convex for every x E A Then

max inf f(x,y) — inf ma,xf(x,y) (2.83)

X&A y£B y£B xEA

If moreover Y is a locally convex space, B is compact and f(x, •) is lower semicontinuous for every x € A, then

maxmmf(x,y) = max f(x,y); (2.84)

xeA y£B yeB x£A

in particular f has saddle points

Proof Let a e K be such that a > maxx£j4 infyeB f(x,y) For every

x S A there exists yx B such that f(x,yx) < a Since f(-,yx) is upper

semicontinuous at x, there exists an open neighborhood Vx of x such that

f(u,yx) < a for every u G Vx Since A is compact and A C UxeA ^ > there

exist x\, , xp € A such that A C U?=i Vxt • Let 2/j := yXi • Consider the

sets

d :=co{{f(x,yi), ,f(x,yp))\xeA}cW,

C2 := {(ui, ,up) \v,i>a\/i€ l,p}

It is obvious that C\ and Ci are nonempty convex sets, and C2 has non-empty interior Moreover C\ fl Ci = In the contrary case there exist

q N, ( A i , , A?) Aq and Xi, , xq € A such that

V i e l p : a < Zf := 2^, >^jf{xj,yi)

Since f(-,yi) is concave, taking xo = 13?=i ^ j2^ ' w e n a v e that i o £ i and

Vi G l , p : a < Zj < f(x0,yi)

There exists io £ l , p such that £0 £ VXi Therefore f{xo,yi0) < a, a

contradiction

Applying Theorem 1.1.3 there exists (/zi, , fj,p) € W \ {0} such that

,mf(x,yi) < V UiUi

t = l * — ' i = l

Letting u; -> 00 we have that /ij > for every i, and so we can suppose that (/xi, , Up) € Ap Taking u = ( a , , a) we obtain that

Let 2/0 : = Y%=i ViVi € B Since f(x, •) is convex, we obtain t h a t f(x, y0) <

a for every x € A Therefore a > i n f j/ eB m a xx g^ / ( a ; , / ) , a n d so (2.83)

holds In Eq (2.83) t h e s u p r e m a with respect t o x are a t t a i n e d because t h e functions f{-,y), y £ B, a n d i n fy es f(-,y) are u p p e r semicontinuous

a n d A is compact

W h e n B is compact and t h e functions f(x,-) are lsc for all x € A we obtain t h a t t h e infima with respect t o y € B are a t t a i n e d in Eq (2.83), i.e

Eq (2.84) holds • Note t h a t t h e convexity assumptions in t h e preceding t h e o r e m can be

weakened More precisely, we can suppose t h a t A is a n o n e m p t y compact subset of a topological space, B is nonempty, and / satisfies t h e following two conditions (similar t o concavity a n d convexity, respectively):

Vxi, ,xp e A, V ( A i , , Ap) e Ap, 3x0 &A,\fyeB :

f(xo,y) > y \ ,Kf(zi,y),

• ' — ' =

\/yi, ,yq GB, V ( / / ! , , / * , ) € A , , 3y0 eB,Vx£A :

v—yQ

f(x,yo) < 2^,.=iHjf{x,yj)

2 1 E x e r c i s e s

E x e r c i s e 2.1 Let X be a linear space, / € A.(X) and x,u £ X Prove that the mapping %p :]0, oo[—• R defined by ip(t) :— t • f(x + t~lu) is convex

E x e r c i s e 2.2 (a) Let I C R be an interval and / : / — > • R Suppose that / is locally nondecreasing, i.e for every a € I there exists e > such that the restriction of / to IC\[a — s, a + e] is nondecreasing Prove that / is nondecreasing (b) Let g : [a, b] —• R (a, € R, a < b) be a continuous function 1) Assume that g'+{x) g R exists for every x € [a,b[; show that there exists x' £ [a,b[ such that g(b) — g(a) < g'+(x')(b — a) 2) Assume that g'-(x) € R exists for every

x S ]a, 6]; show that there exists x" € ]a, b] such that g(b) — g(a) > gL (x")(b — a)

(c) Let X be a locally convex space and A C X be an open convex set If / : A —> R is locally convex, i.e for every a G A there exists a neighborhood V of a such that f\vnA is convex, prove that / is convex

x eRn consider the function fx : Rn ->• R, fx(t) = f(x + tx) Prove that

/ is lsc at i t t V i e l " : fx is lsc at 0,

/ is use at J o V i e R " : fx, is use at

If Rn is replaced by an infinite dimensional normed space then the above

prop-erties not hold, generally

Exercise 2.4 Let X be a linear space, / : X —> R be a convex function and

A > i n fi ex fix)- Prove that

{x G X | f(x) < \y = {x G X | f{x) < A}

Moreover, if X is a topological vector space and / is continuous, then int{x € X | f(x) < A} = {x G X | / ( z ) < A}

Exercise 2.5 Let X be a topological vector space and / : X —> R a convex

function Prove that:

(a) [ / < * ] = cl [ / < *] f°r every t G ] inf / , oo[ if and only if / is lsc;

(b) [ / < * ] = int [ / < *] for every t G ] inf / , cx)[ if and only if / is continuous

on d o m / ;

(c) [/ = *] = b d [ / < t] for every t G ] inf / , oo[ if and only if / is continuous

(on X )

Exercise 2.6 Let / : R" —)• R be a proper convex function for which all the

partial derivatives df/dxi exists at a G i n t ( d o m / ) Prove that / is Gateaux differentiable at a (even Frechet differentiate because / is Lipschitz on a neigh-borhood of a)

Exercise 2.7 Let p G [1, oo[ and <pp : R+ -> R be defined by tpp(t) := |1 - t\p

-|1 + t\p + 2pt Prove that ipp is strictly convex and increasing for p G]l, 2[, ipp

is strictly concave and decreasing for p G ]2, oo[, ip\ is convex and nondecreasing and (f2 is constant

E x e r c i s e 2.8 Take /3 G [1, <x[ and consider the function

/ : R2^ I , f(x,y):={ (^tan|)^V^T^

[ oo

2 for x, y > 0,

otherwise,

where, by convention, arctan | := ^ for a; > Prove that / is a lsc convex function, but not strictly convex

Exercise 2.9 Consider the function

-Prove that / is convex and Frechet differentiable of order on C[0,1] with

V/(x)(«) = f1 -£L= dt, V2f(X)(u, V) = ^ ,7/^3-2 dt

Jo V l + x2 Jo ( l + :2) v l + z

for all u,v £ C[0,1] Moreover, V2/ is continuous on C[0,1]

E x e r c i s e 2.10 Consider the function

/ : L1( , ) - > R , / ( i ) : = /0V l + ( i ( t ) )!* ,

where L1(0,1) is the Banach space of (classes of) Lebesgue integrable functions

on the interval [0,1] Prove that / is a Gateaux differentiable convex function with

xu

Vx.uSL^O.l) : V/(x)(u) = /

Jo

:dt,

%/rn

but / is nowhere Frechet differentiable

E x e r c i s e 2.11 Using properties of convex functions, prove that V a , e R + , V p , g e ] l , o o [ , £ + ^ = : ab < ±ap + \b\

the equality being valid if and only if ap = bq, and

V n € N, V x i , , x „ R + , V a i , , a „ 6]0,1[, a i + + a „ = : a m + + anx „ > X™1 • • -xZn,

the equality being valid if and only if x\ = • • • = xn In particular,

V n e N , V s i , , x „ € R + : £(xi + + xn) > yX\ -xn

E x e r c i s e 2.12 Let / : X —>• R be a proper function Prove that / is convex if and only if / + x* is quasi-convex for every x* G X*

Exercise 2.13 Let <x\, ,an > (n > 1) be such that ai-\ \-an < Prove

that the function / : P " —> R, / ( x ) := x"1 • x%2 • • • x"n, is concave; moreover, if

a\ + • • • + a„ < 1, then / is strictly concave

E x e r c i s e 2.14 Consider / : Rn ->• R, f(xi, ,x„) : = In ( X £= 1 expxk)

Prove that / is convex

E x e r c i s e 2.15 Consider p € R \ {0} and the function

t"

xP"->R, f{t,x):=

nr=i*i'

Let c G Rn and A : = { i £ P " \(x\c)> 0} Prove that the function (x\c)"

g : A -> R, p(x)

niu^

is strictly convex for p > n + (it is possible to prove that g is strictly convex for

p > n) The function g has been used by Karmarkar for establishing his interior

point algorithm

Finally prove that the function

h : Pn ->• R, h(x) := (T[n xA

is strictly convex

Exercise 2.16 Let : R+ —> R+ be a continuous convex function such that

^i(t) = •«• t = Show that

Jo i>'- (V>-x«)) = 7o V - H ^ - H * ) )

for every a >

Exercise 2.17 Let X, Y be normed spaces and T G L(X,Y) Prove that the

function

f:Y^R, f(y):=M{\\x\\\Tx = y},

is a sublinear functional Moreover, if T is an open operator, then dom f = X and / is continuous

Exercise 2.18 Let / : Rk —> R be a strongly coercive proper lsc function

Prove that co(epi/) is a closed set

E x e r c i s e 2.19 Let X be a locally convex space, / G T ( X ) , g : X —> R be a proper function and a, /3 > Prove that co (co(/ + ag) + jig) = co ( / + ( a 4- /3)g) and ( ( / + ag)** + fig)" = (f + (a + fig)*

Exercise 2.20 Let X be a separated locally convex space and A,B,CcXbe

nonempty sets If C is bounded and A + C C B + C prove that A C coB

Exercise 2.21 Let (X, \\-\\) be a normed space, / G T(X) and L > Prove

the equivalence of the following statements:

(i) d o m / = X a n d V x , t / G X : \f(x) - f(y)\ < L \\x - y||; (ii) a GR, V i € X : / ( x ) < L ||x|| + a;

(iii) V M G X : / o o ( u ) < L | | « | |

Exercise 2.22 Let X be a locally convex space and / € T(X) Prove that

E x e r c i s e 2.23 Let X, Y be separated locally convex spaces, / € r ( X ) , F €

F(X x Y), x € dom / and A £ R, e £ R+ be such that [/ < A] and d£f(x) are

non-empty Prove that: [/ < A]oo = [foo < 0], ( ae/ ( x ) )0 0 = N{domf;x), (/*)«, =

Sdom/, foe = Sdom/* and {v* £ y * | (F*)oo(0,t;*) < 0} = ( P r y ( d o m F ) ) In particular, if {v* £ Y* | (F*)o o(0,u*) < 0} is a linear subspace then {0} and

Pry (dom F) are united

E x e r c i s e 2.24 Let X be a separated locally convex space and / £ I \ X ) Consider K := {u £ X | /«,(w) < 0} and X0 := K D (-.K") Prove that:

(i) if is a closed convex cone, Xo is a closed linear subspace and f(x + u) =

f(x) for all a: X and u Xo

(ii) l i n ( d o m / * ) = X 6L

(iii) The function / : X / X0 -»• I , f(x) := f(x), is well defined, fe r ( X / X0) ,

/OO(M) = /oo(«) for every u € X, f* = f*\x± and 5£/ ( £ ) = 9E/(a;)|Xj for all

x € X and £ R+, where x represents the class of x £ X

E x e r c i s e 2.25 Let X be a separated locally convex space and / £ T(X) Assume that there exists u £ X such that /<X>(M) < < fco(-u) Prove t h a t for every e > there exists fe £ T(X) such that f(x) < fc(x) < f(x) + s for every

x e X and argmin/e =

E x e r c i s e 2.26 Let X, Y be separated locally convex spaces and F A(X x

Y) Assume that F satisfies one of the conditions (ii)-(viii) of Theorem 2.7.1

Prove that the marginal function g : X* —¥ R, g(x") := infy*ey* F*(x*,y*)

is convex, iu*-lsc, the infimum is attained for every x* £ X* and goo(u") =

mmv*ey(F*)oo(u* ,v*) Moreover, {v* £ Y* \ (F*)oo(0, «*) < 0} is a linear

subspace

E x e r c i s e 2.27 (Toland-Singer duality formula) Let X be a separated locally convex space, / : X —>• R be a proper function and g £ T(X) Then

inf (/(*) - ff(i)) = inf (g\x') - f*{x')) •

E x e r c i s e 2.28 Let X be a separated locally convex space, / £ T(X) be such that € d o m / a n d p £ P Consider the following assertions:

(i) the mapping P t i - > t~pf(tx) is nondecreasing for every x £ d o m / ;

(ii) f'(x, —x) +pf(x) < for every x £ d o m / ; (iii) pf(x) < f'(x,x) for every x € d o m / ; (iv) {x,x*) >pf(x) for all (x,x*) € g r d / ;

E x e r c i s e 2.29 Let / : X —> IR be a function such that c o / is proper As-sume that for some x € d o m c o / we have that x = ^2i=i ^»^« a n (^ co/(ic) =

^ *= 1 ~\if{x~i) with Xi G d o m / , Ai > for i € ITfc and £ ) *= ^> = 1- Prove that

8c5/(3f) = n t i df(xi)

Exercise 2.30 Let X be a separated locally convex space and / € A(X)

Assume that G d o m / and / | x0 is continuous at 0, where Xo •— aff(dom/)

Prove that

f = f'(0,x) if x G X o , sup{(x,x*) | x * G < / ( ) W < f'(Q,x) = oo if x G X o \ _ X o ,

{ = / ' ( , x ) = oo if xeX\X0,

the supremum being attained for x G Xo Moreover, Xo is closed if and only if V x G X : / ' ( , x ) = s u p { ( x , x * ) | x* / ( ) }

E x e r c i s e 2.31 Let X be a separated locally convex space and / G A(X) be continuous on i n t ( d o m / ) , supposed to be nonempty Prove that for all x,y € i n t ( d o m / ) there exist z £]x,y[and z* G df{z) such that f(y)—f{x) = (y — x, z*) E x e r c i s e 2.32 Let X be a separated locally convex space and / G T(X)

(i) Consider the conditions: a) df is single valued on dom.3/, b) (df)~1(xl)!~)

(df)~1(x2) = for all distinct elements Xi,X2 G X*, c) / * is strictly convex on

every segment [xi,X2] C Imdf, d) dom df = i n t ( d o m / ) Prove that a) <=> b) •<=>• c); moreover, if i n t ( d o m / ) ^ and / is continuous on i n t ( d o m / ) then a) => d)

(ii) If / is continuous on i n t ( d o m / ) and ( / )_ 1 is single-valued on Imdf,

show that / is strictly convex on i n t ( d o m / )

E x e r c i s e 2.33 Let X, Y be separated locally convex spaces and / G A(X x Y) Prove that if / is continuous at (xo,j/o) £ d o m / , then

Prx* (df{x0, j/o)) = df(-,yo)(x0) and P r y ( d / ( x0, y0)) = d / ( x0,

-){yo)-Exercise 2.34 Let X be a separated locally convex space and / o , / i A(X)

Consider

v := inf{/0(x) | / i ( x ) < 0}, v* := sup inf (/o(x) + A/i(x)),

x>o xex

with the convention • oo = oo Suppose that » £ R Prove that

v* =v&[rfe>Q : inf{/i(x) | /0( x ) < y - e} > 0]

E x e r c i s e 2.36 Let (X, ||-||) be a normed linear space and / i , / : X -> R be proper convex functions Prove that there exist x* X* and a € R such that —/i < x" + a < J2 if and only if there exists M > such that fi(xi) + f2(%2) +

M\\xi - x21| > for all on, x2 € X

E x e r c i s e 2.37 Let X be a linear space, (Y, ||-||) be a normed linear space,

T : X -* Y be a linear operator, yo Y and / : I -> R be a proper convex

function Prove that f(x) + \\Tx + yo\\2 > for all x € X if and only if there

exists j / * € y * such that fix) -2(Tx + y0,y*) - ||y*||2 > for all x e X

E x e r c i s e 2.38 Let (X, ||||) be a normed vector space, C,D C X be closed convex cones and x G X, x* € X* Prove that

d(x,C) : = i n f { | | x - x | | | x € C} = m a x { - ( x , x * ) | x* Ux* n C+} ,

d ( x * , C+) = m i n { | | x * - x * | | | a;* C+} = s u p { - ( x , x * ) | x G C/x n C} , a,ndsupx€Uxncd(x,D) = supx,eUx,nD+ d(x* ,C+)

E x e r c i s e 2.39 Let / € A(X) and x € d o m / be such t h a t f(x) > inf/ Consider the sets

d<f(x) := {z* € X* | ( i - x,x*) < f(x) - /(a?) Vx e [/ < / ( * ) ] } ,

d ^ / ( x ) := {x* a ' l f i - f , x*) < f{x) - / ( J ) Vx € [/ < / ( * ) ] } Prove that dKf{x) = d-f(x) = [1, oo[-d/(x) The set a < / ( x ) is Plastria's

sub-differential of / at x

E x e r c i s e 2.40 Let X be a normed space, (an)n>i C X and (An)„>i C [0,oo[

be such that ^ ^ L j An = Consider the function

/ : X - > R , f{x) = Y°° A„||x-a„||2

*•—'n=l

1) Prove that the following statements are equivalent: (a) S^Li-^i|la"ll2 <

oo (i.e € d o m / ) , (b) d o m / ^ 0, (c) d o m / = X

2) Prove that / is finite, convex and continuous when 5Z^Li^illa"l|2 <

°°-3) Suppose that ]C!!!Li^n|lan||2 < °°- Prove that for every x £ X one has 8f{x) = { V0 An||x - On|| xmn I x*n e a|| • IKi - a„) Vn € N )

E x e r c i s e 2.41 Let X be a normed space and / T(X) Prove t h a t the following statements are equivalent:

(a) lim||J.||_too f{x) = 00;

(b) [/ < A] is a bounded set for every A > inf^ex / ( x ) ;

(d) there exists a, £ R, a > 0, such that f(x) > a||x|| + /? for every x G X ; (e) liminf|)x||_foo / ( x ) / | | x | | > 0;

(f) £ i n t ( d o m / * )

Moreover, if dim X < oo then the conditions above are equivalent to (c') [/ < iaf / ] is nonempty and bounded

Furthermore, if p, q £ ]l, oo[ are such that 1/p + 1/q = 1, the following state-ments are equivalent:

(g) l i m i n f |N H o o/ ( x ) / | | x | |p> ;

(h) l i m s u p| | : c,| H o o r(x")/\\x*\\q < oo;

(i) there exists a, /3 £ R, a > 0, such that f(x) > a||x||p + P for every i X ;

(j) there exists a,/3 £ R, a > 0, such that /*(x*) < cx||a;*||9 + /? for every

a;* e X *

E x e r c i s e 2.42 Let f,fn : Rm ->• R (n £ N) be convex functions such that (fn{x)) -¥ f{x) for every x £ Rm Assume that / is coercive Prove that there

exist a,P £ R with a > 0, no £ N such that /n( x ) > a \\x\\ + /3 for all x G Rm

and n > no

Exercise 2.43 Let (X, ||||) be a n.v.s and C C X be a nonempty closed convex

set Consider the following assertions:

(i) there exists XQ G X* such that (x,x*,) > \\x\\ for every x G C;

(ii) there exist xo £ -X", X*, £ X* such that (x — xo, x*>) > ||x — xo|| for every

x eC;

(iii) int(dom s c ) 7^ 0;

(iv) a) there exists x*, G X* such that {u, x*>) > for every u £ Coo \ {0} and b) for every sequence (xn) C C with (||xn||) —• oo and ( | | x „ | |_ 1 xn) -^ u one has

Prove that (i) => (ii) =S> (iii) => (iv) If £ C then (ii) =>• (i) Moreover, if X is a reflexive Banach space then (iv) =>• (ii)

E x e r c i s e 2.44 Let X be a normed space and / £ A(X) be lower bounded Consider A > i n fl €x / ( x ) =: inf / and p > We envisage the conditions:

(a) [f<X}CpU;

(b) V x £ X : f(x) > inf / + A ~l n f f • max{0, l|a?l| - p}; 2p

(c) V x ' G A"2' °f /t / x » : / * ( * ' ) < / * ( ) + p||x'||

Prove that (a) =>• (b) & (c)

Exercise 2.45 Let X be a normed space and / G T(X) Prove the following

(a) liminf||.1.||_+00 / ( x ) / | | x | | > - » £ int(dom/*); in this case f(x)

liminf 4f-if = sup{/i > | /* is upper bounded on (J.U*}

||x||-KX> ||X||

(b) liminf | |B|H o o f(x)/\\x\\ = « B d ( d o m / * )

(c) liminfHJII-KX, f(x)/\\x\\ < O ^ cl(dom/*); in this case

f(x)

liminf ~-r{- = - dd o m/ (0)

llxH-KX) | | Z | |

E x e r c i s e 2.46 Let p ]1, oo[ and

f:£p^R, / ( ( * „ ) „ > i ) := Y°° n\xn\n

Prove that / is finite, convex, continuous and lim||„||_+0O f*(y)/\\y\\ —

E x e r c i s e 2.47 Let X, Y be Hilbert spaces, C C X be a nonempty closed convex set and A G L(X,Y) be a surjective operator Consider the problem ( P ) i | | A c | |2, xeC

A solution x of problem (P) is called a spline function in C associated to A (a) Prove that (P) has optimal solutions if C + ker ^4 is closed

(b) Prove that x G C is an optimal solution of (P) if and only if y := A* (Ax) satisfies the relation (x\y) = min{(x \y) \ x € C}

E x e r c i s e 2.48 Let X,Y be Hilbert spaces, C := {x X | Vi, < i < k :

(x\a,i) < fii}, where a i , , a j b G -X', Pi, ,Pk G R and A G £ ( X , F ) be a

surjective operator Consider the problem (P) | | A E | |2, x G C

Prove that (P) has optimal solutions Suppose, moreover, that there exists x € X such that (x \ a{) < /3i for every i, < i < k Prove that l i s a solution of (P) if and only if there exists (\i)i<i<k C R+ such that

A*(Ax) = A i d H V Xkdk a n d V i , < i < A: : A;((x| a») — /?;) =

E x e r c i s e 2.49 Let X be a Hilbert space and 01,02,03 be three non colinear elements of X (i.e they are not situated on the same straight-line) Prove t h a t there exists a unique element x G X such that

V i £ l : | | x - O i | | + | | x - a | | + | | i - a3| | < ||x - a i | | + ||x - o2|| + ||x - a3||

E x e r c i s e 2.50 Determine the optimal solutions (when they exist) and the value of the problem

( P f ) £ (tx{t) + fiy/1 + (u(t))2) dt, x e Xi,

for every fj, € R+ and every i g {1, 2, 3,4}, where

X i : = C [ , l ] , Xa: = / ( , ) ,

X3 := | x e C [ , l ] | /o 1x ( t ) d t = J , X» := { x G 1/(0,1) | /^ x ( t ) d t = o }

E x e r c i s e 2.51 Consider the (convex) optimization problem

(P) max J,,1 x(t) dt, xeX, x(0) = x(l) = 0, J,,1 y/l +{x'(t))2dt < L,

where L > and X = C^O, 1] := {x : [0,1] —• R | x derivable, x continuous on [0,1]} or X = AC[0,1] := {x : [0,1] -¥ R | x absolutely continuous} Determine the optimal solutions of problem ( P ) , when they exist, and its value (using, eventually, the dual problem)

2 B i b l i o g r a p h i c a l N o t e s

Many results of this chapter are well-known and can be found in several books treating convex analysis: [Rockafellar (1970); Hiriart-Urruty and Lemarechal (1993)] (for finite dimensional spaces), [Ekeland and Temam (1974); Ioffe and Tikhomirov (1974); Barbu and Precupanu (1986); Castaing and Valadier (1977); Phelps (1989); Aze (1997)] In the sequel we point only those results that are not contained in these books but the last one

Sections 2.1-2.5: The assertion (vii) of Theorem 2.1.5 was established in

(1982) in the general case; one can find another proof in [Aze (1997)] The local boundedness of df in Theorem 2.4.13 and Corollary 2.4.10 can be found in many books on convex analysis The assertions (iv) and (vii) of Theorem 2.4.14 are established in [Zalinescu (1980)]; for (i) see also [Anger and Lembcke (1974)] Theorem 2.4.18 is from the book [Ioffe and Tikhomirov (1974)] Theorem 2.5.2 was proved by Polyak (1966) under the more stringent conditions that / is a quasi-convex function which is bounded and Lipschitz on bounded sets and at-tains its infimum on every closed convex subset of X Theorem 2.5.5 and Lemma 2.5.3 are established by Borwein and Kortezov (2001), while Lemma 2.5.4 and other results on non-attaining convex functionals are established by Adly et al (2001a)

Sections 2.6-2.9: The systematic use of perturbation functions for

cal-culating conjugates and subdifferentials was done for the first time by Rock-afellar (1974) The author of this book continued this approach in [Zalinescu (1983a); Zalinescu (1987); Zalinescu (1989); Zalinescu (1992a); Zalinescu (1992b); Zalinescu (1999)]; this permitted, for example, to give simpler proofs to several results stated in [Kutateladze (1977); Kutateladze (1979a); Kutateladze (1979b)] Theorem 2.6.2(i) was established by Moussaoui and Seeger (1994), but Theorem 2.6.2(h) and Theorem 2.6.3 are new Corollaries 2.6.4-2.6.7 can be found in [Hiriart-Urruty and Phelps (1993)] and [Moussaoui and Seeger (1994)]; for other results in this direction see the survey paper [Hiriart-Urruty et al (1995)]

The sufficient conditions for the fundamental duality formula and for the va-lidity of the formulas for conjugates and e-subdifferentials are, mainly, those from author's survey paper [Zalinescu (1999)], where one can find detailed historical notes; here we mention only the first use (to our knowledge) of them; actually, all these sufficient conditions are mentioned in that paper, excepting those which use li-convex or lcs-closed functions So, conditions (iii) and (viii) of Theorem 2.7.1 and the corresponding ones in Theorems 2.8.1, 2.8.3, 2.8.7 and 2.8.10 are the classical ones and can be found in all the mentioned books which treat them Condition (i) of Theorem 2.7.1 was used by Rockafellar (1974) when Yo = Y and by Zalinescu (1998) in the present form (see also [Combari et al (1999)]), (ii) and (vi) were introduced in [Zalinescu (1983a)] for Yo = Y (and R replaced by a separated lcs ordered by a normal cone) and in [Zalinescu (1999)] in the present form, (iv) was introduced in [Zalinescu (1992b)] with (Hwi) replaced by (Hx), (v) is stated by Amara and Ciligot-Travain (1999) for X, Y Frechet spaces, $ a lcs-closed function and ,b replaced by lc, (vii) was introduced by Rockafellar

(1974) for X, Y Banach spaces (X even reflexive) and Yo = Y, and by Zalinescu (1987) in the present form, while (ix) was used by Joly and Laurent (1971) for $ lsc and by Moussaoui and Voile (1997) for arbitrary 3>; note that condition (b) of [Cominetti (1994), Th 2.2] implies condition (2.54) Theorem 2.7.4(iv) was established in [Zalinescu (1992b)]; the other conditions were introduced in [Zalinescu (1999)] (but in (v) it was assumed that F is lsc and C is closed)

intro-duced in [Zalinescu (1992b)], (i) and (viii) in [Zalinescu (1998)] for X,Y normed spaces, while conditions (ii) and (vi) were introduced in [Zalinescu (1999)]

Condition (vii) of Theorem 2.8.3 was introduced by Borwein (1983) for Yo = Y (see also [Zalinescu (1986)]), (x) was introduced by Borwein and Lewis (1992), (ix) was introduced by Moussaoui and Voile (1997), (i) by Zalinescu (1998) for X, Y normed spaces, (ii) was introduced by Combari et al (1999), (iv) was introduced by Zalinescu (1999) (a slightly stronger form was used in [Simons (1990)]) as well as conditions (v) and (vi)

For C G L(X, Y), generally, Theorem 2.8.6 is obtained under the correspond-ing conditions of Theorem 2.8.3 (takcorrespond-ing / = 0) For C a densely defined closed linear operator, Rockafellar (1974) and Hiriart-Urruty Hiriart-Urruty (1982) ob-tained the results for g lsc, X, Y Banach spaces and Yo = Y (in [Rockafellar (1974)] X is reflexive and e = 0); Aze (1994) obtained the formula for the conju-gate under condition (ii) in normed spaces For general C condition (iv) was used by Zalinescu (1992b); the other conditions (excepting (v)) were used in [Zalinescu (1999)]

Condition (ix) of Theorem 2.8.7 was used by Joly and Laurent (1971), (vii) for X a Banach space was introduced by Attouch and Brezis (1986), (vi) was in-troduced by Zalinescu (1992b), (i) was inin-troduced by Aze (1994) in an equivalent form in normed spaces, while (ii) was introduced by Combari et al (1999)

Corollary 2.8.8 was obtained by Voile (1994) for X a Banach space, / , g lsc and '* replaced by !C Formula (2.65) from Proposition 2.8.9 was obtained by Aze

(1994) and Simons (1998b) for x = € ( d o m / — domg)1 when / , g are lsc

The case / = of Theorem 2.8.10 was considered by several authors; condi-tion (iii) was used in [Hiriart-Urruty (1982); Lemaire (1985); Zalinescu (1983a); Zalinescu (1984)]; the algebraic case was considered in [Kutateladze (1979a); Kutateladze (1979b); Zalinescu (1983a)]; Zalinescu (1984) considered a stronger version of (v) (for lsc functions and Yo = Y); note that in these papers g is as-sumed to be Q-increasing on the entire space The general case was considered in [Combari et al (1994)] under (iii) and a condition stronger than (v) (/, g, H lsc and lc instead of lb) and in [Combari et al (1999)] under condition (i) and

a variant of (iv); it was also considered by Moussaoui and Voile (1997) under condition (vii)

The formula for d(p(x) in Corollary 2.8.11 can be found in [Combari et al (1994)] The formula for d<p(x) in Corollary 2.8.13 is stated in [Voile (1992); Voile (1993)] under condition (vii) of Theorem 2.8.7 and in [Combari et al (1994)] for / , g lower semicontinuous (even for an arbitrary finite number of functions) and (0,0) £ ic({(x,x) \ x € X} - d o m / x domg) The formula for (/1O/2)* in

Corollary 2.8.14 is stated by Seeger and Voile (1995); the formulas for d(fi§fi)(x) are stated in this paper for /1 and /a continuous at x\ and X2, respectively

Section 2.10: The minimax theorem is also classical and can be found in

the books [Barbu and Precupanu (1986); Simons (1998a)]

E x e r c i s e s : Exercise 2.2 is from [Penot and Bougeard (1988)], Exercise 2.3 is from [Crouzeix (1981)], Exercise 2.6 is from [Marti (1977)], Exercise 2.15 is from [Crouzeix et al (1992)] (there in a more complete form), Exercises 2.18 and 2.29 are from [Hiriart-Urruty and Lemarechal (1993)], Exercise 2.19 is from [Lions and Rochet (1986)], Exercise 2.20 is the celebrated cancellation lemma from [Hormander (1955)], Exercise 2.21 is from [Hiriart-Urruty (1998)], Exercise 2.22 is from [Jourani (2000)], the assertions in Exercise 2.23 are well-known (see also [Aze (1997)]), Exercise 2.25 can be found in [Borwein and Kortezov (2001)] (in normed spaces), the formula for goo under condition (vii) of Theorem 2.8.1 in Exercise 2.26 is obtained in [Amara and Ciligot-Travain (1999)] and [Amara (1998)], Exercise 2.27 is the well-known Toland-Singer duality formula (see [Toland (1978)] and [Singer (1979)]), Exercise 2.30 is from [Zalinescu (1999)], the equivalence of a) and c) (for Banach spaces) in Exercise 2.32 can be found in [Bauschke et al (2001)] (see also [Barbu and Da Prato (1985)]), Exercises 2.33 and 2.34 are from [Eremin and Astafiev (1976)], Exercises 2.35, 2.36 and 2.37 are from [Simons (1998a)], the last formula in Exercise 2.38 is from [Walkup and Wets (1967)], Exercise 2.39 is from [Penot (1998a)], the formula for the subdifferential of the function considered in Exercise 2.40 and its proof are from [Aussel et al (1995)], the equivalences (e) •£> (f) and (g) & (h) of Exercise 2.41 are from [Zalinescu (1983b)], the equivalence of conditions (ii)-(iv) in Exercise 2.43 are established in [Adly et al (2001b)] in reflexive Banach spaces, a weaker variant of the implication (a) => (c) of Exercise 2.44 is stated in [Aze and Rahmouni (1996)], Exercise 2.45(c) is from [Borwein and Vanderwerff (1995)], Exercises 2.47, 2.48 axe from [Laurent

Some Results and Applications of Convex Analysis in Normed Spaces

3.1 Further Fundamental Results in Convex Analysis

Throughout this chapter X, Y are normed spaces and X*,Y* are their duals endowed with the dual norms

As an application of Ekeland's variational principle and of some results relative to convex functions, we state the following multi-propose general-ization of the Br0ndsted-Rockafellar theorem

Theorem 3.1.1 (Borwein) Let X be a Banach space and f £ r ( X )

Consider e £ P, XQ £ d o m / , XQ £ def{xa) and /3 € K+ Then there exist

xe £ X, y* € Ux- and Ae £ [—1, +1] such that

\\xe - Xo\\ + P • \(X£ - X0,X*0)\ < y/E, (3.1)

x% := x*0 + VE(y* + /3\ex*) £ df{xe) (3.2)

Moreover

||*e - soil < Ve, \te-xl\\<y/i(\ + P\\xl\\), (3.3)

\(xe-x0,x*)\<e + JilP, (3.4)

x*e£d2ef(x0), \f(xe)-f(x0)\<e + ^/fi, (3.5)

with the convention 1/0 = oo

Proof The function

|| • Ho • x - • K, \\x\\0~\\x\\ + p-\(x,x*)\,

is, obviously, a norm on X, equivalent to the initial norm Therefore

{X, ||-||0) is a Banach space Consider the function g := / — XQ It is obvious

that Xo £ dom<? = d o m / = dom<9£/ and g is lsc and lower bounded:

WxGX : g{x)>g(x0)-e [& x*0 £ dsf{x0)]

We apply Ekeland's theorem (Theorem 1.4.1) to g, y/e and the metric d given by d(x,y) := \\x — y\\o So there exists xe £ domg such that

g{xE) + y/e- \\x£ -x0\\o < g(x0), (3.6)

\/x£X, x^xe : g(x£) < g(x) + y/e • \\x - xe\\0 (3.7)

From Eq (3.6) we obtain that

g(xe) + Ve(\\xs - ar0|| +0-\{xe- x0, XQ)\) < g(x0) < g(xe) + e,

whence Eq (3.1) follows immediately Let us consider the function h : X -> E,

h(x) := g(x)+y/e-\\x-xE\\0 - f{x)-{x,xl)+y/E-\\x-xE\\+Py/e-\(x-xe,xl)\;

by Eq (3.7) xe is a minimum point of h Therefore £ dh(xe) Since

h is the sum of four convex functions, three of them being continuous,

and taking into account the expression of the subdifferential of a norm (Corollary 2.4.16) and of the absolute value of a linear functional (at the end of Section 2.8), we have that

0 £ dh(xe) = df(x£) - x* + y/i • Ux- + PVe • [ - , +1] • x%

Therefore there exists x* £ df(xs), y* £ Ux* and Xe £ [—1,1] such that

Eq (3.2) is verified

The estimations from Eq (3.3) follow immediately from Eqs (3.1) and (3.2) Using again Eqs (3.1) and (3.2) we obtain that

|(a;e -x0,x* -XQ)\ = y/e-\{xE - xQ,y* + p\Ex%}\

< v/i(||ze - x0\\ • W\ + 0\Xe\ -\(xe- x0,x*0)\)

< y/e(\\xs - x0\\ + fi • \{x£ -x0,x*0)\)

<y/e-y/e = e (3.8)

From Eqs (3.1) and (3.8) we get

i.e the estimation in Eq (3.4) holds, too Since x$ G dEf(xo) and x* G

df(xE), we get

(X0 ~Xe,X*£ -XQ) + (X0 -X£,X*Q) = (X0 ~Xe,X*) < f(x0) ~f(Xe)

< (X0 -XE,XQ) +£

Using relations (3.1) and (3.8), we obtain that

\f(x0) - f(xe)\ < \(x0-xe,x*0)\+s<e + VE//3,

i.e the second relation in (3.5) holds Using Eq (3.8) and the fact that XQ G dEf(x0), x* G df(xE), we obtain that

(x-X0,X*e) = (X-X£,X*) + (Xe -X0,X* -XQ) + (XE -X0,XQ)

< f{x) - f(xe) +s + f(x£) - f{x0) + e

- f(x) - f(x0) + 2e

for every x G X, i.e x£ G d2ef(x0); hence the first relation in Eq (3.5)

holds, too • The next result is well-known The first part is an immediate

conse-quence of Borwein's theorem, while the density part, which follows easily from the first one, will be reinforced in Theorem 3.1.4 below

Theorem 3.1.2 (Br0ndsted-Rockafellar) Let X be a Banach space and

f € r ( ^ ) - Consider e > and (XO,XQ) G grd6f Then there exists

(x£,x*) G gr<9/ such that \\x£— XQ\\ < ^Jl and \\X*—XQ\\ < y/e Inparticular

domf C cl(dom<9/) and d o m / * C cl(Im<9/) D

Another consequence of Borwein's theorem is the following result which will be completed in Proposition 3.1.10 below

Proposition 3.1.3 Let X be a Banach space and f G T(X) Then

f(x) = s u p ^ z - z,z*) + f(z) I (z,z*) G grdf}

= sup{(x, z*) - /•(**) | z" G lm(df)} (3.9) for every x G d o m /

Proof The second equality in Eq (3.9) is obvious because f(z)+f*(z*) =

{z, z*) for any (z, z*) G grdf Let x G dom / be fixed When (z, z*) G grdf

we have (a; — z, z*) + f(z) < f(x) for every i £ l ; hence f(x) > sup{(a;

exists x* € de/2f(x) Applying Borwein's theorem we get {xe,x*) £ grdf

such that x* def(x) Therefore (xe — x,x*) < f{xe) - f(x) + e, whence

f(x) — e < {x — xe,x*) + f(xe) Hence relation (3.9) holds •

As announced before, the density results in Theorem 3.1.2 can be stated in a stronger form In the next result (xn) —>/ x means that (a;n) —> x and

{f(xn)) -* f(x), and similarly for (xn) ->•/• x*

Theorem 3.1.4 Let X be a Banach space and f € r ( X ) Then

(i) Vx e d o m / , 3((xn,xn)) Cgrdf : (a;„) -+/ x;

(ii) Vx* E dom/*, ( ( x „ , < ) ) C gvdf : ô ) - ã , x*

Proof, (i) Consider x € d o m / Because / is lsc, by Theorem 2.4.4

(hi), for every n S N there exists x* € dn-2f(x) Applying Borwein's

theorem for (x,x*), e = n~2 and /? = 1, we get (xn,xn) € grdf such that

\\xn — x\\ < n_ 1, \f(xn) — f(x)\ < n~2 + n~x Hence (xn) - ^ / x

(ii) Because / * is lsc, it is sufficient to show that for every e > there exists (y, y*) e df such that \\y* - x*\\ < e a n d / * ( y * ) < f*(x*)+e Fixx £ d o m / Let £> and take r := \\x\\ + 2e-1 (e + f(x) + f*(x*) - (x,x*)) >

0 Consider also the function g := fO(x* + irux) € A(X) Then g(x) >

(fnx*)(x) = (x,x*) - f*(x*) for every x € X It follows that g € T(X)

and f(x) + (—x,x*) > g~(0) > —f*(x*) By Proposition 3.1.3 there exists

(z,z*) € dg such that -g*(z*) = g(z) - (z,z*) > -f*(x*) - e/2 But

g*(z*) = f*(z*) + (x* + trUx)*(z*) = f*(z*) + r l|ar* - z*\\ Therefore

f*(z*) + r\\x*-z*\\<r(x*)+e/2

Because f*(z*) > (x,z*) - f{x) > - \\x\\ • \\z* - x*\\ + (x,x*) - f(x), from the preceding inequality we obtain that (r — \\x\\) \\z* — x*\\ < e + f(x) +

f*(x*) — (x,x*), and so \\z* - x*\\ < e/2 The inequality above shows also

that f*(z*) < f*(x*)+e/2 On the other hand, because (z,g~(z)) € cl(epi^), there exists (zn) C dom / and (un) C rUx such that (zn + un) —> z and

l i m s u p ( / ( zn) + (u„,a;*» <g(z) = (z,z*) - g*(z*)

= (z, z*) - f*{z*) - r ||ar* - z*|| (3.10)

Setting en := f(z„) + f*{z*) — (zn,z*), we have that

0 < £„ = f{Zn) + (U„, X*) - (Zn + Un, Z*) + (Un, Z* - X*) + f*(z*)

Taking into account that (un,z* — x*) < r \\x* — z*\\, from Eq (3.10) we

is (zn) Therefore there exists r' > such that (zn) c r'Ux- Take n E N

so that S := en < and VS(r' + 2)(1 + \\z*\\) < e/2; set z := zn Of

course, we have that z* G dsf(z) By Borwein's theorem (for /? = 1) there exists (y,y*) G df such that \\z - y\\ < y/S, \\z* - y*\\ < VS{1 + ||z*||) and |/(«) ~ f(v)\ <S + VS Hence ||j/* - x*\\ < y/S{l + \\z*\\) + e/2 < e and

/*(»*) = <V,V*>-/(V)

= (y,y* - z*) + (y-z,z*)-6 + f*(z*) + f{z) - f(y)

< VS(V5 + r') (1 + ||z*||) + \f5 \\z*\\ -5 + f(x*) +s/2 + + V6 < f*(x*) + e/2 + VS(r' + 2)(1 + ||«*||) < f*(x*) + e

The proof is complete • Using Br0ndsted-Rockafellar's theorem we can add other sufficient

con-ditions for the validity of the conclusions of Theorems 2.8.1 and 2.8.7

Proposition 3.1.5 Let X,Y be Banach spaces, F £ T(X x Y), A G

L(X,Y), D = {Ax - y \ (x,y) € d o m F } and E = {Ax - y \ (x,y) €

dom&F} Then

icE = iiE = ic(coE) = ri(coE) = icD = riD

Moreover, if one of the above sets is nonempty then icE = E — D; in

particular the sets ICE and E are convex

Proof Consider the linear operator T : X xY ->• Y defined by T(x, y) :=

Ax — y Since domdF C dom.F and d o m F is convex, we have that E = T(domdF) CcoE = T(co{domdF)) CD = T ( d o m F )

By Theorem 3.1.2 we have that d o m F C domdF Hence, using the conti-nuity of T, we have that

E C D C T ( d o m d F ) C T ( d o m d F ) = E

Using the properties of the affine hull (see p 2), it follows that aff E C aff D c aff E Therefore, if aff E is closed then aff D — aff E; the above relation shows that %CE c %CD Let us show the converse inclusion Let

2/o G %CD and consider the function F0 T(X x Y), F0(x,y) :— F(x,y —

yo)-Of course, domF0 = d o m F + (0,j/o) and dom9F0 = d o m F + {0,y0)

Therefore € i c( P ry( d o m F0) ) Taking <p0(x) := F0(x,Ax), <p0 G T(X)

such that dF0(x,Ax) = dF(x,Ax — y0) ^ Therefore yo € E So we

obtained that *D = ICD C E C D, which shows that aff E = aff-D and

*D C iE It follows that icE = icD = ic(coE) As observed after the

proof of Theorem 2.8.1, in our situation, if %CD is nonempty we have that icD = rintD

Suppose that %CD 9^ (for example) From what was proved above, we

have that

rinti? = lD = icD = rintE = lE = icE C E C D C E

Hence E~ = T°E = D: The conclusion follows •

Remark 3.1.1 Taking into account the preceding proposition, for X, Y

Banach spaces and F G T(X x ) , we may add to the sufficient conditions in Theorem 2.8.1 the following conditions:

0 G ri{Ax - y \ (x,y) £ domdF}, £ ic{Ax -y\(x,y)€ domdF},

0 € ic{Ax -y\(x,y)e co(domdF)},

y0 = cone{ Ar - y | (x, y) dom OF} is a closed linear space

Indeed, the first three conditions are, evidently (using the preceding proposition), equivalent to £ lc{Ax — y \ (x,y) £ d o m F } In the fourth

case y0 = aff {Ax — y\(x,y)e co(dom OF)}, and so e lc{Ax — y\(x,y)e

domF}

An important particular case of the preceding proposition is when X =

Y, A = Idx and F(x,y) = f(x) + g(y) with f,g T(X); in this case D — d o m / — domg and E = domdf — domdg

Using the Borwein theorem one obtains a formula for the subdifferential of a composition g o A of a lsc convex function and a continuous linear operator without qualification conditions

Theorem 3.1.6 Let X,Y be Banach spaces, A e C{X,Y), g £ T(Y),

f = g o A and x d o m / Then x* df(x) if and only if there exists a net ((yi,yt))ieI C g r d s such that (yt) ->• y := Ax, (g(yi)) ->• g{y),

((yi-y,y*))^Oand(A*y*)^x*

Proof Let (fa,y*))i€l C gidg be such that fa) -> y, (fa - y,y*)) ->•

0, {gfa)) -> g(y) and (i4'i/r) ^ x* Then (y - yhy*) < gfa-gfa) for all

i G I and t / S F It follows that

(x' - x,A*y*) + (y- yhy*) = (Ax' - yuy*) < f(x') - gfa)

for a l H G / and x' G X Taking the limit we obtain that (x' — x,x*) < / ( x ' ) - f(x) for all x' G X, and so x* G d / ( x )

Let now x* G df(x) and consider (£„)„6N 4- 0- Using Corollary 2.6.5 we have that x* G w*-clA*(d£ng(y)) for every n G N Let A/" be a base of

w*-neighborhoods of x* and consider / = Nx M with (n, V) y (n1, V) iff n > n'

and V CV Then for all i = (n, V) G / there exists z* G dEng(y) such that

A*£* G V Taking /? = and e = e^ in Theorem 3.1.1, there exists (yi,y*) G 3g such that \\yi - T/|| < e„, ||y* - ^ | | < en, \g(yt) - g(y)\ < en(en + l) and

\(Vi ~ V,V*)\ < en(en + 1) Because \\A*z* - A*y*\\ < \\A*\\ • \\y* - z*\\ <

en \\A*\\ and (A*z*)ieI ^> x*, we obtain that (A*y*)i€l ^> x*

If X is reflexive then w*-cli*(9fBg(j/)) = cl A* (dSng(y)), and so, for

every n G N, we can take z*n G dEng(y) such that ||A*z* — x*|| < e„ The

conclusion follows • Using the preceding result one obtains a similar formula for the

sub-differential of the sum of two lsc proper convex functions

Theorem 3.1.7 Let X be a Banach space, / i , / G T(X) and x G

dom / i fl dom /2 Then x* e S f / i + M (x) if and only if there exist two nets

{(xk,i>xt,i))ieI C gr<9/fc, k = l,2, such that ( xM)i e / ->• x, (fk(xk,i))ieI ->

/fc(x), ( ( xM - x , x ^i) ) e / -^0 for k = 1,2 and (x*^ + x*2<i) £ / ^>x*

7/X is reflexive one can take sequences instead of nets and impose norm

convergence instead of w* -convergence

Proof We apply the preceding result iorY ~ XxX, A: X -> Y defined

by Ax := (x,x) and# : Y -> E defined by #(xi,x2) := / i ( x i ) + / ( x ) Then

/ := / + / = g°A The sufficiency is immediate by taking y = (x, x) = Ax,

Vi = (xi,i,X2,i) and y* ~ (xl^xlj) for i G i" Let x* G df(x) By Theorem

3.1.6 there exists a net ((yi,y*))ieI C gvdg verifying the conditions of the

theorem with y := (x,x) Taking yt = (xi)j,x2,i) and y* = ( x ' ^ x ^ j )

and taking into account that dg(x\,X2) — dfi(x{) x 0f2(x2), we have that

((xk,i,x*ki))ieI C gvdfk and (xk,i)iei -»• x for k = 1,2 Assume that

there exists S > such that J := {i £ I \ fi(xiti) — fi(x) > 5} is

co-final It follows that g(yi) - g(y) > + (f2{x2,i) — f2{x)) for all i £ J Taking the limit inferior, we obtain the contradiction > S Therefore

{fk(%k,i))ieI -^ fk(x) for k — 1,2 The inequalities

fi(xi,i) - fi(x) < (xij -x,x{A) < (yi -y,y*) - {h(x2,i) - f2(x))

for i £ I imply that ({xij — x,x{ti}) -> D

Using Br0ndsted-Rockafellar theorem we obtain another famous result

Theorem 3.1.8 (Bishop-Phelps) Let X be a Banach space and C C X,

C 7^ X, be a nonempty closed convex set Then

(i) The set of support points of C is dense in B d C

(ii) The set of support functionals of C is dense in the set of continuous

linear functionals which are bounded above on C Moreover, ifC is bounded, then the set of support functionals of C is dense in X*

Proof, (i) Let x0 Bd C and e £ ]0,1[ Taking into account that C ^ X,

there exists xi £ X \C such that \\xi — x0|| < £2- Applying a separation

theorem, there exists XQ £ Sx> such that SVLPX£C{X,XQ) < (XI,XQ) SO, for

every x £ C,

{x - X0,XQ) = (x - XI,XQ) + (xi - X0,XQ) < (x - xi,xl) + \\x\ - x0\\ < e2,

whence XQ £ de2tc(xo)- Applying the Br0ndsted-Rockafellar theorem we

have that

3xe £ C, 3a;* £ dbc{xe) : \\x£ - x0\\ < e, \\x* - XQ\\ < e <

Since \\XQ\\ = 1, we have that x*e ^ 0, and so xE is a support point of C

with \\x£ — xo\\ < e The conclusion holds

(ii) Using the second part of Theorem 3.1.2 we have that

dom(tc)* C cKJmdic) = c l ( I m d tc \ {0}),

which shows that the conclusion of the theorem is true •

Theorem 3.1.9 (Simons) Let X be a Banach space, f € T(X) and x €

X, n G R be such that inf / < 77 < f(x) Consider

L := sup -fizz rr,

x€X\{x} ||ar — ar|| and dx • X -V R, dj(x) := | | i — x|| Then:

(i) < -L < oo and inf ( / + Ldx) > n;

(ii) V e e ] , l [ , i / X : ( / + L«fc)(i/) < i n f ( / + i d * ) + e L | | i - y\\; (iii) V e € ] , l [ , (ô,*ã) G g r d / : {x - z,z*) > (I - e)L\\x - z\\ >

and\\z*\\ < (l + s)L;

(iv) Ve €]0,1[, 3(z,z*) € g r S / : (z - z,z*> + / ( * ) > 17 and L <

\\z*\\<(l + e)L

Proof, (i) Because 77 > inf / , it is clear that L > Since 77 < /(a;) and / is lsc at x, there exists p > such that / ( x ) > 77 for every a; € B(x,p) Furthermore, since / € r ( X ) , there exist x* € X* and a R such that

f(x) > (a;!a;*) ~ Q f°r every a; £ X Therefore

77 — f(x) <n — {x,x*) + a < 77 + \\x — x\\ • \\x*\\ — (x,x*) + a

for every x £ X with ||a; — a?|| > p Let := max{0,r\ + a — (x, x*)} So,

V* £ X, \\x - x\\ > p : 5_jlZg) < ||x*|| + _ ] _ < li^H +

IF ~ x\\ \\x ~ x\\ P

This relation and the choice of p show that L < 00 From the expression of L we obtain immediately that inf ( / + Ldx) > 77

(ii) Let e e]0,1[ Since (1 - e)L < L, from the definition of L there exists y £ X, y ^ x, such that (77 — f{y))/\\x — y\\ > (1 — e)L, and so

( / + Ldx)(y) < 77 + eLdi(y) < inf(/ + Ldx) + eL\\y - x\\

(iii) Let e €]0,1[ be fixed The function / + Ldx 1S proper, lsc and

bounded from below, while the element y from (ii) is in d o m ( / + Ldx) = d o m / Taking the metric d o n X defined by d(xi,X2) := L\\xi—X2\\, (X,d) is a complete metric space Using Ekeland's variational principle we get the existence of z € X such that

and

(f + Ldw)(z) < {f + Ld¥)(x) + eL\\x - z\\ VxeX

The first relation and (ii) give ||^ — yj| < \\x — y\\, and so z ^ I The second relation says that z is a minimum point of the function / + Ld^ + sLdz

Taking into account that cfe- and dz are continuous convex functions, it

follows that

0 G d(f + Ldw + eLdz)(z) = df(z) + Ldd^{z) + eLddz(z)

But ddz{z) = d\\ • ||(0) = Ux- and dd^(z) = {x* € Ux* | (z - x,x*) =

\\z — x\\} Hence there exist z* G df(z) and x*, y* £ Ux* such that z* = Lx* + sLy* and (x - z, x*) — \\x - z\\ Therefore ||z*|| < (1 + e)L and

(x — z, z*) = {x — z, Lx* + eLy*) = L\\x — z\\ + eL(x — z, y*) > L\\x - z\\ - eL\\x - z\\ = L(l - e)\\x - z\\ >

(iv) Let e £]0,1[ be fixed and consider e' := e/3 Let us take M := (1 + 2e')L Since / + Md% > f + Ld% > r], we can apply Ekeland's theorem for / + Md-x, an element xo of dom / , e' > and the metric defined at (iii) We get so the existence of z e X such that

V i e l : (f + Mtk)(z)<(f + Mds)(x)+e'L\\x-z\\

As in the proof of (iii), there exist z* € df(z), x*, y* € Ux* s u ch that

z* = Mx* +e'Ly* and (x-z,x*) = \\x - z\\ Thus ||z*|| < (1 + s)L and (x - z, z*) > (M - e'L)\\x - z\\ = (1 + e'L)\\x - z\\

So, for x = z we have that (x — z, z*) + f(z) = f(x) > rj, while for ~x yi z we have

(x-z,z*) + f(z) > (l + e')L\\x-z\\ + f(z) = (f + Ldw)(z)+e'L\\x-z\\ > v

Therefore (x — z, z*) + f(z) > T) Moreover

| | i - z|| • \\z*\\ >(x-x,z*) = [(x - z, z*) + f(z)] - [(x - z, z*) + f(z)]

>V~ f(x)

for every x € X; the last inequality holds because z* G df(z) Dividing by

Using the preceding theorem we reinforce slightly Proposition 3.1.3

Proposition 3.1.10 Let X be a Banach space and f T(X) Then

relation (3.9) holds for every x X

Proof Taking into account Proposition 3.1.3 we have to show that oo = sup{(a; — z,z*) + f(z) \ (z,z*) € gr<9/} when x £ d o m / So, let x £

X \ d o m / and take inf / < 77 < 00 Using Theorem 3.1.9 (iv), there exists {z,z*) E gr<9/ such that (a; - z,z*) + f(z) > rj The conclusion follows •

The maximal monotone operators are of a great importance in the the-ory of evolution equations A significant example of such operators is the subdifferential of a proper lsc convex function on a Banach space

Theorem 3.1.11 (Rockafellar) Let X be a Banach space and f T(X)

Then df is a maximal monotone operator

Proof Let (x,x*) EXxX*\gvdf Then x* £ df{x), and so i df(x), where / := / — x* It follows that inf/ < fix) Applying assertion (iii) of Simons' theorem, there exists (z, z*) € g r / such that (x — z,z*} > Consider z* := z* + x*\ we have that (z, z*) £ df and (z -x,z*— x*) < 0, which shows that the set df U {(x,x*)} is not monotone Therefore df is

a maximal monotone operator •

3.2 Convexity and Monotonicity of Subdifferentials

The aim of this section is to show that the monotonicity of an abstract sub-differential of a lower semicontinuous function ensures its convexity; among these abstract subdifferentials one can mention the Clarke subdifferential we introduce below

Throughout this section (X, ||.||) is a normed vector space Consider | / M c I and x e c l M In the sequel we shall denote by (xn) - > M X

a sequence (xn) C M with (xn) -» x; more generally, x - > M X will mean

x £ M and x —> x The Clarke's tangent cone of M at x is defined by Tc{M,x):={ueX\V{tn) 10, V(xn)-*Mx, 3(un)^u,

V n e N : xn + tnun M)

Recall another cone introduced in Section 2.3:

Because clM -x C C(M,x) C C ( c l M , i ) , we have C{M,x) = C(clM,:r) Note (Exercise!) that

0ETc(M,x)cC(M,x) (3.11)

Several properties of the tangent cone in the sense of Clarke are collected in the following proposition

Proposition 3.2.1 Let ^ M C X and xEc\M Then:

(i) Tc{M,x) is a nonempty closed convex cone;

(ii) TC(M, x) = Tc(cl M, x) = Tc(M D V, x) for all V E M{x);

(iii) if M is a convex set then Tc(M,x) = C(M,x)

Proof, (i) It is obvious that Xu E Tc(M,x) for A > and u E Tc(M,x);

hence Tc(M,x) is a cone Let u,v € Tc(M,x) and consider (tn) 4- and

(xn) ~>M x Then there exists (un) -> u such that x'n := xn + tnun G M for

every n E N Of course (xJJ —>• 3; Hence there exists (vn) —> u such that

£n + in(w„ + u„) = x^ + inwn G M for every n Therefore u + v E Tc(M,x)

Consider now (uk)ke^ C Tc(M,x) with (ufc) -+ u E X Let us show

that u E Tc{M,x) For this take (tn) I and (xn) - > M £• Because ufc E

Tc{M,x), there exists (u*) ->• ufc such that xn + tnukn E M for every n G N

Hence xn+tnukn E M for all k, n E N For every n EN there exists k'n E N

such that ||u* — uk\\ < n~l for every k > k'n Let (fc„) C N be an increasing

sequence such that kn > k'n; then ||u* - uk\\ < n_ 1 for all k,n E N with

k > A;„ Let u„ := u*n Of course, £„ + £nu„ = xn + tnuknn G M for every n

As ||M„ - u|| = \\unn - u|| < ||u*n - ukn || + \\ukn - u\\ < n_ + ||u*n - u||,

we have that (un) ->• u Hence u G Tc(M,x) Therefore Tc(M,x) is a

closed convex cone

(ii) Let u E Tc(M,x) and take (tn) I and (xn) - >CI M £• For every n G

N there exists xn E M such that \\xn — xn\\ < tn/n, i.e xn = xn + tnn~1u'n

with u'n E Ux- Hence (xn) ->M x Therefore there exists («„) -> u with

xn + tnun = xn + tn(n~lu'n + un) G M C c l M for every n E N As

( n_ 1u ^ + u„) —> u, we have that u G I b ( c l M , x )

Conversely, let u G Tc(c\M,x) and take (tn) I and (zn) -tM x

There exists (un) -> u with xn + tnun G cl M for every n Like above, for

every n E N there exists u^ G Ux such that a;„ + tnun + tnn~lu'n G M

Because ( u „ + n_ 1u ^ ) —> ii, we have that w E TQ^M^ X) Hence TQ^M, X) —

The equality Tc{M,x) = b ( M n V,x) is obvious when V is a neigh-borhood of x

(hi) Let M be convex Taking into consideration (ii) we may assume that M is closed Consider x E M and take (£„) 4- and (xn) —»M X

There exists no € N such that tn < 1/2 for n > no Take un := x +

x - 2xn for n > n0 and un = otherwise Of course, (un) -¥ x — x

As xn + tnun = (1 — 2tn)xn + tnx + tnx S M for n > no, we have that

x-x € Tc{M,x) It follows that C(M,x) C Tc( M , x ) Using Eq (3.11)

we obtain the conclusion D

From (ii) and Eq (3.11) we obtain that

Tc(M,x) C n C(MHV,x) =: TB{M,x);

' 'V'gJV(x)

TB{M,X) is the well known tangent cone in the sense of Bouligand of M

atxed M

Let now / : X —> M be a proper function and x dom / It is natural to consider the tangent cone Tc (epi/, (x, f(x))) This cone is related to the Clarke—Rockafellar directional derivative of / at x introduced as follows:

,*,_ , „ fix + tv) — a

f'{x,u):=sup limsup mi

e>0 tiQ,(x,a)->epi,(x,f(x)) ll«-«ll<e *

. f f(x + tv)-a

:= sup ml sup ml

e>0 *>0 0<t<5,\\x-x\\<6, f(x)<a<f(x)+6 II"-'"ll<£ *

It follows that

,t, _ , , f{x + tv) - f(x)

/ ' ( a ; , w ) = s u p inf sup mi ,

£> <5>0 o<t<<5, \\x~x\\<6,f(x)<f{x)+6 l k - " l l <E *

and even

/t( x , u ) = s u p limsup inf / (*+ tV} ~~f{x) (3.12)

e>0 (4.0, x->/5 ll«-«ll<e '

if / is lower semicontinuous at x, where x —>/ x~ means x —>• x and f(x) —>•

f(x) It is obvious that /t( x , 0) < and /''"(xjAu) = A/1'(x, u) for A >

and u G X We may have f^(x, 0) = — oo even if / is continuous at x Take for example / : R -»• E, f(x) := — \/\x\, and x = (Exercise!)

Proposition 3.2.2 Let f : X —> R be a proper function and x G d o m /

Then epi ft(x, •) = Tc (epi / , (x, f(x))) In particular ft(x, •) is a Isc con-vex function; ft(x, •) is a Isc sublinear function if and only if ft(x, 0) = Moreover, if f is convex then epi ft(x, •) = cl (epi f'(x, •))

Proof Let (u,A) G Tc (epi f, (x, f(x))); assume that ft(x,u) > A, and take ft(x, u) > A' > A Then there exist EQ > 0, ((xn,an)) -»epi/ (x,f(x))

and (tn) 4- such that inf„6c(U)£o) t~* (f(xn + tnv) — an) > A' Because

(u, A) Tc (epi f, (x, f(x))), there exists the sequence ((un,Xn))

converg-ing to (u, A) such that (xn,an) + tn(un,\n) G e p i / , i.e f(xn + tnun) <

&n + tnXn for all n G N Since (un) -» u, there exists no such that

un G D(u,eo) for n > no- Hence

v ^ -r f(xn + tnv)-an f(xn + tnun)-an

A < inf — < — < A„ Vn > n0

||w—u||<e0 tn tn

Taking the limit we get the contradiction A' < A

Let now f^(x~,u) < A and take ((xn,an)) —>epif (x,f(x)) and (tn) I

Let us fix (ek) I Because

• r • t f(x + tv) - a

inf sup inf — < A

5>0 0<t<5,(x,a)€epif, \\x-x\\<S, \a-f(x)\<S \\v-u\\<ek t

for every k G N, there exists 8k > such that

f(x + tv) -a sup inf — < A

0«<<5fc,||x-x||<(5fc,/(x)<a</(x)+<5fc ll»-ull<£* *

There exists n'k G N such that < t„ < 8k, \\xn — x\\ < 8k, \an — f(x)\ <

8k for all n > n'k Therefore inf„€£>(U)efc) t "1 (f(xn + tnv) - an) < A, which

shows that for every k and every n>n'k there exists u* D(u, e^) such that

f(xn + tnUn) < a „ + Xtn Consider an increasing sequence (n^) C N such

that nk > n'k for every k G N Taking un := u* for n^ <n < nk+\ we have

that ((un,X)) —> (u, A) and (x„,Q!n) + £„(u„,A) G e p i / for every n G N

Hence (w,A) G Tc (epi / , (x,f(x))) Because Tc (epi / , (x,f(x))) is closed, we have that epi ft (x, •) C Tc (epi f,(x, f(x))) Therefore epi ft(x,-) =

Tc(epif,(x,f(x))) From this relation and Proposition 3.2.1 (i) we have

that ft(x, •) is Isc and convex The other statement follows from Proposi-tion 2.2.7

Assume now that / is convex Because

we have that epip(x, •) = C (epi/, (x,f(x))) = cl(epi/'(:r, •))• D

+

Note that when / is not lsc at x, f^(x, •) and / (x, •) may be different, where, as usual, / is the lower semicontinuous envelope of / Take for example / : R ->• R, f(x) = for x ^ 0, /(0) = 1; /t( , •) = - c o , but

/ (0, •) = When / is lsc at x then f*(x, •) and / (x, •) coincide When / has additional properties, Z1" has a simpler expression

Proposition 3.2.3 Let f : X —> R be a proper function and x d o m /

Then for every u € X we have:

(i) limsupj.^ xP(x,u) < f^(x,u); moreover, if f is lsc atx then

, - l v - N / • r • f f(x + tv)- f(x)

f'(x,u)<sup inf sup inf

£>0 S>0 |j^—^||<<S,0<t<<5 ll«-«ll<e *

< inf sup (3.13)

*>° \\z-x\\<S,0<t<5 *

(ii) / / / is continuous at x then

,-iv- x • r t f{y + tv) - f(x)

f'{x,u)— sup inf sup mi

e>0 <5>0 \\x-x\\<S,0<t<S l l " -ul l <e *

(hi) If f is finite and L-Lipschitz on B(x,r) for some r > and L > 0,

then

ttr- \ • ( f(x + tu) - / ( : r )

/ ' ( a ; , u ) = inf sup

s>0 ||x-x||<(5,0<t<<5 t

f(x + tu) — f(x) , ,

= limsup^—-—'-—J-±-L < L -\\u\\ (3.14)

(iv) If f is finite and Gateaux differentiate on B(x,r), and V / is

con-tinuous at x then f^(x, •) = V/(3;)

Proof Throughout the proof u € X is a fixed element

(i) Let f*(x, u) < X and consider e > 0; there exists > such that f(x + tv)-f(x) ^ sup inf < A

\\x-x\\<8,0<t<S,f(x)<f(x)+5 l k - « l l <e *

0 < t < and f(x) < f(x) + It follows that

sup {mt^D^r1 (f{x + tv) - f(x)) | t G ]0, 8'}, x e D{x', 8'),

f(x)<f(x')+8'} < sup {miv€D(u>e)rl (f(x + tv) - f{x)) \te]0,8], x £ D(x,8),

f(x) < f(x) + 5} < A

for all x' e B(x,8') with \f(x') - f(x)\ < 8' Therefore f^(x',u) < A for such x', and so lizn supx^ w f^(x,u) < f^(x,u)

The first inequality in Eq (3.13) follows immediately from Eq (3.12), while the second is obvious

(ii) Taking into account Eq (3.13), one must show the inequality > For this take e > and > There exists 8' € ]0,8] such that f(x) < f(x) + for every x € D(x,8') Then

sup inf ^ H M > sup inf ^ + ^ ~ ^ ,

0<£<<5 llu-ull^e t 0<t<6' l l " -ul l <e t

\\x~x\\<i ||x-x||<<5'

f(x)<f(x)+S

whence the conclusion follows

(iii) The inequality < is proved in Eq (3.13) Assume that / is Lipschitz on B(x,r) with Lipschitz constant L Take A > lim supx_,.s ^0 Hx+tu)~Hx)

and fix e > Consider 80 > such that sup||;c_j||<5o)0<t<(50 f\x+tu)-f\x) <

A Let :=mm{80,r/(l+e + \\u\\)} Then for all x £ B(x,8), t e]0,8] and

v e D(u,e) we have that \\x + tv - x\\ < + S\\v\\ < 8(1 + e + \\u\\) < r

Therefore for such x, t, v we have

f(x + tv) - f{x) < f(x + tv) - f(x + tu) f{x + tu)-f{x)

<L\\v- u\\ + f{x + tU] ~ fix) < / (* + tU] ~ ™ + Le

Hence

, fix + tv) - fix) fix + tu) - fix) sup inf ^ '-—i±-± < sup — —i- ^ -L + Le,

\\x-x\\<5 ll"-«ll<£ t \]x-x\\<6 t

and so

/(a; + fa) - / ( a ; )

mi sup inf —

*>° ||i-i||<<5,0<t<<5 ll«-«ll< £ t

fix + tu) - f(x)

< sup J-^——-—J-^-L +Le<\ + Le

||a:-x||<<So,0«<<5o *

Taking the limit for e -> 0, we obtain

\ ^ r -t -t f(x + tv)-f(x) ttf- \

A > hm inf sup inf —- ^—L = f]{ x , u )

e4-0 d>0 \\x-x\\<6,0<t<6 ll"-«ll<e t

Taking into account (ii), we get Eq (3.14) Taking = r / ( l + ||w||), for

x € B(x,S) and t £]0,S] we have that x + tu,x € B(x,r), and so f{x + tu) — f(x) < tL \\u\\, whence f^(x,u) < L \\u\\

(iv) Let r > be such that / is Gateaux differentiable on B(x, r) Since V / is continuous at x, V / is bounded on a neighborhood of x So we may assume that ||V/(x)|| < L for x € B(x,r) Using the mean-value theorem we obtain that / is i-Lipschitz on B(x,r), and so Eq (3.14) holds Let

(tn) i and (xn) -»• x be such that f^(x,u) = \imt~1 (f(xn + tnu) — f(xn))

But xn+tnu, xn £ 5(3;, r) for n > no (for some no £ N); applying the mean

value theorem, there exists 6n £]0,i„[ such that f(xn + tnu) — f(xn) —

V / ( xn + 6nu)(tnu) for every n > no- Because V / is continuous at x, we

obtain that ft(x,u) = Vf(x)(u) O Let us introduce now the Clarke subdifFerential of / : X —> K at

xeX with f(x) £ E This is

dcf(x) = {x* € X* | {u,x*) < f\x,u) Vu € X } ;

if f(x) £ E we consider that dcf(x) =

Taking into account Theorem 2.4.14, dcf(x) ^ if and only if /^(af, 0) = 0 Moreover, dcf(x) is a nonempty w*-compact subset of X* if / is Lips-chitz on a neighborhood of x Other properties of dc are collected in the following result

Theorem 3.2.4 Let f,g : X —> R be proper functions and x £ d o m / n

domg

(i) If / and g coincide on a neighborhood ofx then dcf(x) — dcg(x) (ii) If f is convex then dcf(x) = df(x)

(iv) \imsupx_>x dcf(x) C dcf(x), where kmsupx^ dcf(x) is the set

{x* G X* | (xn) -+f x, « ) % x*, Vn G N : < G 3c/ ( a ;n) } •

(3.15) (v) If g is finite and Lipschitz on a neighborhood of x then

(f + g)Hx, •) < fHx, •) + g\x, •), dcU + g)(?) c dcf(x) + dcg(x)

(vi) / / g is Gateaux differentiate on a neighborhood of x and Vg is

con-tinuous at x then

(f + g)H*, •) = fHx, •) + V5(af), dc(f + g)(x) = dcf(x) + Vg(x)

Proof, (i) and (iii) are obvious because p{x, •) = (flv^ix, •) for every

V G N(x), while (ii) follows from the second part of Proposition 3.2.2

(iv) Let x* G limsupx^ fX-dcf(y); t n e n there exist (xn) ->/ x and

(a£) ^ x* such that x*n G dcf(xn) for all n G N Let M £ I be fixed

Then (u,a;*) < p{xn,u) for every n G N, whence, by Proposition 3.2.3 (i),

(u,x*) < ft(x,u), and so x* G dcf(x)

(v) Let r > be such that g is L-Lipschitz on B(x, r); we may as-sume that L > Let u £ X, e > and > be fixed and take

6' = mm{S/{2L),r/(l + e + \\u\\)} > Let v, x and t be such that \\v-u\\ < e , \\x-x\\ <5',0<t<5' andf(x)+g(x) < f(x)+g(x)+6' It

follows that f(x) < f(x) + (g(x) - g{x)) + 6' < f(x) + LS' + 6' < f(x) + Furthermore

(f + g)(x + tv)-(f + g)(x) f{x + tv) - f{x) g{x + tu) - g{x)

t ~ t t

whence, taking first the supremum with respect to v G D(u,e), we obtain that a < Ai(5) + A2(5) + Le, where

(f + g)(x + tv)-(f + g)(x)

a : = sup mi ,

||a!-s||<a',0<t<<5' H"-«ll<£ t U+9)(x)<(f+gm+s'

Ai(S) : = sup inf — — - ^ -L,

\\x-x\\<S,0<t<S,f(x)<f(x)+S Il«-«II<£ t

Therefore /3 < Ai(S) + A2{5) + Le for every > 0, where

/ : = i n f sup inf if + 9)(x+ tv) - jf + g)(X) ^ 5'>0 ||a!-»||<<?',0<t<(?' Ilu-"II<£ *

(f+9)(x)<(f+g)(x)+S'

Taking the limit of Ai (S) + A2 (5) for S —> (taking into account that A\

and A2 are nondecreasing for > 0), we get

inf sup inf ( / + g)(* + * - ( / + g)(*)

5>°\\x-x\\<5,0<t<5,(f+g)(x)<{f+g){x)+5\\v-u\\<e t

S • f • f f(X + tv) ~ /(*)

< inf sup inf —- - —L

s>0 \\x-x\\<6,0<t<6,f(x)<f(x)+S l l " - " l l <£ *

- 9(x + tu) — g(x)

+ inf sup — '-—^^- + Le

<5>°||x-x||<(5 t

Taking now the limit for e —> 0, we get the desired conclusion The relation for the subdifferentials follows from the preceding inequality and the fact that g^(x, •) is continuous

(vi) As mentioned in the proof of Proposition 3.2.3 (iv), g is Lipschitz on a neighborhood of x and so the conclusion of (iv) holds with g^(x, •) =

Vg(x) Applying again (iv) for f + g and — g (we may assume that g is

finite on X; otherwise take g = outside a neighborhood of x), we obtain that f*(x, •) < (f + g)^(x, •) — Vg(x) The conclusion is now obvious •

In the rest of this section we use an abstract subdifferential Before introducing this notion let us consider the following class of finite-valued convex functions:

C(X) := {g : X -> E | g is convex and Lipschitz};

of course, €(X) is a convex cone in the vector space Rx of all functions from

X into E Let d ( X ) C Ex be the cone generated by X*U{d[a,6] | a, b € X)

(C €{X)) and €2{X) C Ex be the cone generated by

X*u{dfaM\a,beX}WD0,

where

Do := { ^2k>QVkdlk (Hk) C K+, ^2k>Ql*k = *' (u*)*>° c o n v e rSe n t} •

We call an abstract subdifferential on the nonempty class T C M a multifunction d : X x T =4 X*, which associates to (x,f) a set denoted by df(x), satisfying condition (PI) below:

(PI) G limsupy_>xd/(y) + dg(x) if / € T, g € <£(X), f(x) £ and a;

is a local minimum of / + g, where limsup _yX df{y) is defined as

in Eq (3.15) and dg(x) is the Fenchel subdifferential of g at x

A stronger form of (PI) is

(P2) € ~8f(x) + dg(x) if / T, g e €(X), f(x) and a; is a local minimum of / + g

Sometimes one asks

(P3) df(x) = df(x) if / e Tn€(X)

Of course, (P2) holds if (P4) and (P5) below are satisfied:

(P4) df (x) if / J-, f(x) € E and x is a local minimum of / , (P5) T + €(X) C T and 8{f + g)(x) C df(x) + dg(x) when / G T and

9 e €(X)

Note that

WX + €(X) = RX, A(X) + €(X) = A(X), T(X) + €(X) = T(X)

Remark 3.2.1 From the preceding theorem we observe that Clarke's

sub-differential dn and the Fenchel subsub-differential d are abstract subsub-differentials

x

on R and A(X), respectively; in fact they satisfy conditions (P1)-(P5) There are many other subdifferentials which satisfy condition (PI)

Remark 3.2.2 If the abstract subdifferential d on T satisfies the stronger

condition (P2) then for any proper function / T one has df(x) C df(x) for all x € d o m /

Indeed, if x* df(x) then a; is a (local) minimum point of / + (—x*), and so, by (P2), £ df(x) + d(-x*)(x) = df(x) - x* Hence x* € df(x)

Theorem 3.2.5 (Zagrodny) Let (X, ||-||) be a Banach space and d be

x

an abstract subdifferential on T C E Let f T be Isc, a,b G X with a G d o m / and a ^ b, and r l with r < f(b) Then there exist (xn) —>•/

c € [a, b[ and x*n G df(xn) for every n G N such that

(i) r — f(a) < lim inf (6 - a, x*n),

(ii) < lim inf (c - xn,x*n),

(iii) f ^ ( r - / ( a ) ) < l i m i n f ( & - x „ , < ) , (iv) ||& - a|| (/(c) - / ( a ) ) < ||c - a|| (r - / ( a ) )

Proof There exists x* G -X"* such that (6 - a, x*) = r — f(a) Consider

h := / — x*; then /i(a) < h(b) Because h is Isc, there exists c G [a,b[

such that h(c) < h(x) for all a; G [a,b] Therefore c = (1 — /x)a + fib for some /i G [0,1[ It follows that c — a = fi(b — a) and ||c - a|| = fi \\b — a\\ Therefore /(c) - f(a) = h(c) - h(a) + (c — a,x*) < fi(r - / ( a ) ) , whence (iv) follows

Let < h(c); then there exists r > such that < h(x) for every

x G [a, b]+rUx- Otherwise, for every n G N there exists xn G [a, b ] + n- 1{ / x

such that h(xn) < Hence arn = dn + yn with d„ G [a, 6] and yn G n~1

Ux-As the segment [a,b] is a compact set, there exists a subsequence (dnk)

converging to d G [a,b] It follows that (xnh) —> d, and so, by the lower

semicontinuity of h, we get the contradiction h(d) <

Let r > correspond to := h(c) — 1, and take C/ := [a, b] + rUx', of course, U is closed Like above, for any n G N there exists r„ G ]0, r[ such that /i(z) > h(c) — n~2 for 3; G [a,b] + rnUx', choose tn > n such that

7 + tnrn > h(c) — n~2 Then one has

Vz G U : /i(c) < h(x) + tnd[atb](x) + n~2 (3.16)

Indeed, the inequality is obvious for x G [a, b] + rnUx- If x £ U \ ([a, b] +

r „ [ / x ) then d[0i6](x) > r„, and so /i(x) + t„d[a)ft](a;) > + inrn > ft.(c)-n-2

Consider i?„ — h + iu + *nd[0)(,] Prom Eq (3.16) we have that Hn(c) <

infx Hn + n- 2; moreover, Jf„ is Isc and bounded from below Applying

Corollary 1.4.2 for Hn, c, e ~ n~2 and A := n_ 1, we get un G X such that

Hn(un) < Hn(c), | | c - u „ | | < n_ 1, (3-17)

Hn{un) < Hn(x) + n -1 ||z - un|| Va; G X (3.18)

suc/