DSpace at VNU: Generalized convex functions and generalized differentials

This article was downloaded by: [Florida Atlantic University] On: 02 August 2013, At: 06:48 Publisher: Taylor & Francis Informa Ltd Registered in England and Wales Registered Number: 1072954 Registered office: Mortimer House, 37-41 Mortimer Street, London W1T 3JH, UK Optimization: A Journal of Mathematical Programming and Operations Research Publication details, including instructions for authors and subscription information: http://www.tandfonline.com/loi/gopt20 Generalized convex functions and generalized differentials a Nguyen Thi Hong Linh & Jean-Paul Penot b a International University , Vietnam National University, Quarter 6, Linh Trung Ward, Thu Duc District, Ho Chi Minh City , Vietnam b Laboratoire J.-L Lions, Université Pierre et Marie Curie , place Jussieu, 75005 Paris , France Published online: 19 Sep 2011 To cite this article: Nguyen Thi Hong Linh & Jean-Paul Penot (2013) Generalized convex functions and generalized differentials, Optimization: A Journal of Mathematical Programming and Operations Research, 62:7, 943-959, DOI: 10.1080/02331934.2011.611882 To link to this article: http://dx.doi.org/10.1080/02331934.2011.611882 PLEASE SCROLL DOWN FOR ARTICLE Taylor & Francis makes every effort to ensure the accuracy of all the information (the “Content”) contained in the publications on our platform However, Taylor & Francis, our agents, and our licensors make no representations or warranties whatsoever as to the accuracy, completeness, or suitability for any purpose of the Content Any opinions and views expressed in this publication are the opinions and views of the authors, and are not the views of or endorsed by Taylor & Francis The accuracy of the Content should not be relied upon and should be independently verified with primary sources of information Taylor and Francis shall not be liable for any losses, actions, claims, proceedings, demands, costs, expenses, damages, and other liabilities whatsoever or howsoever caused arising directly or indirectly in connection with, in relation to or arising out of the use of the Content This article may be used for research, teaching, and private study purposes Any substantial or systematic reproduction, redistribution, reselling, loan, sub-licensing, systematic supply, or distribution in any form to anyone is expressly forbidden Terms & Downloaded by [Florida Atlantic University] at 06:48 02 August 2013 Conditions of access and use can be found at http://www.tandfonline.com/page/termsand-conditions Optimization, 2013 Vol 62, No 7, 943–959, http://dx.doi.org/10.1080/02331934.2011.611882 Generalized convex functions and generalized differentials Nguyen Thi Hong Linha* and Jean-Paul Penotb a International University, Vietnam National University, Quarter 6, Linh Trung Ward, Thu Duc District, Ho Chi Minh City, Vietnam; bLaboratoire J.-L Lions, Universite´ Pierre et Marie Curie, place Jussieu, 75005 Paris, France Downloaded by [Florida Atlantic University] at 06:48 02 August 2013 (Received 10 February 2011; final version received August 2011) We study some classes of generalized convex functions, using a generalized differential approach By this we mean a set-valued mapping which stands either for a derivative, a subdifferential or a pseudo-differential in the sense of Jeyakumar and Luc Such a general framework allows us to avoid technical assumptions related to specific constructions We establish some links between the corresponding classes of pseudoconvex, quasiconvex and another class of generalized convex functions we introduced We devise some optimality conditions for constrained optimization problems In particular, we get Lagrange–Kuhn–Tucker multipliers for mathematical programming problems Keywords: colinvex; generalized differential; mathematical programming; optimality conditions; protoconvex function; pseudoconvex function; quasiconvex function AMS Subject Classifications: 26B25; 46G05; 49K27; 90C26; 90C32 Introduction Various needs have led mathematicians to introduce and study several notions of generalized convexity or concavity Among the tools used to define or study these notions are the various subdifferentials of nonsmooth analysis [1,8–10,12–17,25– 28,30,32–38,40, etc.], the convexificators of [11], the pseudo-differentials of Jeyakumar and Luc [19], the normal cones to sublevel sets [2,4–6] and the generalized directional derivatives [20,21,23,41] In this article, we use a concept of generalized derivative which can encompass all these notions but the last one We call it a generalized differential in contrast with the notion of subdifferential because it is not necessarily a one-sided concept It allows much flexibility It also leads us to get rid of some assumptions required in previous works such as smooth renorming of the space In Section we establish comparisons with the notions obtained using directional derivatives or their substitutes *Corresponding author Email: honglinh98t1@yahoo.com ß 2013 Taylor & Francis 944 N.T.H Linh and J.-P Penot Downloaded by [Florida Atlantic University] at 06:48 02 August 2013 In the two last sections of this article, we investigate some necessary and sufficient optimality conditions for constrained problems and mathematical programming problems The case of pseudo-affine functions (also called pseudo-linear functions) is studied elsewhere [24] with similar concepts and methods It requires some results of this article Characterizations of generalized convex functions We devote this preliminary section to review some concepts of generalized convexity; we also introduce new ones Some elementary properties and characterizations are given In the sequel X, Y are normed vector spaces, X* is the dual space of X, C is a nonempty subset of X and f : C ! R1 :¼ R [ {ỵ1} We extend f to X\C by ỵ1; alternatively, we may consider f : X ! R1 and take for C a subset of X containing the domain of f We denote by P (resp Rỵ) the set of positive numbers (resp nonnegative numbers) and R stands for R [ {1, ỵ1} We assume that a set! X* is given which stands as a substitute to the derivative of f; we valued map @f : C ! call it a generalized differential of f Among possible choices for @f are the subdifferentials of f in the various senses of nonsmooth analysis and the pseudodifferentials of f in the sense of [19] These cases will be considered below We also deal with some other cases, such as normal cones set-valued maps We recall that the visibility cone V(C, x) of C at x cl C is the cone generated by C x: VC, xị :ẳ PC xị :ẳ frc xị : r P, c Cg: It contains the radial tangent cone to C at x which is the set T r ðC, xÞ :ẳ fu X : 9rn ị ! 0ỵ , x ỵ rn u C 8ng: The visibility bundle (resp the radial tangent bundle) of C is the set ẩ ẫ [ fxg VC, xị VC :ẳ x, uÞ C  X : 9r P, w C, u ẳ rw xị ẳ x2C resp: T r C :ẳ fx, uị C X : 9rn ị ! 0ỵ , x ỵ rn u C 8ng ¼ [ ! fxg  T r ðC, xÞ : x2C If C is starshaped at x (in the sense that for all w C and t ]0,1] one has (1 t)w ỵ tx C) one has V(C, x) ¼ T r(C, x) Thus, if C is convex, VC ¼ T rC The upper and the lower radial derivatives (or upper and lower Dini derivatives) of f at x C in the direction u T r(C, x) are defined by dỵ f x, uị ẳ dỵ f x, uị ẳ ẵ f x ỵ tuị f xị; t!0ỵ, xỵtu C t lim sup lim inf t!0ỵ, xỵtu C ẵ f x ỵ tuị f xị: t 945 Optimization The upper and the lower radial subdifferentials of f at x are defined, respectively, by @r f xị :ẳ fx : hx , ui @r f xị :ẳ fx : hx , ui dỵ f x, uị dỵ f x, uị 8u T r C, xÞg; 8u T r ðC, xÞg: The definitions of @f-pseudoconvexity and @f-quasiconvexity we adopt here for a generalized subdifferential @f of f are similar to the ones used for a subdifferential by several authors; see [30,35] and the references therein We also introduce definitions of @f-protoconvexity and strict @f-pseudoconvexity of f as natural variants of the two preceding concepts Downloaded by [Florida Atlantic University] at 06:48 02 August 2013 Definition Let f : C ! R1, x dom f A function f is said to be (a) @f-pseudoconvex at x if for all w C: f ðwÞ f ðxÞ ) for all xà @f ðxÞ : hxà , w À xi 0: ð1Þ (b) @f-quasiconvex at x if for all w C: f ðwÞ f ðxÞ ) for all xà @f ðxÞ : hxà , w À xi 0: ð2Þ 0: ð3Þ f ðxÞ ) for all xà @f ðxÞ : hxà , w À xi 0: ð4Þ (c) @f-protoconvex at x if for all w C: f ðwÞ f ðxÞ ) for all xà @f ðxÞ : hxà , w À xi (d) strictly @f-pseudoconvex at x if for all w C \{x}: f ðwÞ We add the word ‘eventually’ when in the preceding implications ‘for all’ is changed into ‘there exists’ Thus, for instance, f is eventually @f-pseudoconvex at x if for all w C satisfying f (w) f (x) there exists x* @f (x) such that hx*, w À xi Remark that the definition of @f-strict pseudoconvexity is irrealistic if x is a nonunique minimizer of f Clearly f is strictly @f-pseudoconvex at x ¼) f is @f-protoconvex and @f-pseudoconvex at x, ð5Þ f is @f-pseudoconvex at x ẳ) f is @f-quasiconvex at x, 6ị f is @f-protoconvex at x ẳ) f is @f-quasiconvex at x: 7ị We say that f is @f-pseudoconvex on C (in short @f-pseudoconvex) if it is @f-pseudoconvex at every x C We use a similar convention for the other concepts introduced above Let us observe that @f-quasiconvexity is different from full @f-quasiconvexity (in the terminology of [30]) in the sense (due to Aussel [1]) that dom f is convex and for any w, x C, x* @f (x) with hx*, w À xi 0, one has f (w) ! f (z) for any z [x, w] Clearly, if f is @f-quasiconvex and quasiconvex then f is fully @f-quasiconvex while a fully @f-quasiconvex function is obviously @f-quasiconvex 946 N.T.H Linh and J.-P Penot The following examples show that versatility is gained by taking for @f a multimap which may differ from usual subdifferentials such as the Fenchel, the Fre´chet or the Dini–Hadamard subdifferentials Example Let f : R ! R be given by f (x) ¼ 2x for x [1,1], f (x) ẳ x ỵ for x 1, f (x) ¼ x À for x À1 We can take @f (x) ¼ @rf (x) ¼ @Df (x) when x 6¼ and @f (1) ¼ À@r(Àf )(1) ¼ À@D(Àf )(1) (where @D f is the Dini–Hadamard subdifferential of f ) Observe that @Df (1) ¼ ; and that f is @f-strictly pseudoconvex Downloaded by [Florida Atlantic University] at 06:48 02 August 2013 Example Let f : X ! R be such that for some generalized differential @] one has @](Àf )(x) 6¼ ; for all x X Then it may be convenient to take @f :¼ À @](Àf )(x) Example Let f : X ! R be such that f ẳ f1 ỵ f2 Then one may take @f :ẳ @] f1 ỵ @] f2, where @] is some generalized differential, although it may be different from @] ( f1 ỵ f2) Example Let f : X ! R be such that f ¼ g À h for some functions g, h Then one may take @f (x) :¼ @]g(x) - @]h(x), where @] is some generalized differential, and for subsets A, B of X*, A - B :ẳ {x* X* : B ỵ x* & A}, although @f may be different from @](g À h) For some subdifferentials the preceding definitions may be automatically satisfied This is the case for the notions of Definition but @f-strict pseudoconvexity if @f is the Fenchel–Moreau subdifferential @FMf given by @FM f xị :ẳ fx X : f ! x ỵ f xị x xịg: Let us consider some other classical subdifferentials We recall that the original GreenbergPierskallas subdifferential is defined by @GP f xị ẳ fxà Xà : hxà , w À xi ! ) f ðwÞ ! f ðxÞg: The lower subdifferential, or Plastria subdifferential of f at some point x of its domain dom f :¼ {x X : f (x) R} is the set à @5 f ðxÞ :ẳ fx X : 8w S5 f xị, f ðwÞ À f ðxÞ ! hx , w À xig, ẵ1, f xịẵị is the strict sublevel set of f at x where S5 f xị :ẳ Sf f xịị :ẳ f We also recall the following variant called the infradifferential or Gutie´rrez subdifferential: @ f xị :ẳ fx X : 8w Sf xị, f ðwÞ À f ðxÞ ! hxà , w À xig, where Sf (x) :¼ Sf ( f (x)) :¼ f À1([À1, f (x)]) is the sublevel set of f at x For such subdifferentials one has the following obvious statement LEMMA Let f : X ! R1 and x dom f Then (a) f is @GPf-pseudoconvex at x; (b) f is @5f-pseudoconvex at x; (c) f is @ f-protoconvex and @ f-pseudoconvex at x We denote by N(C, x) the normal cone at x X to a subset C of X given by NC, xị :ẳ fxà Xà : hxà , w À xi 8w Cg 947 Optimization even when C is nonconvex; of course, such a cone is mainly of interest in the case C is convex Some normal cone multimaps can be associated with a function f : X ! R1 as follows: for x dom f, Nf ðxÞ :ẳ NSf xị, xị :ẳ fx X : hx , w À xi à à à N5 f xị :ẳ NSf xị, xị :ẳ fx X : hx , w À xi 8w Sf ðxÞg, 8w S5 f ðxÞg, Downloaded by [Florida Atlantic University] at 06:48 02 August 2013 (see, e.g [4,6,30] for the case f is quasiconvex) Clearly, any function f is Nf -protoconvex and N5 f -quasiconvex at any x dom f Moreover, for any function f, @GPf (resp Nf, N5 ) is the greatest generalized differential @f of f, such that f is f @f-pseudoconvex (resp @f-protoconvex, @f-quasiconvex) Recall that f is said to be semi-strictly quasiconvex if it is quasiconvex and if for all w, x, y C, x ]w, y[ :¼ [w, y]\{w, y} one has f (x) f ( y) whenever f (w) f ( y) LEMMA Let f : X ! R1 be a quasiconvex function and let x C such that f (x) inf f (X ) If there is no local minimizer of f on Lf (x) :¼ f À1( f (x)), in particular f is semistrictly quasiconvex, then N5 f xị ẳ Nf xị Moreover, if f is radially upper semicontinuous, then f is Nf \{0}-pseudoconvex at x This last assertion is given in [2, Proposition 3] and [5, Proposition 2.7]; we give a direct, simple proof for the convenience of the reader Proof Since there is no local minimizer of f on Lf (x) (that occurs when f is semi5 strictly quasiconvex), S5 f xị is dense in Sf (x), hence Nf xị ẳ Nf ðxÞ Suppose f is radially upper semicontinuous Then, for all x* Nf (x)\{0} and w S5 f ðxÞ we have hx*, w À xi since if hx*, w À xi ¼ 0, taking u X with hx*, ui we would have w þ tu S5 f ðxÞ for t small enough, and hx*, w ỵ tu xi 0, a contradiction Thus f is Nf \{0}-pseudoconvex at x g We say that @f is a valuable generalized differential if for all a X and b X with f (b) f (a) there exist c [a, b[ :ẳ [a, b]\{b} and sequences cn ị, cn Þ such that ðcn Þ !f c, cÃn @f ðcn Þ for each n and hcÃn , d À cn i for all d b ỵ Rỵ b aị and all n N: The following result shows that for a valuable subdifferential, quasiconvexity, @f-quasiconvexity and @f-protoconvexity coincide under a mild continuity assumption The last assertion follows from [35] PROPOSITION (a) If f : X ! R1 is quasiconvex and @f  @ rf, then f is @f-protoconvex hence @f-quasiconvex (b) Let f : X ! R1 be a radially continuous l.s.c function and let @f be a valuable generalized differential If f is @f-quasiconvex, then it is quasiconvex Proof Let w, x C and x* @f (x) be such that hx*, w À xi Then there exists (tn) ! 0ỵ such that f (x ỵ tn(w x)) f (x) Since f is quasiconvex, one has f (w) f (x) (b) It follows from [35] with a similar proof g The following proposition is well-known when @f is the derivative of f or a subdifferential of f 948 N.T.H Linh and J.-P Penot PROPOSITION 2= @f (x) Let C be open, let f : C ! R be radially upper semicontinuous and (a) If f is @f-quasiconvex at x then it is @f-pseudoconvex at x (b) If f is eventually @f-quasiconvex at x then it is eventually @f-pseudoconvex at x Proof Let w, x C be such that f (w) f (x) (a) If f is @f-quasiconvex at x then for all x* @f (x), hx*, w À xi Suppose that there exists x* @f (x) such that hx*, w À xi ¼ Since x* 6¼ 0, we can find u X such that hx*, ui ¼ By radial upper semicontinuity of f, there exists " such that y :ẳ w ỵ "u C and f ( y) f (x) Since f is @f-quasiconvex, Downloaded by [Florida Atlantic University] at 06:48 02 August 2013 ! hxà , y À xi ¼ hx , w xi ỵ "hx , ui ẳ " 0, a contradiction Hence, one has hx*, w À xi for all x* @f (x): f is @f-pseudoconvex at x (b) If f is eventually @f-quasiconvex at x, there exists x* @f (x), such that hx*, w À xi With a similar proof to the one in (a), one gets hx*, w À xi g A comparison between @f-pseudoconvexity, quasiconvexity (in the usual sense) and semi-strict quasiconvexity is given in the next proposition Such comparisons have been made in [15, Prop 11.5], [31,35,41] in case @f is contained in the dag subdifferential @yf defined there We use the conditions: (C) If for some w, x C, f is constant on [w, x], then one has hx*, w À xi for some x* @f (x) (M) If x is a local minimizer of f, then @f (x) Condition (C) is satisfied when @f (x) \ @Df (x) 6¼ ;, where @Df is the Dini– Hadamard subdifferential of f, in particular when f is Gaˆteaux-differentiable at each point of C and f (x) @f (x) Condition (M) is a natural condition which is satisfied by all sensible generalizations of the derivative used for minimization problems Let us recall that a function f is radially nonconstant if one cannot find any line segment on which f is constant The first assertion of the following proposition can be claimed when f is a radially continuous l.s.c function and @f is a valuable generalized differential since if f is @f-pseudoconvex, then it is @f-quasiconvex and, by Proposition 4, it is quasiconvex; see also in [1, Theorem 4.1] for the case X admits a smooth renorming A comparison of assertions (b) and (c) with [1, Proposition 5.2] is made in the example below The proof of the last assertion is similar to the one of [1, Proposition 5.2]; note that one does not need that @ be valuable PROPOSITION Let C be convex, let f : C ! R1 be @f-pseudoconvex and @f (x) 6¼ ; for all x C Then (a) f is quasiconvex (b) If f satisfies condition (C), then it is semi-strictly quasiconvex (c) If f satisfies condition (M), then every local minimizer is a global minimizer Moreover, if f is radially continuous and l.s.c., then it is semi-strictly quasiconvex (d) If f is radially nonconstant, then f is strictly @f-pseudoconvex Optimization 949 Downloaded by [Florida Atlantic University] at 06:48 02 August 2013 Proof (a) Let f be @f-pseudoconvex Suppose that f is not quasiconvex Then there exist w, x, y C such that x (w, y) and f (x) f (w), f (x) f ( y) By @f-pseudoconvexity of f, for all x* @f (x) one has hx*, w À xi and hx*, y À xi 0, a contradiction with the linearity of x* (b) Now, suppose that f is @f-pseudoconvex and satisfies condition (C) If f is not semi-strictly quasiconvex then there exist w, x, y C with x 2]w, y[ such that f (w) f ( y) f (x) Thus, by (a), f (x) ¼ f ( y) and x is a maximizer of f on [w, x] Now, if f is not constant on [x, y] there exists z ]x, y[ such that f (z) f (x), a contradiction with the quasiconvexity of f and f (x) max( f (w), f (z)) Hence, f ([x, y]) is constant Since f is @f-pseudoconvex and f (w) f (x), one has hx*, w À xi for all x* @f (x) On the other hand, by condition (C), there exists xà @f ðxÞ such that ! hxà , y À xi: a contradiction (c) The first assertion follows from the definition of @f-pseudoconvexity of f and condition (M) For the second assertion, see [15, Prop 11.5 (iii)] Note that for this implication, one does not need that @ be valuable (d) See [1, Proposition 5.2] g Example Let f : R ! R be given by f (x) ¼ for x and f (x) ¼ x for x Since f is not radially nonconstant, we cannot use [1, Proposition 5.2] However, appropriate choices of @f allow us to apply Proposition 6: (a) Suppose @f (x) for x and @f (x) & P for x ! Then f is @f-pseudoconvex and condition (C) is satisfied Thus, f is semi-strictly quasiconvex Note that condition (M) is not satisfied since is a local minimizer of f but 2= @f (0) (b) If @f (x) for x 0, then condition (M) is satisfied A simple stability property is given in the following lemma (here the convexity of C is not needed) ! X* and let LEMMA Let I be a finite set For i I, let fi : C ! R1, let @fi : C ! f :¼ supi2Ifi For x C, let I(x) :¼ {i I : fi(x) ¼ f (x)} Suppose @f (x) & co( [ i2I(x)@fi(x)) If for i I(x), the function fi is @fi-pseudoconvex (resp @fi-protoconvex, @fi-quasiconvex) at x, then f is @f-pseudoconvex (resp @f-protoconvex, @f-quasiconvex) at x Proof Let w C be such that f (w) f (x) Then for all i I(x) one has fi(w) fi(x) hx*, w À xi Now, Since fi is @fi-pseudoconvex, for all x* [ Pi2I(x) @fi(x), one has à [0, 1], t ẳ and x @fi xị for i I(x) such let x* @f (x) Then there exist t i i2I(x) i i P that xà ¼ i IðxÞ ti xÃi , hence hx*, w À xi A similar proof can be given to show that f is @f-protoconvex or @f-quasiconvex at x g Links with previous works In [23], assuming a generalized directional derivative h : VC ! R of f is given, we defined concepts which seem to be related to the present notions Let us make a precise comparison Recall that f is h-pseudoconvex (resp h -quasiconvex, 950 N.T.H Linh and J.-P Penot h-protoconvex) at x if for all w C f ðwÞ f ðxÞ ) hðx, w À xÞ ðresp: f ðwÞ f ðxÞ ) hðx, w À xÞ resp: f ðwÞ f ðxÞ ) hðx, w À xÞ 0, 0Þ: Let us first consider the case we dispose of a general differential @f which is a generalized pseudo-differential of f The concept of pseudo-differential of f has been introduced by Jeyakumar and Luc in [19] when hỵ and hỵ are the Dini derivatives of f We introduce a slight variant which enables to encompass [18, Definitions 2.1, 2.2] and [19] Downloaded by [Florida Atlantic University] at 06:48 02 August 2013 Definition Let f : C ! R and h : VC ! R: We say that a subset @f (x) of X* is: (a) an upper h-pseudo-differential of f at x if for all (x, u) VC, we have hðx, uÞ supfhxà , ui : xà @f ðxÞg: ð8Þ (b) a lower h-pseudo-differential of f at x if for all (x, u) VC, we have hðx, uÞ ! inffhxà , ui : xà @f ðxÞg: ð9Þ We say that @f (x) is upper exact (resp lower exact) if for all u X one has supfhxà , ui : x @f xịg ẳ maxfhx , ui : xà @f ðxÞg (resp inf{hx*, ui : x* @f (x)} ¼ min{hx*, ui : x* @f (x)}) whenever the supremum (resp infimum) is finite Clearly, if @f (x) is weak* compact then @f (x) is upper exact and lower exact PROPOSITION Let f : C ! R1, h : VC ! R and let @f (x) be an upper h-pseudodifferential of f at x If f is @f-quasiconvex (resp @f-protoconvex) at x, then f is h-quasiconvex (resp h-protoconvex) at x If f is @f-pseudoconvex at x and if @f (x) is upper exact, then f is h-pseudoconvex at x Proof Let f be @f-quasiconvex (resp @f-protoconvex) at x and let w C be such that f (w) f (x) (resp f (w) f (x)) Since h(x, w À x) sup{hx*,w À xi : x* @f (x)} 0, f is h-quasiconvex (resp h-protoconvex) at x Now let f be @f-pseudoconvex at x and let w [ f f (x)] Then for all x* @f (x), one has hx*, w À xi Thus, if @f (x) upper exact, one has max{hx*, ui : x* @f (x)}50, hence h(x, w À x) and so f is h-pseudoconvex at x g We have a kind of converse PROPOSITION 10 Let @f (x) be a lower h-pseudo-differential of f at x If f is h-pseudoconvex then f is eventually @f-pseudoconvex If f is h-quasiconvex (resp h-protoconvex) at x and if @f (x) is lower exact, then f is eventually @f-quasiconvex (resp eventually @f-protoconvex) at x Proof Since f is h-pseudoconvex at x, for any w C, f (w) f (x), one has hðx, w À xÞ ! inffhxà , w À xi : xà @f ðxÞg: 951 Optimization Hence, there exists x* @f (x) such that hx*, w À xi 0: f is eventually @f-pseudoconvex at x Now, let f be h-quasiconvex (resp h-protoconvex) at x For any w C, w [ f f (x)] (resp w [ f f (x)]), one has ! hðx, w À xÞ ! inffhxà , w À xi : xà @f ðxÞg: Since @f (x) is lower exact, there exists x* @f (x) such that hx*, w À xi eventually @f-quasiconvex (resp eventually @f-protoconvex) at x Hence, f is g We dispose of another form of converse when @f is the generalized differential of f associated with a bifunction h in the following way: Downloaded by [Florida Atlantic University] at 06:48 02 August 2013 @f ðxÞ :ẳ @h f xị :ẳ @hx, ị0ị :ẳ fx Xà : 8u X, hxà , ui hðx, uÞg: We will use the following conditions on h: ðSÞ 8w, x C hx, w xị ẳ supfhx , w À xi : xà @h f ðxÞg: ðS0 Þ 8w, x C hðx, w À xÞ ¼ maxfhxà , w À xi : xà @h f ðxÞg: Condition (S) (resp (S0 )) is satisfied when C is open and for all x C the function h(x, Á) is sublinear and l.s.c (resp continuous) PROPOSITION 11 (a) If the function f is h-pseudoconvex (resp h -quasiconvex, h-protoconvex) at x, then f is @hf-pseudoconvex (resp @hf-quasiconvex, @h f-protoconvex) at x (b) Conversely, if f is @hf-pseudoconvex at x and if h satisfies condition (S0 ), then f is h-pseudoconvex at x (c) If f is @hf-quasiconvex (resp @hf-protoconvex) at x and if h satisfies condition (S), then f is h-quasiconvex (resp h-protoconvex) at x Proof (a) If f is h-pseudoconvex at x, then for w [ f f (x)] and x* @hf (x) one has hx*, w À xi h(x, w À x) Thus f is @hf-pseudoconvex Now, if f is h-quasiconvex (resp h-protoconvex) at x then for w [ f f (x)] (resp w [ f f (x)]), x* @hf (x), one has hx*, w À xi h(x, w À x) and then f is @hf-quasiconvex (resp @hf-protoconvex) at x (b) The assertion follows from Proposition 9, since when h satisfies condition (S0 ), @hf (x) is an upper h -pseudo-differential of f at x which is upper exact (c) Since h satisfies condition (S), @hf is an upper h-pseudo-differential of f at x Thus, the conclusion follows from the first assertion of Proposition Optimality conditions for problems with constraints In the present section, we apply the preceding concepts to the constrained minimization problem ðCÞ f ðxÞ subject to x C, 952 10 N.T.H Linh and J.-P Penot where C is a subset of X, and f : C ! R1 Let S be the set of solutions to (C) and let ! X* be a generalized differential of f @f : C ! Let us give a sufficient optimality condition for (C) PROPOSITION 12 Downloaded by [Florida Atlantic University] at 06:48 02 August 2013 (a) Let f be @f-pseudoconvex at a C If @f (a) ỵ N(C, a), then a is a solution to (C) (b) Let f be eventually @f-pseudoconvex at a C If À@f (a) & N(C, a), then a is a solution to (C) (c) If f is @f-pseudoconvex at a, if @f (a) is upper exact and if the following condition holds, then a is a solution to (C): sup haà , ui ! 8u VðC, aÞ: ð10Þ aà 2@f ðaÞ Proof (a) Let us note that the assumption of (a) is stronger than the assumption of (c) However, we give a direct proof If @f (a) ỵ N(C, a), then there exists a* @f (a) such that ha*, x À ! for all x C Since f is @f-pseudoconvex at a, one has f (x) ! f (a) for all x C : a S (b) Suppose À@f (a) & N(C, a) and f is eventually @f-pseudoconvex at a If for some x C one has f (x) f (a) then there exists a* @f (a) such that ha*, x À 0, a contradiction with Àa* N(C, a) (c) Supposing that a 2= S we show that (10) does not hold Let x C be such that f (x) f (a) Since f is @f-pseudoconvex at a, for all a* @f (a), we have ha*, x À Since @f (a) is upper exact, we get max{ha*, x À ai: g a* @f (a)} and (10) does not hold It follows from Lemma that the preceding proposition implies the result in [22, Prop 5] The following example shows that with some suitable choices of @f, the @f-pseudoconvexity assumption of the preceding proposition is easily fulfilled Example Let us take for @f the subdifferentials @0f and @6f given in [31, Example 7.2] as follows: xà @0 f xị , f xị ẳ f wị, hx , xi 1, hx , wi41 à ^ f ðwÞ, hxà , xi ! 1: x @ f xị , f xị ẳ hx , wi!1 Then, any function f is @f-pseudoconvex: if x* @0f (x) is such that hx*, w À xi ! for some w X, then hx*, wi ! hx*, xi 1, hence f (w) ! f (x) Similarly, we have that f is g @6f-pseudoconvex The next proposition is similar to [4, Prop 4.1] Note that here, we not need the quasiconvexity of f and moreover @f (x) may be different from N5 f ðxÞnf0g: PROPOSITION 13 Let C be convex and let f be @f-quasiconvex and radially upper semicontinuous at a C Assume that 2= @f (a) and C? :¼ {x* X* : 8w, x C, hx*, xi ¼ hx*, wi} ¼ {0} If @f (a) ỵ N(C, a), then a is a solution to (C) Optimization 953 11 Proof Since @f (a) ỵ N(C, a), there exists a* @f (a) such that ha*, x À ! for all x C Since C? ¼ 0, there exists w C such that ha*, w À Now, for any given x C, t ]0,1[, we define xt :¼ (1 À t)x ỵ tw so that , xt ẳ tịha , x ỵ tha , w À 0: Since f is @f-quasiconvex at a, one has f (xt) ! f (a) and then f (x) ! f (a) by radial upper semicontinuity of f at a Thus, a S g Downloaded by [Florida Atlantic University] at 06:48 02 August 2013 When f is @f-quasiconvex but is not quasiconvex, [4, Proposition 4.1, Theorem 4.3] cannot be applied Note that when f attains its minimum on C at a, the assumption that f is @f-quasiconvex at a is satisfied whatever @f is Such a situation occurs in the next example Example Let C :¼ [0, 2], a ¼ and f (x) ¼ min{x, jx À 2j} for all x R Let us take for @f (0) any nonempty subset of P, for instance @f (0) ¼ { f (0)} Then f is @f-quasiconvex at a on C and radially upper semicontinuous at a and @f (a) ỵ N(C, a) Thus, by the preceding proposition, a is a solution to (C) When a suitable bifunction h is available and @f is an upper h-pseudo-differential of f, condition (10) turns out to be necessary PROPOSITION 14 Let C be convex, h : VC ! R be such that h(a, ) ! dỵ f (a, ) and let a subset @f (a) of X* be an upper h-pseudo-differential of f at a If a S then condition (10) holds Proof Let a S and u V(C, a) Let x C be such that u ¼ x a Then dỵ f a, x aị hða, x À aÞ supfhaà , x À aiaà @f ðaÞg: g Under some additional convexity assumptions, the sufficient condition of Proposition 12(a) also is necessary PROPOSITION 15 Let C be convex, let h : VC ! R be such that h(a, ) ! Dỵf (a, ) and let @f (a) be an upper h-pseudo-differential of f at a Assume that h(a, Á) is sublinear and continuous If a S then w* cl(co(@f (a))) ỵ N(C, a) In particular, if @f (a) is weak* closed convex, one has @f (a) ỵ N(C, a) Note that the assumption that @f (a) is weak* closed convex is satisfied when @f (a) ¼ @hf (a) Proof Let a S One has h(a, v) ! for all v V(C, a) by [23, Proposition 27] or, equivalently, h(a, ) ỵ V(C,a)() ! Since C is convex, V(C, a) :ẳ P(C a) ẳRỵ(C a) is a convex set Hence we have @(h(a, ) ỵ V(C,a)())(0) so that @h(a, )(0) þ @V(C,a)(0) by a classical rule of convex analysis, h(a, Á) being sublinear and continuous Now @V(C,a)(0) ¼ N(C, a) and since @f (a) is an upper h-pseudodifferential of f at a, the set @h(a, Á)(0) is contained in the weak* closed convex hull of @f (a) Hence, we get w* cl(co(@f (a))) ỵ N(C, a) and, when @f (a) is weak* closed convex, @f (a) ỵ N(C, a) g Let us give some other optimality conditions in the spirit of Minty variational ! X* is said to be upper inequalities and [4] Recall [3,14] that a multimap T : C ! 954 12 N.T.H Linh and J.-P Penot sign-continuous at x C if, for every w C, the following implication (in which xt :ẳ tx ỵ (1 t)w) holds: 8t Š0, 1½, inffhxÃt , w À xi : xÃt Tðxt Þg ! ) supfhxà , w À xi : xà TðxÞg ! 0: Downloaded by [Florida Atlantic University] at 06:48 02 August 2013 Given a C, we will use the following assumption taken from [3]: à (A) There exists a convex neighbourhood V of a and an operator T : V \ C ! 2X with nonempty values, satisfying T(w)  @f (w) for all w V \ C and upper signcontinuous at a This condition is rather mild since it is satisfied when f is eventually T-colinfine [24] or when the restriction of T to all straight lines through a is upper semicontinuous at a w.r.t the w*-topology in X* It is also satisfied when a S and sup{ha*, w À : a* T(a)} ! dỵf (a, w a) for all w C ! X* and let f be @f-protoconvex on C PROPOSITION 16 Let C be convex, let @f : C ! If a S, then for any x C one has inffhxà , x À : xà @f ðxÞg ! 0: If moreover, assumption (A) holds for some V and T, then condition (10) is satisfied If furthermore T(a) is convex and w*-compact, then @f (a) ỵ N(C, a) In view of Proposition 4(a), the assumption that f is @f-protoconvex on C is satisfied whenever f is quasiconvex and @f is contained in @ rf Proof Suppose that there exist x C and x* @f (x) such that hx*, x À Since f is @f-protoconvex at x, one has f (a) f (x): a contradiction with a S Thus, inf{hx*, x À : x* @f (x)} ! for all x C When assumption (A) holds, so that there exist an upper-sign continuous multimap T and a convex neighbourhood V of a such that for any w V \ C, T(w)  @f (w), one has inf inffhwà , w À : wà TðwÞg ! 0: w2V\C Since T is upper sign-continuous and V \ C is a convex set, for w V \ C and t ]0,1[, we have wt :ẳ (1 t)a ỵ tw V \ C and inffhwÃt , wt À : wÃt Tðwt Þg ! hence inffhwÃt , w À : wÃt Tðwt Þg ! 0: By upper sign-continuity of T, we deduce that sup{ha*, w À : a* T(a)} ! for all w V \ C Using the convexity of C, we get condition (10) Moreover, when T(a) is convex, weak* compact, one has inf max haà , x À ! 0: x2C aà 2TðaÞ Now, applying the Sion minimax theorem or [3, Lemma 2.1(iii)], we get max inf fhaà , x À aig ! 0: aà 2TðaÞ x2C Hence there exists some a* T(a) & @f (a) such that Àa* N(C, a) g Example Let C :¼ [0, 2] and let f : R ! R be given by f (x) ¼ max(x, 0) for x 1, f (x) ¼ for x Then a ¼ is a solution to (C) Take @f (x) ¼ {1} if x [0,1] and @f (x) ¼ {0} if x R \ [0, 1] Then f is @f-protoconvex on C and all the Optimization 955 13 assumptions in the preceding proposition are satisfied with T :¼ @f Hence one has @f (a) þ N(C, a) Since f is neither radially continuous, upper semicontinuous, nor semi-strictly quasiconvex and a is a global minimizer, [4, Theorem 4.3], [5, Theorem 4.1, Theorem 4.7, Theorem 4.8, Theorem 4.9] cannot be applied g Now let us turn to optimality conditions formulated in terms of normal cones to sublevel sets The following result can be couched in terms of the adjusted normal cone to f as in [5, Prop 5.1]; here, we give a short proof which avoids this concept Downloaded by [Florida Atlantic University] at 06:48 02 August 2013 PROPOSITION 17 Let f be an upper semicontinuous, semi-strictly quasiconvex function on a convex C and let a X be such that f (a) infX f Then a S if, and only if, Nf (a)\{0} ỵ N(C, a) where Nf (a) :ẳ N(Sf (a), a) Proof Since a S we have that S5 f aị :ẳ fx X : f xị f ðaÞg is disjoint from C Since these two sets are convex and S5 f ðaÞ is open, the Hahn–Banach theorem yields some a* X*\{0} and some  R such that 8w S5 f ðaÞ 8x C haà , wi  ! haà , xi: Since f is semi-strictly quasiconvex, a belongs to C and the closure of S5 f ðaÞ: Thus ha*, ¼  and 8w S5 f ðaÞ 8x C haà , w À ! haà , x À ai: Therefore a* N(C, a) and a* Nf (a)\{0} since N5 f aị ẳ Nf ðaÞ by Lemma Now, assume that Nf (a)\{0} ỵ N(C, a) By Lemma 3, f is Nf\{0}-pseudoconvex at a Hence, we get a S by Proposition 12(a) g Under special assumptions, one has a stronger necessary condition PROPOSITION 18 (a) If Àf is (À@f )-protoconvex at a S, then À@f (a) & N(C, a) (b) If Àf is eventually (À@f )-protoconvex at a S, then condition (10) is satisfied Proof (a) Let a S Since Àf is À@f-protoconvex at a and, for all x C, Àf (x) À f (a), for all a* @f (a) one has h Àa*, x À for all x C so that Àa* N(C, a) Thus, À@f (a) & N(C, a) (b) Let a S Since Àf is eventually À@f-protoconvex at a and, for all x C, Àf (x) À f (a), there exists a* @f (a) such that ha*, x À ! Thus, we get g infx2C supaà 2@f ðaÞ haà , x À ! 0: With a similar proof, we have the following proposition PROPOSITION 19 Let a C be a strict solution of (C) (i.e for all x C, f (a) f (x)) (a) If Àf is (À@f )-quasiconvex at a, then À@f (a) & N(C, a) (b) If Àf is (À@f )-pseudoconvex at a, then À@f (a) & N(C, a)\{0} Example shows that if we take @f :¼ À@0(Àf ) or @f :¼ À@6(Àf ), the pseudoconvexity assumption of the preceding proposition is automatically satisfied 956 14 N.T.H Linh and J.-P Penot Mathematical programming problems Let us consider now the case in which the constraint set C is defined by a finite family of inequalities, so that problem (C) turns into the mathematical programming problem ðMÞ Minimize f xị subject to x C :ẳ fx X : g1 ðxÞ 0, , gn ðxÞ 0g, Downloaded by [Florida Atlantic University] at 06:48 02 August 2013 where f : W ! R1, gi : W ! R1 and W is an open convex subset of X Let a C, and let I :¼ {1, , n}, I(a) :¼ {i I : gi(a) ¼ 0} Let us first present sufficient optimality conditions PROPOSITION 20 Let f be @f-pseudoconvex at a C and gi be @gi-protoconvex at a and ð yi Þ RIðaÞ for i I(a) If for all x C, there exist a* @f (a), @gi aị (i I(a)) P ỵ P such that ỵ i Iaị yi , x ! (in particular, if a ỵ i Iaị yi ẳ for some a* @f (a) and some aÃi @gi ðaÞ for i I(a)), then a is a solution to problem (M) Proof Suppose on the contrary that there exists some x C such that f (x) f (a) Since f is @f-pseudoconvex at a and a* @f (a), we have ha*, x À Since for i I(a), gi is @gi-protoconvex at a and aÃi @gi ðaÞ, we have haÃi , x À Taking yi ị RIaị ỵ as in our assumption, we get a contradiction with the relation X , x ỵ yi hai , x À i2IðaÞ deduced from the preceding inequalities g Remark (a) According to Proposition 4, the condition that gi is @gi-protoconvex at a can be replaced by quasiconvexity of gi and @gi(a)  @ rgi(a) for i I(a) (b) By Lemma 2(b) and (c), the preceding proposition implies [22, Thm 11] A variant with a similar proof can be given PROPOSITION 21 Let f be eventually @f-pseudoconvex at a C and let gi be eventually all x C, a* @f (a) and aÃi @gi ðaÞ (i I(a)) @gi-protoconvex at a for i I(a) If forP IðaÞ Ã there exists ð yi ị Rỵ such that ỵ i2Iaị yi aÃi , x À ! 0, then a is a solution to problem (M) Another sufficient condition can be given PROPOSITION 22 Let f be @f-quasiconvex and radially upper semicontinuous at a C ? r and let gi be quasiconvex for i I Assume that 2= @f P(a), C ¼Ã {0} and @gi(a)  @ gi (a) IðaÞ Ã for i I(a) If there exists yi ị Rỵ such that a ỵ i Iaị yi ẳ for some a* @f (a) and some aÃi @gi ðaÞ (i I(a)), then a is a solution to problem (M) Proof By Proposition 4(a), gi is @g at a Let a* @f (a), aÃi @gi ðaÞ and Pi-protoconvex à à for all x C, i I(a), one has yi Rỵ (i I(a)) be such that a ỵ i Iaị yi ẳ 0: Since P gi(x) gi (a), hence haÃi , x À 0, we obtain a ẳ i Iaị yi  NðC, aÞ: Then Proposition 13 yields the conclusion g Now let us turn to necessary optimality conditions for (M) Let us first consider the case of a single constraint which is quasiconvex Optimization 957 15 ! X* and let f be @f-protoconvex Let g1 be quasiconvex, PROPOSITION 23 Let @f : W ! a S and g1(a) ¼ If assumption (A) holds for some V and T such that T(a) is convex w*-compact, then @f aị ỵ NSg1 aị, aị: Downloaded by [Florida Atlantic University] at 06:48 02 August 2013 Proof One has @f (a) ỵ N(C, a) by Proposition 16 On the other hand, since g1(a) ¼ 0, we have C ¼ Sg1 aị and NC, aị ẳ NSg1 aị, aị So, we have the conclusion g Now, let us turn to the general case We shall use the following lemma in which D :¼ {(xi)i2I(a) : 8j, k I(a), xj ¼ xk} is the diagonal of XI(a) Its proof is a consequence of the rule for computing the subdifferential of a sum of convex functions, applied to the case the functions are the indicator functions of the sets Ci :¼ gÀ1 i ððÀ1, 0ŠÞ: LEMMA 24 Assume gi is quasiconvex for all i I, gi is u.s.c at a for all i I\I(a) and let Ci :¼ gÀ1 i ððÀ1, 0ŠÞ, a X Assume that one of the following two conditions holds: (a) there exist some k I(a) and some z Ck such that gi(z) for each i I(a)\{k} (Slater condition) Q (b) X is complete, gi is l.s.c for all i I(a) and Rỵ D i Iaị Ci Þ ¼ XIðaÞ : Then NðC, aÞ ¼ P NðCi , aÞ: i2I This lemma and Proposition 16 entail the following result ! X*and let f be @f-protoconvex Suppose g1, , gn PROPOSITION 25 Let @f : W ! satisfy the assumptions of the preceding lemma, assumption (A) holds for some V and T such that T(a) is convex w*-compact Then P @f aị ỵ NðSgi ðaÞ, aÞ: i IðaÞ Note that this optimality condition can be formulated in terms of Gutie´rrez functions to the constraints Recall from [22] that f is a Gutie´rrez function at a if its sublevel set Sf (a) is convex and such that NSf aị, aị ẳ Rỵ @ f ðaÞ: By the preceding proposition and [22, Lemma 9], if the constraints (gi)i2I(a) are Gutie´rrez functions at a in the preceding proposition as in the preceding lemma, then P there exists yi ị RIaị ỵ such that @f aị ỵ i Iaị yi @ gi ðaÞ: Using representations of normal vectors to sublevels sets such as [7, Theorem 3.3.4], [29, Theorem 4.1], [39, Proposition 5.4] under appropriate assumptions, one can derive from the last proposition necessary conditions in fuzzy or limiting forms References [1] D Aussel, Subdifferential 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Linh and J.-P Penot, Generalized affine maps and generalized convex functions, Pac J Optimization (2008), pp 353–380 [24] N.T.H Linh and J.-P Penot, Generalized affine functions and generalized differentials, ... h-protoconvex) at x If f is @f-pseudoconvex at x and if @f (x) is upper exact, then f is h-pseudoconvex at x Proof Let f be @f-quasiconvex (resp @f-protoconvex) at x and let w C be such that f... @f-pseudoconvex at x ẳ) f is @f-quasiconvex at x, 6ị f is @f-protoconvex at x ẳ) f is @f-quasiconvex at x: ð7Þ We say that f is @f-pseudoconvex on C (in short @f-pseudoconvex) if it is @f-pseudoconvex