This article was downloaded by: [Heriot-Watt University] On: 07 March 2015, At: 01:44 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 An approximate Hahn–Banach theorem for positively homogeneous functions a b c d N Dinh , E Ernst , M.A López & M Volle a Department of Mathematics, International University, Vietnam National University, Ho Chi Minh City, Vietnam b UMR7353, Aix-Marseille University, Marseille, France c Click for updates Faculty of Sciences, Department of Statistics and Operations Research, Alicante University, Alicante, Spain d EA2151, Université d’Avignon et des Pays de Vaucluse, Avignon Cedex 1, France Published online: 17 Dec 2013 To cite this article: N Dinh, E Ernst, M.A López & M Volle (2015) An approximate Hahn–Banach theorem for positively homogeneous functions, Optimization: A Journal of Mathematical Programming and Operations Research, 64:5, 1321-1328, DOI: 10.1080/02331934.2013.864290 To link to this article: http://dx.doi.org/10.1080/02331934.2013.864290 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 [Heriot-Watt University] at 01:44 07 March 2015 Conditions of access and use can be found at http://www.tandfonline.com/page/termsand-conditions Optimization, 2015 Vol 64, No 5, 1321–1328, http://dx.doi.org/10.1080/02331934.2013.864290 An approximate Hahn–Banach theorem for positively homogeneous functions N Dinha , E Ernstb∗ , M.A Lópezc and M Volled a Department of Mathematics, International University, Vietnam National University, Ho Chi Minh City, Vietnam; b UMR7353, Aix-Marseille University, Marseille, France; c Faculty of Sciences, Downloaded by [Heriot-Watt University] at 01:44 07 March 2015 Department of Statistics and Operations Research, Alicante University, Alicante, Spain; d EA2151, Université d’Avignon et des Pays de Vaucluse, Avignon Cedex 1, France (Received 10 April 2013; accepted 24 October 2013) This note provides an approximate version of the Hahn–Banach theorem for non-necessarily convex extended-real valued positively homogeneous functions of degree one Given p : X → R∪{+∞} such a function defined on the real vector space X , and a linear function defined on a subspace V of X and dominated by p (i.e (x) ≤ p(x) for all x ∈ V ), we say that can approximately be p-extended to X , if is the pointwise limit of a net of linear functions on V , every one of which can be extended to a linear function defined on X and dominated by p The main result of this note proves that can approximately be p-extended to X if and only if is dominated by p∗∗ , the pointwise supremum over the family of all the linear functions on X which are dominated by p Keywords: approximate Hahn–Banach theorem; non-convex Hahn–Banach theorem; Fenchel–Legendre conjugate; positively homogeneous functions of degree one AMS Subject Classifications: 46A22; 46A20 Introduction Let us consider X , a non-zero real vector space, and p : X → R ∪ {+∞}, an extended-real valued function which is positively homogeneous of degree one (in short, a ph-function) Given V , a linear subspace of X , we call the linear function : V → R such that (x) ≤ p(x) for all x ∈ V , a p-dominated linear function As customary, we say that a p-dominated linear function : V → R can be p-extended to X if there is a p-dominated linear function g : X → R such that g(x) = (x) for all x ∈ V The Hahn–Banach theorem (the Crown Jewel of Functional Analysis, as called in [1]), simply says that, provided that the ph-function p is real-defined and convex (in other words, sublinear), then any p-dominated linear function : V → R can be p-extended to X , generally in more than one way A surprising fact occurs when, as requested in many constraint optimization problem, p is allowed to take the value +∞ When the dimension of the underlying space X is ∗ Corresponding author Email: emil.ernst@univ-amu.fr © 2013 Taylor & Francis Downloaded by [Heriot-Watt University] at 01:44 07 March 2015 1322 N Dinh et al infinite, S Simons provided (see the paragraph of the article [2] entitled ‘Counterexample to 4’, at p.114) a highly counterintuitive example: a hypolinear function (that is a convex ph-function) p : X → R ∪ {+∞} such that the inequality g ≤ p is false for all the linear mappings g : X → R Arguably, the better illustration of the difficulty to address this case is the unusually large number of flawed Hahn–Banach type theorems for hypolinear functions which can be found in the mathematical literature; in the articles,[3–5] the reader can find examples and criticism of as much as seven such incorrect results published between 1969 and 2005 The first correct Hahn–Banach theorem for hypolinear functions have been achieved by Anger and Lembcke, [3, Theorem 2.4, p.135] (result generalized for multifunctions in [6]): the authors prove that a p-dominated linear function : V → R can be p-extended to X if and only if the largest hypolinear minorant of both the linear and the hypolinear functions, namely p : X → R ∪ {+∞}, p (x) = inf { p(x + y) − (y) : y ∈ V }, is lower semi-continuous at with respect to the finest locally convex topology on X However, as remarked by the authors themselves ([7, p.251]): ‘[not even the] continuity of the linear form and lower semicontinuity of the hypolinear domination functional […] guarantee the existence of a dominated […] linear extension, even in a finite dimensional setting’ In order to overcome this difficulty and achieve a result easier to use in applications, we address, in very much the same spirit as the ε-strategies in mathematics analysed in [8], an approximate version of the Hahn–Banach theorem for extended-real valued functions which are positively homogeneous of degree one but non-necessarily convex Namely, given : V → R, a p-dominated linear function, we say that can approximately be p-extended to X , if is the pointwise limit of a net ( i )i∈I of p-dominated linear functions on V , every one of which can be p-extended to X A simple remark states that if a p-dominated linear function : V → R can approximately be p-extended to X , then must be dominated not only by p, but also by p ∗∗ , where p ∗∗ : X → R ∪ {+∞} is the pointwise supremum over the family of p-dominated linear functions on X (equivalently, p ∗∗ is the bi-conjugate of p with respect to the finest locally convex topology defined on X ) Our main result, Theorem 2.3, proves that the simple necessary condition previously defined is sufficient for approximately extending a p-dominated linear function Namely, we prove that given p : X → R ∪ {+∞}, a (non-necessarily convex) ph-function and V a linear subspace of X , then a p-dominated linear function : V → R can approximately be p-extended to X if and only if is also dominated by p ∗∗ On this ground, we address the class of ph-functions with the V -approximate extension property in the setting of a Hausdorff locally convex space X : Proposition 3.1 proves that, given a ph-function p, and a closed subspace V of X , any p-dominated linear function ∈ V ∗ can approximately be p-extended to X , if and only if p ∗∗ |V = ( p|V )∗∗ Finally, Theorem 4.1 addresses the case of ph-functions with the approximate extension property, and proves that, given a ph-function p, then, for any closed subspace V of X , any p-dominated linear function : V → R can approximately be p-extended to X , if and only if p and p∗∗ coincide Optimization 1323 The main result Let X be a non-zero real vector space endowed with a locally convex topology τ ; its continuous dual, X ∗ , is the vector space of all the linear and continuous functions on X Of a particular interest for our study is the σ (X ∗ , X )-topology on X ∗ (the weak topology), customary defined as the coarsest locally convex topology on X ∗ which makes continuous all the mappings Downloaded by [Heriot-Watt University] at 01:44 07 March 2015 Tx : X ∗ → R, Tx (g) = g(x), g ∈ X ∗ , x ∈ X Any closed linear subspace V of X is tacitly considered as endowed with the topology induced by τ ; similarly, we define V ∗ , the continuous dual of V , and endow it with the σ (V ∗ , V )-topology ([9, Chapter 6] provides an excellent short introduction to this topic) Finally, let us pick p : X → R ∪ {+∞}, a ph-function, that is an extended-real-valued function which fulfills the relation p(s x) = sp(x) ∀s ≥ 0, x ∈ X; we make the convention that · ∞ = So any ph-function vanishes at Definition 2.1 Let ∈ V ∗ be a p-dominated continuous linear function We say that can approximatively be p-extended to X with respect to τ , if is the σ (V ∗ , V )-limit of a net ( i )i∈I ⊂ V ∗ of p-dominated linear functions on V , every one of which can be p-extended to X by an element L i in X ∗ Remark Let us consider the particular case when V = {0} and the linear function : V → R is defined by (0) = As is the only linear function on V , saying that can approximately be p-extended to X with respect to τ simply means that can be p-extended to X (any of the elements of the net ( i )i∈I which σ (V ∗ , V )-converges to is necessarily equal to ) Moreover, as any linear and continuous function on X is an extension of , we may conclude that = can approximately be p-extended to X with respect to τ if and only if p dominates at least one element of X ∗ Before proceeding to our main technical result, we recall the definition of the Fenchel– Legendre conjugate for functions defined on X and on X ∗ Let us consider h : X → R ∪ {+∞} and j : X ∗ → R ∪ {+∞}, two extended-real-valued function on X and X ∗ The conjugate of h is defined as h ∗ : X ∗ → R ∪ {−∞, +∞}, h ∗ ( f ) = sup{ f (x) − h(x) : x ∈ X }, while relation j ∗ : X → R ∪ {−∞, +∞}, j ∗ (x) = sup{ f (x) − j ( f ) : f ∈ X ∗ } defines the conjugate of j Proposition 2.2 Let us consider (X, τ ), a non-zero Hausdorff locally convex real vector space, p : X → R ∪ {+∞}, a ph-function, V , a closed linear subspace of X and ∈ V ∗ , a p-dominated continuous linear function The two following sentences are equivalent: 1324 (i) (ii) N Dinh et al 0 can approximately be p-extended to X with respect to τ , is dominated by p ∗∗ , the bi-conjugate of p Proof Let us first remark that p ∗ : X ∗ → R ∪ {+∞} is the conjugate of a ph-function, so it is a convex function which can only take two values: and +∞ Moreover, relation p∗ (g) = holds if and only if the function g ∈ X ∗ is p-dominated Let us define the mapping v : V ∗ → R ∪ {+∞} as Downloaded by [Heriot-Watt University] at 01:44 07 March 2015 v( ) = inf { p ∗ (g) : g ∈ X ∗ , g|V = } It is straightforward that v is also a convex function which can take only the values and +∞; moreover, the definition of v implies that the linear function ∈ V ∗ can be p-extended to X by an element in X ∗ if and only if v( ) = Set now A = { ∈ V ∗ : v( ) = 0}; Definition 2.1 reads that can approximatively be p-extended to X with respect to τ if and only if it lies in the σ (V ∗ , V )-closure of the set A But v is the indicator function of the set A, v( ) = ι A ( ) = ∈A , ∈ / A +∞ so, by virtue of a well-known convex analysis result (see for instance [9, p.3, below Proposition 6.2]), it follows that its bi-conjugate is the indicator function of the σ (V ∗ , V )closure of the set A: v ∗∗ = ι σ (V ∗ ,V ) A To the purpose of computing v ∗∗ , let us consider x ∈ V Relation g|V = means that (x) = g(x); it results that v ∗ (x) = sup ( (x) − v( )) = sup ∈V ∗ = sup ∈V ∗ ∈V ∗ (x) + sup (x) − inf g∈X ∗ , g|V = p∗ (g) (− p ∗ (g)) g∈X ∗ , g|V = ∗ ∗∗ = sup g(x) − p (g) = p (x) g∈X ∗ ∗ ∗ Accordingly, v ∗∗ = ( p∗∗ )|V , that is ι σ (V ∗ ,V ) = ( p∗∗ )|V A We have thus proved the following facts: first, that the linear and continuous function : V → R can approximatively be p-extended to X with respect to τ if∗and only if σ (V ,V ) belongs to the σ (V ∗ , V )-closure of the set A, second, that belongs to A if and only ∗ ∗ ∗∗ ∗∗ if ( p )|V ( ) = 0, and third, that ( p )|V ( ) = if and only if is dominated by the hypolinear function p ∗∗ The conclusion of Proposition 2.2 follows by combining these three sentences Let us consider the case when τ is the finest locally convex topology defined on X Then, every linear subspace V of X is closed, and any linear function : V → R is continuous Moreover, as for any locally convex topology τ , and for any ph-function p : X → R∪{+∞} it holds that p ∗∗ (x) = sup{g(x) : g ∈ X ∗ , g ≤ p} Optimization 1325 (see for instance [9, Proposition 6.1]), we deduce that, for the finest locally convex topology on X , the bi-conjugate of the function p is the pointwise supremum over the family of p-dominated linear functions g : X → R We conclude that, when specified to the case of the finest locally convex topology, Proposition 2.2 yields the following approximate version of the Hahn–Banach theorem Theorem 2.3 Let us consider X , a non-zero real vector space, together with p : X → R ∪ {+∞}, a ph-function, V , a linear subspace of X , and : V → R, a p-dominated linear function The two following sentences are equivalent: Downloaded by [Heriot-Watt University] at 01:44 07 March 2015 (i) (ii) can approximatively be p-extended to X and is dominated by the pointwise supremum over the family of linear functions on X which are p-dominated Positively homogeneous functions of degree one with the V -approximate extension property The main result of this section uses Proposition 2.2 in order to provide an approximate version of the Hahn–Banach theorem valid for all the linear functions defined on some subspace V of X Proposition 3.1 Let us consider (X, τ ), a locally convex real vector space, p : X → R ∪ {+∞}, a ph-function and V , a closed linear subspace of X The two following sentences are equivalent: (i) (ii) any linear, continuous and p-dominated function : V → R can approximately be p-extended to X with respect to τ and p∗∗ |V = ( p|V )∗∗ ≤ p} be the set of all the linear and Proof (i) ⇒ (ii) Let P(V ) = { ∈ V ∗ : continuous functions defined on V which are dominated by p As already noticed, for any x ∈ V it holds that ( p|V )∗∗ (x) = sup{ (x) : ∈ P(V )}; moreover, statement (i) says that any ∈ P(V ) can approximately be p-extended to X with respect to τ , so, in virtue of Proposition 2.2, (x) ≤ p∗∗ (x) ∀x ∈ V, ∈ P(V ) Accordingly, ( p|V )∗∗ (x) ≤ p ∗∗ (x) ∀x ∈ V ; since the opposite inequality is clear, statement (ii) follows (ii) ⇒ (i) It is well-known that the bi-conjugate of an extended-real valued function defined on X which is not identically equal to +∞, either takes only the value −∞, or is an extended-real valued lower semi-continuous and convex function which takes at least one real value (the class of those functions is usually called (X )) 1326 N Dinh et al Downloaded by [Heriot-Watt University] at 01:44 07 March 2015 If p ∗∗ amounts to −∞ X , the constant function on X taking the value −∞, relation (ii) implies that the same holds for ( p|V )∗∗ ; thus, the set P(V ) is void, so statement (i) obviously holds Let us now assume now that p ∗∗ belongs to (X ), and let ∈ V ∗ be such that ≤ p on V By Proposition 2.2 one has to check that ≤ p ∗∗ on V Now, ≤ p|V , a fact which implies that ≤ ( p|V )∗∗ By relation (ii), we know that ( p|V )∗∗ = ( p∗∗ ) |V , hence ≤ ( p∗∗ ) |V , that is ≤ p ∗∗ on V Remark Recently, an analytical result involving affine functions dominated by an extended-real valued mapping and their approximate extensions, has been used by Dinh et al [10] to derive a series of subdifferential calculus rules, different generalizations of Farkas lemma for non-necessarily convex systems, optimality conditions and duality theory for infinite optimization problems Theorem 1, the main result from the article [10] (for a convex version, the reader is refereed to [11, Proposition 1]), reads as follows: [10, Theorem 1]: Let U and X be two locally convex spaces, and consider a function F : U × X → R ∪ {+∞} such that F is finite at some point of × X , and F ∗ is finite at some point of U ∗ × X ∗ Then, the following statements are equivalent: (a) F ∗∗ (0, ·) = (F(0, ·))∗∗ (b) For any h : X → R ∪ {+∞}, convex and lower semi-continuous function, the two following statements are equivalent: (b1 ) F(0, x) ≥ h(x), ∀x ∈ X (b2 ) if, for some x ∗ ∈ X ∗ , h ∗ (x ∗ ) is finite, then there exists a net (u i∗ , xi∗ , εi )i∈I ⊂ U ∗ × X ∗ × R+ such that F ∗ (u i∗ , xi∗ ) ≤ h ∗ (x ∗ ) + εi , ∀i ∈ I , and limi∈I (xi∗ , εi ) = (x ∗ , 0) This result may be used in order to give an alternative proof of the implication (ii) ⇒ (i) of Proposition 3.1, by setting U = X , F(u, x) := p(x + u) + ιV (x), h(x) = (x) +∞ (u, x) ∈ X × X, x∈V , x ∈ X\V and by noticing that statements (a) from [10, Theorem 1] and (ii) from Proposition 3.1 are equivalent, while the statement (i) from Proposition 3.1 is implied by statement (b2 ) from [10, Theorem 1] Positively homogeneous functions of degree one with the approximate extension property The following result characterizes the class of hypolinear functions with the approximate extension property, that is hypolinear functions p such that, for any subspace V of X , any p-dominated linear function : V → R can approximately be p-extended to X Optimization 1327 Theorem 4.1 Let us consider (X, τ ), a Hausdorff locally convex real vector space, and p : X → R ∪ {+∞}, a positively homogeneous function of degree one The two following sentences are equivalent: Downloaded by [Heriot-Watt University] at 01:44 07 March 2015 (i) (ii) p is lower semi-continuous and convex, for any closed subspace V of X , any p-dominated continuous linear function ∈ V ∗ can approximately be p-extended to X with respect to τ Proof (i) ⇒ (ii) Any lower semi-continuous and convex function which takes at least one real value coincides with its bi-conjugate Accordingly, if p ∈ (X ), then any p-dominated function is also p ∗∗ -dominated, and Proposition 2.2 proves the desired implication (ii) ⇒ (i) Let us consider p : X → R ∪ {+∞}, a ph-function with the approximate extension property Then it is impossible for p ∗∗ to take the value −∞ (proven as in Remark 2, by taking V := {0}) To the aim of achieving a contradiction, let us assume that there exists x ∈ X such that p ∗∗ (x) < p(x), and let us pick α such that p ∗∗ (x) < α ≤ p(x) Let us first notice that −α ≤ p(−x); indeed, if −α > p(−x), then p ∗∗ (−x) ≤ p(−x) < −α, whence the contradiction = p∗∗ (0) ≤ p ∗∗ (x) + p ∗∗ (−x) < α − α = Take V := R x (a closed linear subspace of X because τ is Hausdorff) and : V → R defined by (t x) := t α Clearly, (y) ≤ p(y) for every y ∈ V (this is because p is positively homogeneous, α ≤ p(x), and −α ≤ p(−x)) By Proposition 2.2, we have that ∗∗ ∗∗ ≤ p ; in particular, α = (x) ≤ p (x), a contradiction References [1] Narici L, Beckenstein E The Hahn–Banach theorem: the life and times Topology Appl 1997;77:193–211 [2] Simons S Extended and sandwich versions of the Hahn–Banach theorem J Math Anal Appl 1968;21:112–122 [3] Anger B, Lembcke J Hahn–Banach type theorems for hypolinear functionals Math Ann 1974;209:127–151 [4] Z˘alinescu C On zero duality gap and the Farkas lemma for conic programming Math Oper Res 2008;33:991–1001 [5] Z˘alinescu C Hahn–Banach extension theorems for multifunctions revisited Math Methods Oper Res 2008;68:493–508 [6] Malivert C, Penot J-P, Thera M Un prolongement du théorème de Hahn-Banach C R Acad Sci., Paris 1978;286:165–168 [7] Anger B, Lembcke J Extension of linear forms with strict domination on locally compact cones Math Scand 1980;47:251–265 [8] Hiriart-Urruty J-B, López MA, Volle M The ε-strategy in variational analysis: illustration with the closed convexification of a function Rev Mat Iberoamericana 2011;27:449–474 [9] Moreau J-J Fonctionelles Convexes, Séminaire “Equations aux dérivées partielles,” Collège de France, 1966 and Edizioni del Dipartimento di Ingegneria Civile dell’Università di Roma Tor Vergata, Roma, 2003 1328 N Dinh et al Downloaded by [Heriot-Watt University] at 01:44 07 March 2015 [10] Dinh N, López MA, Volle M Functional inequalities in the absence of convexity and lower semicontinuity with applications to optimization SIAM J Optim 2010;20:2540–2559 [11] Dinh N, Goberna MA, López MA, Volle M Convex inequalities without constraint qualification nor closedness condition, and their applications in optimization Set-Valued Anal 2010;18: 423–445 ... http://dx.doi.org/10.1080/02331934.2013.864290 An approximate Hahn–Banach theorem for positively homogeneous functions N Dinha , E Ernstb∗ , M.A Lópezc and M Volled a Department of Mathematics, International University, Vietnam National... in applications, we address, in very much the same spirit as the ε-strategies in mathematics analysed in [8], an approximate version of the Hahn–Banach theorem for extended-real valued functions. .. property, and proves that, given a ph-function p, then, for any closed subspace V of X , any p-dominated linear function : V → R can approximately be p-extended to X , if and only if p and p∗∗