DSpace at VNU: A closedness condition and its applications to DC programs with convex constraints

This article was downloaded by: [University of California Santa Barbara] On: 31 July 2013, At: 07:23 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 A closedness condition and its applications to DC programs with convex constraints a b N Dinh , T.T.A Nghia & G Vallet c a Department of Mathematics, International University, Ho Chi Minh City, Vietnam b Department of Mathematics and Computer Science, Ho Chi Minh City University of Pedagogy, Vietnam c Laboratory of Applied Mathematics, UMR-CNRS 5142 University of PAU IPRA, BP 1155, 64013 Pau Cedex, France Published online: 31 Mar 2008 To cite this article: N Dinh , T.T.A Nghia & G Vallet (2010) A closedness condition and its applications to DC programs with convex constraints, Optimization: A Journal of Mathematical Programming and Operations Research, 59:4, 541-560, DOI: 10.1080/02331930801951348 To link to this article: http://dx.doi.org/10.1080/02331930801951348 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, Downloaded by [University of California Santa Barbara] at 07:23 31 July 2013 systematic supply, or distribution in any form to anyone is expressly forbidden Terms & Conditions of access and use can be found at http://www.tandfonline.com/page/termsand-conditions Optimization Vol 59, No 4, May 2010, 541–560 A closedness condition and its applications to DC programs with convex constraints N Dinha*, T.T.A Nghiab and G Valletc a Department of Mathematics, International University, Ho Chi Minh City, Vietnam; Department of Mathematics and Computer Science, Ho Chi Minh City University of Pedagogy, Vietnam; cLaboratory of Applied Mathematics, UMR-CNRS 5142 University of PAU IPRA BP 1155, 64013Pau Cedex, France Downloaded by [University of California Santa Barbara] at 07:23 31 July 2013 b (Received 24 October 2006; final version received November 2007) This paper concerns a closedness condition called (CC) involving a convex function and a convex constrained system This type of condition has played an important role in the study of convex optimization problems Our aim is to establish several characterizations of this condition and to apply them to study problems of minimizing a DC function under a cone-convex constraint and a set constraint First, we establish several so-called ‘Toland–Fenchel–Lagrange’ duality theorems As consequences, various versions of generalized Farkas lemmas in dual forms for systems involving convex and DC functions are derived Then, we establish optimality conditions for DC problem under convex constraints Optimality conditions for convex problems and problems of maximizing a convex function under convex constraints are given as well Most of the results are established under the (CC) condition This article serves as a link between several corresponding known ones published recently for DC programs and for convex programs Keywords: DC programs; closedness conditions; closed-cone constraint qualification; Farkas lemmas; Fenchel–Lagrange duality; Toland–Fenchel–Lagrange duality AMS Classifications: 90C25; 90C26; 90C46; 49K30 Introduction Let us consider the DC optimization problem with convex constraints: Pị inf ẵfxị gxị subject to x C, hðxÞ ÀS: Throughout this article, we assume that: X, Z are real locally convex Hausdorff topological vector spaces, X* (resp Z*) denotes the topological dual of X (resp Z), endowed with the weak*-topology; C is a closed convex subset of X; f, g : X ! R [ fỵ1} are functions such that f is proper lower semicontinuous (l.s.c.) convex and g is proper convex and satisfies g** ¼ g on the feasible set A :¼ hÀ1(ÀS) \ C of (P) *Corresponding author Email: ndinh@hcmiu.edu.vn ISSN 0233–1934 print/ISSN 1029–4945 online ß 2010 Taylor & Francis DOI: 10.1080/02331930801951348 http://www.informaworld.com 542 N Dinh et al Further assume that S is a closed convex cone of Z (not necessarily with non-empty interior) and h : X ! Z is an S-convex mapping, i.e 8u, v X, 8t ½0, 1, htu ỵ tịvị thuị À tÞhðvÞ ÀS, such that   h is l.s.c for each  Sỵ, the dual cone of S, defined by: Downloaded by [University of California Santa Barbara] at 07:23 31 July 2013 Sỵ :ẳ f Z j ð, sÞ ! 0, for all s Sg: This last property, called star S-l.s.c in [23], corresponds to an extension of the notion of lower semi-continuity to vector-valued functions Other related notions already exist in the literature, e.g in the sense of Penot–Thera in [28] or in a sequential sense in [12] Since any  Sỵ is a S-nondecreasing continuous function, Lemma 1.7 in [28] yields that any l.s.c function in the sense of Penot–Thera is star S-l.s.c., notion equivalent to the one given in [12] for metric spaces (see [12, Proposition 3.6]) For convenience, for any  Z*, the composition of mappings   h would be denoted by h We define, by convention, that þ1 À (þ1) ¼ þ1 Recall that a function p : X ! R [ fỵ1} is called a DC function, if it can be decomposed as a difference of two convex functions Such a class of functions covers the classes of convex functions, concave functions, and many other non-convex functions (see, e.g [33,35]) The DC problem (P) has been studied by many authors since the last decades (see [1,18,22,26,29,31–35] and references therein) Many real world problems possess this mathematical model (see [2–4]) and several numerical methods have been developed for this class of problems as well (see [2,19,33,35] for an overview) It is well-known that for convex and DC optimization problems, a constraint qualification is an essential ingredient for the Lagrange multiplier rule and for the duality theory The well-known constraint qualifications for convex and DC optimization are often of Slater-type conditions (see [12,26,27] for instance) However, these conditions are often not satisfied for many problems in applications In recent years, a constraint qualification called closed-cone constraint qualification (or (CCCQ) for short), has been developed and used in [9,10,13,14,21,25] for convex (infinite) optimization problems Moreover, in the cases where the cost functional is not continuous at any point in the feasible set, another condition, often called the closedness condition [10,11,13,14], should be imposed We are interested in a condition called closedness condition (CC) [9] (see also [13,14]) that replaces both the mentioned conditions We will give several characterizations of this condition These characterizations will pave the way to derive strong duality and optimality conditions for the DC problem (P) Concerning the problem (P), we consider the system  :¼ fx C, hðxÞ ÀSg and the set of its solution A which is the feasible set of (P), A :ẳ fx X j x C, hxị Sg ẳ C \ h1 Sị: Throughout this article, we assume that A \ dom f 6¼ ; and we denote by [ K :ẳ epihị ỵ epiC , 2Sỵ where * stands for the conjugate of the function ’ ð1Þ Optimization 543 Downloaded by [University of California Santa Barbara] at 07:23 31 July 2013 The system  is said to satisfy the closed clone constraint qualification ((CCCQ), in brief) when K is weak*-closed The assumption that  satisfies, CCCQ serves as a constraint qualification in the study of convex optimization problems It was proposed in [21] and was used in [8,25] and in [13,14] to establish optimality conditions, duality and stability for convex (infinite) programming problems Various sufficient conditions for (CCCQ) were given in these mentioned papers In particular, it was shown that the constraint qualification (CCCQ) is strictly weaker than several generalized Slater type ones, and weaker than the Robinson-type one stating that Rỵ [S ỵ h(C)] is a closed subspace (see [8,21] for more details) Let us introduce the following closedness condition: CCị epif ỵ K is weakà -closed It involves the function f and the system  and will play a crucial role in the sequel of this article This closedness condition was proposed for the first time by Burachik and Jeyakumar in [9] Then, it has been used in [24] to establish optimality conditions of Karush–Kuhn– Tucker form for convex cone-constrained programs Several sufficient conditions for (CC) were given in [9] and [24] A relaxed version, stating that epif * ỵ clK is weak*-closed, was introduced in [13] and [14] where this last condition, with the (CCCQ) implies (CC), were used to establish optimality conditions, duality and stability results, for convex infinite programs Let us mention too that the (CC) condition will be satisfied if  satisfies (CCCQ) and: (i) if on the one hand f is continuous at least at one point in A (see [9,13]), or (ii) on the other hand, if cone(dom f À A) is a closed subspace of X Indeed, if cone(dom f À A) ¼ cone(dom f À domA) is a closed subspace, by [9, Proposition 3.1], epi f ỵ epi A is weak*-closed and so, thanks to (4), epi f ỵ epi A ẳ epi f ỵ clK ẳ epi f ỵ K is weakà -closed Note that if the set S constraint ‘x C’ is absent, i.e if  :¼ fh(x) S}, the condition (CC) becomes: epif * ỵ 2Sỵ epi(h)* is weak*-closed In this article, characterizations for (CC) in dual forms and in terms of approximate sub-differentials will be established These results will serve as main tools to establish duality results and optimality conditions for (P) We first consider various dual problems of (P) which will be called ‘Toland–Fenchel–Lagrange’-type dual problems (see, [26] e.g.) It is, in some sense, a ‘combination’ of Toland dual for DC problem in [31], and Fenchel and Lagrange dual problems (see [7,8,21]) We establish several duality theorems which extend Laghdir’s one in [26] In particular, these results would yield the stable strong duality for convex programs under linear perturbations [16,23] Moreover, various versions of generalized Farkas lemmas in dual forms for systems involving convex and DC functions are derived Applied to convex systems or to convex programs, these results give strong Lagrange duality and Fenchel–Lagrange duality results, and extend known ones established, for example in [7,18,13,15,21] Optimality conditions for DC problem are also obtained Our results are obtained under the (CC) condition and by using its characterizations given in Section They serve as a link between several corresponding known ones published recently for DC programs and for convex programs Downloaded by [University of California Santa Barbara] at 07:23 31 July 2013 544 N Dinh et al This article is organized as follows: in Section 2, we fix some notations and recall results needed in the sequel of this article Characterizations of the condition (CC) are given in Section 3, and corollaries are derived from simple cases Characterizations of (CCCQ) are also proposed In particular, a representation for the approximate normal cones to convex constrained sets is given In Section 4, we establish several duality results of ‘Toland–Fenchel–Lagrange’ type for the problem (P), which extend some recent ones given in [26] Corollaries for concrete classes of problems, including the classes of convex and concave programs, are obtained, which are compatible with the ones given in [7,13,14] We establish in Section 5, various versions of generalized Farkas lemma in dual forms for systems involving convex and DC functions, which go back to the ones established in [7] for convex systems In Section 6, optimality conditions for (P), as well as for convex problems, are proved The problems of maximizing a convex function under convex constraints is also treated as a special case Preliminaries Let us fix some notations used in the sequel of this paper For a set D & X, the closure (resp the convex hull) of D will be denoted by cl D (with suitable topology) (resp co D) and cone D stands for the convex cone generated by D The indicator function of a set D & X is defined by: D(x) ¼ if x D and D(x) ẳ ỵ1 else Moreover, the support function  D is given by  D(u) ¼ supx2Du(x) Let f : X ! R [ fỵ1} be a proper l.s.c convex function Then: (i) The conjugate function of f, f * : X* ! R[ fỵ1}, is defined by f vị ¼ supfvðxÞ À fðxÞ j x dom f g, where the domain of f is given by dom f :ẳ fx X j f(x)5ỵ1}, (ii) If a dom f then, following [20], [ÈÀ É Á v, vðaÞ þ  À fðaÞ j v @ fðaÞ , epi f ẳ 2ị !0 where, for a given  ! 0, the -subdifferential of f at a dom f, @f(a), is defined as the possibly empty closed convex set: @ faị ẳ fv X j fxị À fðaÞ ! ðv, x À aÞ À , 8x dom fg: T (iii) If 40 then @f(a) 6¼ ; Moreover, 40 @f(a) ¼ @f(a), where @f(a) denotes the usual convex subdifferential of f at a (for more details, see [36]) For a closed convex subset D & X and an arbitrary  ! 0, the -normal cone to D at a point x D is defined by (see [17,18])  :ẳ @ D xị  ẳ fx Xà j ðxà , x À xÞ  N D, xị , 8x Dg:  ẳ N(D, x)  is the normal cone to D at x in the sense of convex analysis When  ¼ 0, N0(D, x) Following [30] and [9], it is worth noting that for two proper l.s.c convex functions f1 and f2 with dom f1 \ dom f2 6ẳ ;, epi f1 ỵ f2 ị ẳ clepi f ỵ epi f Þ: ð3Þ Note moreover, that a sufficient condition for epi f1 ỵ epi f2 to be weak*-closed, is that at least one of the functions f1 or f2 is continuous at some point of dom f1 \ dom f2 Optimization 545 Note that, if, as it is assumed, h is a S-convex mapping such that h is l.s.c for each  Sỵ, then h1(S) :ẳ fx X j Àh(x) S} is convex and A :¼ C \ hÀ1(ÀS) is closed This last T assertion comes from the remark that (Sỵ)ỵ ẳ S (by separation theorem) and A ẳ 2Sỵ fx X, h(x) 0} \ C Moreover, [2Sỵ epi(h)* is a convex cone (see [22]) and Downloaded by [University of California Santa Barbara] at 07:23 31 July 2013 epiA ẳ clK 4ị (note that the equality (4) has been proved in [13,21,24]) It is worth noticing that, if the mapping h is sequentially l.s.c in the sense given in [12], then h is l.s.c for each  Sỵ, provided that X is metrizable (see [12, Proposition 3.7]) Let us conclude this section by recalling some results on duality and optimality conditions for DC programs established by Toland in [31,32] and by Hiriart-Urruty in [17] LEMMA 2.1 [31,32] Let X be a locally convex Hausdorff topological vector space with X* its topological dual Let further F, G : X ! R [ fỵ1} Assume that G is a proper, convex and l.s.c function and F is an arbitrary function Then inf fFxị Gxịg ẳ inf fG ðuÞ À Fà ðuÞg: x2X u2X LEMMA 2.2 [17] Let X be a locally convex Hausdorff topological vector space and F, G : X ! R[ fỵ1} be l.s.c, proper and convex functions Then (i) A point a dom F \ dom G is a global minimizer of the problem infx2X fF(x) À G(x)} if and only if for any  ! 0, @G(a) & @F(a) (ii) If a dom F \ dom G is a local minimizer of infx X fF(x) À G(x)} then @G(a) & @F(a) It is worth observing that the conclusions of Lemma 2.1 and Lemma 2.2 still hold if the condition that G is l.s.c is replaced by ‘G(x) ¼ G**(x) for all x domF ’ The proofs of these conclusions are modifications of the original ones given in [31] and [17] and so they are omitted Characterizations of the (CC) In this section, we will establish necessary and sufficient conditions for the condition (CC) These conditions will be crucial in the sequel and they also deserve some attention for their independent interest Then, characterizations of (CCCQ) and of approximate normal cones to convex constrained sets are given at the end of the section THEOREM 3.1 The following statements are equivalent: (i) Condition (CC) holds, (ii) For all x* X*, f ỵ A ị x ị ẳ þ 2S , u, v2Xà  à fà ðuÞ þ hị vị ỵ C x u vị , (the infimum on the right-hand side is attained at some  Sỵ and u, v X*), (iii) For any x A \ dom f and each  ! 0, [ [ ẩ ẫ  ẳ  ỵ @2 hxị  ỵ N3 C, xị  : @ f ỵ A ịxị @1 fxị 2Sỵ 1 , 2 , 3 !0  1 ỵ2 ỵ3 ẳỵhxị 5ị ð6Þ 546 N Dinh et al Proof [(i) () (ii)] Let  : X  X  Z  X ! R[ fỵ1} be the function defined by & x, y, z, tị ẳ fx ỵ yị, if hxị ỵ z S, x ỵ t C, ỵ1 otherwise and let Y ¼ X  Z  X For each (x*, y*, z*, t*) X*  Y* ¼ X*  X*  Z*  X*, by an easy calculation (see [5,11]), using the definition of conjugate function, we get Downloaded by [University of California Santa Barbara] at 07:23 31 July 2013 à ðxà , yà , zà , t ịf y ị ỵ z hị x y t ị ỵ S z ị ỵ C t ị: Since domS ẳ Sỵ and S z ị ẳ if z* Sỵ, it follows from the last equality that & f y ị ỵ z hị x y t ị ỵ C t ị, if z Sỵ à  ðx , y , z , t Þ ẳ ỵ1 otherwise: 7ị By Theorem 3.1 [11], the following statements are equivalent () PrX*  R (epi*) is weak*-closed () supx2X ððxà , xÞ À ðx, 0, 0, 0ÞÞ ¼ minðyà , zà , tà Þ2Yà à ðxà , yà , zà , tÃ Þ 8xà Xà Note that sup ððxà , xÞ À ðx, 0, 0, 0ịị ẳ supx , xị fxịị ẳ f þ A Þà ðxà Þ: x2X x2A This fact, together with (7), shows that () is none other than (ii) On the other hand, it is easily seen from (7) that PrX*R(epi*) ẳ epif *ỵK The equivalence between (i) and (ii) follows [(ii) ¼) (iii)] Suppose that (ii) holds Let  be an arbitrary non-negative number and x* be a point in the set of the right-hand side of (6) Then, there exist  Sỵ, 1, 2, 3 ! 0,  v @2(h)(x)  and w N3 (C, x),  such that 1 ỵ 2 ỵ 3 ẳ  ỵ h(x)  and u @1f(x), x* ẳ u ỵ v ỵ w Since   1 fðxÞ À fðxÞ ðu, x À xÞ  À 2 hðxÞ À hðxÞ  ðv, x À xÞ  À 3 ðw, x À xÞ  À we get that (x*, x À x) 8x X, 8x X, 8x C,  ỵ h(x) for all x C This yields f(x) À f(x)  À x , x xị  ỵ hxị fxị fðxÞ  fðxÞ À fðxÞ, 8x A, and hence,  À ðxà , x À xÞ  ð f þ A ÞðxÞ À ð f þ A ÞðxÞ, 8x X,  Thus, we have proved which proves that x* @(f ỵ A)(x) [ [ ẩ ẫ  '  ỵ @2 hxị  ỵ N3 C, xị  : @ f ỵ A ịxị @1 fxị 2Sỵ 1 , 2 , 3 !0  1 ỵ2 ỵ3 ẳỵhxị  Since x dom (f ỵ A), it follows from (2) For the converse inclusion, let x* @(f ỵ A)(x) that  fxị  A xị  ẳ  ỵ x , xị  fxị  ! f ỵ A ị x ị:  ỵ x , xị 8ị 547 Optimization Thanks to (ii), there exist  Sỵ and u, v, w X*, such that u ỵ v ỵ w ẳ x* and that f ỵ A ị x ị ẳ f uị ỵ hị vị ỵ C ðwÞ This and (8) gives  À fðxÞ  ! f uị ỵ hị vị ỵ C wị,  þ ðxà , xÞ ð9Þ ÃC Moreover, since (u, f * which implies that u dom f *, v dom(h)* and that w dom  ỵ f(x)  ! 0, and by the construction u0 @1 f(x)  (u)) epi f *, 1 ¼ f * (u) À (u, x)  and that w @03 C xị  ẳ N03 C, xị  where 2 ¼ (h)* Similarly, it comes that v @2(h)(x)  ! Since x* ẳ u ỵ v þ w, it follows from (9)  þ h(x)  ! and 03 ẳ C wị w, xị (v) À (v, x) that Downloaded by [University of California Santa Barbara] at 07:23 31 July 2013  ỵ fxịg  ỵ fhị vị v, xị  ỵ hxịg  ỵ fC wị w, xịg,   ! ff uị u, xị  ỵ hxị  ! 1 ỵ 2 ỵ 03 which means that  ỵ hxị  1 2, then  þ h(x)  ¼ 1 þ 2 þ 3 and w N03 ðC, xÞ  & N3 ðN, xÞ  Let 3 ẳ  ỵ h(x) as 3 3 Consequently, [ [ ẩ ẫ  ỵ @2 hxị  ỵ N3 C, xị  : @1 fxị x ẳ u ỵ v ỵ w 2Sỵ 1 , 2 , 3 !0  1 ỵ2 ỵ3 ẳỵhxị Thus, (ii) implies (iii) [(iii) ¼) (i)] Assume (iii) and consider any (x*, r) cl(epi f * ỵ K) Since cl epi f ỵ Kị ẳ epi f ỵ A Þà (by (3) and (4)), it comes that (x*, r) epi(fỵA)* As x dom(f ỵ A), it follows from (2) that  and r ¼ (x*, x)  f(x)  ỵ  Now, from (iii), there exists  ! exists such that x* @(fỵA)(x)  with  Sỵ, u, v, w X* and 1, 2, 3 ! satisfying x* ¼ u þ v þ w, 1 þ 2 þ 3 ¼  ỵ h(x)  v @2 h(x)  and w N3(C, x)  u @1 f(x),  f(x)  ỵ 1, t ẳ (v, x)h(   ỵ 2 and k ẳ (w, x)  ỵ 3 Once again, thanks Set s ¼ (u, x) x) to (2), one gets (u, s) epi f *, (v, t) epi (h)* and ðw, kÞ epi C Moreover, one has  hxị  ỵ 2 ỵ w, xị  ỵ 3  fxị  ỵ 1 ỵ v, xị s ỵ t ỵ k ẳ u, xị   hxị  ỵ 1 ỵ 2 ỵ 3 ẳ x , xị  fxị  ỵ  ẳ r, ẳ ðx , xÞ À fðxÞ and hence, ðxà , rÞ ẳ u, sị ỵ v, tị ỵ w, kị epi f ỵ [2Sỵ epi hị ỵ epi C ẳ epi f ỵ K, which proves that epi f * ỵ K is weak*-closed In other words, (CC) holds and the theorem is completely proved Let us present in the following corollary, some useful results for the forth-coming demonstrations g COROLLARY 3.2 Suppose that the condition (CC) holds Then, for each x* X*, f ỵ A ị x ị ẳ minỵ f ỵ h ỵ C ị x ị, 2S f ỵ A ị x ị ẳ f ỵ A ị x ị ẳ ỵ , uị2S ỵ , uị2S ẩ X X ẩ 10ị ẫ f uị ỵ h þ C Þà ðxà À uÞ , ð11Þ É ð f ỵ C ị uị ỵ hị x uị : ð12Þ 548 N Dinh et al Proof Let x* X* Since (CC) holds, it follows from Theorem 3.1-(ii) that  Sỵ and u, v X* exists such that  vị ỵ C x u vị: f ỵ A ị x ị ẳ f uị ỵ hị Thus, for each x X, one gets  ỵ x u v, xị C xị f ỵ A ị x ị ! u, xị fxị ỵ v, xị hxị  ỵ C ịxị, ! x , xị f ỵ h Downloaded by [University of California Santa Barbara] at 07:23 31 July 2013 which implies  ỵ C ị x ị: f ỵ A ị x ị ! f ỵ h 13ị Similarly, for each x, y X, one gets  f ỵ A ị x ị ! u, xị fxị ỵ v, yị hyị ỵ x u v, yị C yị  ỵ C ịyị, ! u, xị fxị ỵ x u, yị h which implies  ỵ C ị x uị: f þ A Þà ðxÃ Þ ! f à ðuÞ þ h 14ị  x uị: f ỵ A ị x ị ! f ỵ C ị uị ỵ hị 15ị As well, one has that On the other hand, for each  Sỵ, f ỵ h ỵ C ị x ị ẳ supfx , xị f ỵ hịxịg x2C ! supfx , xị f ỵ hịxịg x2A ! supfx , xị fxịg ẳ f ỵ A ị x Þ: ð16Þ x2A Combining (13) and (16), we get  þ C Þà ðxà Þ: ð f þ A Þà x ị ẳ minỵ f ỵ h ỵ C ị x ị ẳ f ỵ h 2S ỵ Since, for each  S and u X*, f ỵ A ị x ị f ỵ A ị x ị f ỵ h ỵ C ị x ị f ỵ h ỵ C ị x ị f uị ỵ h ỵ C ị x uị, f ỵ C ị uị ỵ ðhÞà ðxà À uÞ: g The proof is complete Note that in the absence of the set constraint ‘x C’ (i.e C ¼ X), characterizations of (CC) are given in the following corollary, whose proofs are the same as those of Theorem 3.1 and hence, will be omitted COROLLARY 3.3 Suppose that C ¼ X The following statements are equivalent: (i) Condition (CC) holds, (ii) for each x* X*, f ỵ A ị x ị ẳ þ ÈÃ É Ã Ã f ðuÞ þ ðhÞ ðx À uÞ , à ð, uÞ2S ÂX 549 Optimization (iii) For each  ! and each x dom f \ hÀ1(ÀS), [ È [ É  ¼  þ @2 hðxÞ  : @ ð f þ A ịxị @1 fxị 2Sỵ 1 , 2 !0  1 ỵ2 ẳỵhxị Downloaded by [University of California Santa Barbara] at 07:23 31 July 2013 Now, let us give representations of the approximate normal cones to the convex constrained set A These representations are useful in optimization problems when one needs to establish the optimality conditions or the characterization of approximate solutions of the mentioned problems The result also gives a characterization of the (CCCQ) condition for the system  For more details and other results on the approximate normal cones, see [16] COROLLARY 3.4 The following statements are equivalent: (i) The system  satisfies (CCCQ), (ii) For each x* X*, A x ị ẳ hị uị ỵ C x uị ẳ minỵ ẵhị ẩ C x ị, ỵ , uị2S X 2S (iii) For any a A and each  ! 0, [ N A, aị ẳ 2Sỵ [ ẩ ẫ @1 haị þ N2 ðC, aÞ : 1 , 2 !0 1 ỵ2 ẳỵhaị Here hị ẩ C denotes the infimum convolution defined, for any x* in X* by: ẵhị ẩ C x ị :ẳ hị uị ỵ C ðxà À uÞ : u2X Proof Let us consider f  Then, it comes that f * (0) ¼ 0, dom f * ¼ f0} and that epi f ỵ K ẳ f0g ẵ0, ỵ 1ị ỵ K ¼ K: Since for each  ! 0, @A(a) ¼ N(A, a), the conclusion follows from Theorem 3.1 Toland–Fenchel–Lagrange duality for DC programs with convex constraints Duality results are useful in the study of the primal problems In particular, for DC programs, they have been used successfully in building numerical methods for the primal problems (see, for instance, [2,3] and references therein) In this section, we are interested in a dual problem for the DC program (P) called ‘Toland–Fenchel–Lagrange’ dual problem This type of dual problem was considered in [26] for problems of model (P) and in [27] for DC programs with a finite number of DC constraints It is, in some sense, a ‘combination’ of Toland dual problem (for DC problem) in [31] and Fenchel and Lagrange dual problems (see [7,8,21]) We propose several duality results of this type for (P) which extend Laghdir’s one in [26] As consequences of these results, we obtained various corresponding results for convex programs which go back, and in some cases extend, the Fenchel–Lagrange duality or Lagrange duality results in [7,8,15,21] So, ‘Toland–Fenchel–Lagrange’ duality serves as a generalization of these types of dual problems to DC programs 550 N Dinh et al 4.1 Toland–Fenchel–Lagrange duality for DC programs In this section, we are interested in the duality results for the DC problem (P) given by: inf ẵfxị gxị Pị subject to x C, hðxÞ ÀS: Let us consider this first result Then, other forms of duality would be derived from this one Downloaded by [University of California Santa Barbara] at 07:23 31 July 2013 THEOREM 4.1 (Toland–Fenchel–Lagrange duality) Suppose (CC) holds Then È Ã Ã É g ðx ị f ỵ h ỵ C ị x ị : infPị ẳ inf max ỵ x 2X 2S Proof Thanks to the Toland dual theorem recalled in Lemma 2.1, ẩ ẫ infPị ẳ inf f ỵ A gịxị ẳ inf g x ị f ỵ A ị x ị : x2X 17ị x 2X Since (CC) holds, it follows from Corollary 3.2 that f ỵ A ị x ị ẳ minỵ f ỵ h ỵ C ị x ị: 2S This equality and (17) leads to ẩ ẫ infPị ẳ inf g x ị f ỵ A ị ðxÃ Þ : x 2X È Ã Ã É g x ị f ỵ h ỵ C ị x ị , ẳ inf max ỵ x 2X 2S g which completes the proof Theorem 4.1 was proved in [27, Corollary 4.1 (p 666)] where C ¼ X, h ¼ (g1, g2, , gm) and gi, i ¼ 1, 2, , m are extended real-valued convex functions and under Slater constraints qualification Other forms of strong Toland–Fenchel–Lagrange duality are given in the following theorem THEOREM 4.2 Then (Toland–Fenchel–Lagrange duality) Suppose that the condition (CC) holds infPị & max ẳ inf 2Sỵ , u, v2Xà x 2X & ¼ Ãinf à x 2X x 2X Proof à à à fg ðx Þ À f ðuÞ À ðhÞ ðvÞ À ÃC ðxà ' u vịg 18ị max ỵ ' ẩ Ã É Ã Ã Ã g ðx Þ À f uị h ỵ  ị x uị C 19ị max ỵ ' ẩ ẫ g x ị f ỵ  ị ðuÞ À ðhÞ ðx À uÞ : C à ð20Þ , uị2S X & ẳ inf , uị2S ÂX Since (CC) holds, by Theorem 3.1, for each x* X*, f ỵ A ị x ị ẳ 2Sỵ ;u, v2X f uị ỵ hị vị ỵ C x u vị : 21ị 551 Optimization Combining (17) and (21), we get that È ẫ infPị ẳ inf g x ị f ỵ A ị x ị x 2X & ' Àà Á à à à à à ¼ Ãinf à g x ị ỵmin f uị ỵ hị vị ỵ C x u vị x 2X 2S ;u, v2X & ¼ Ãinf à Downloaded by [University of California Santa Barbara] at 07:23 31 July 2013 x 2X max 2Sỵ ;u, v2X fg ðx Þ À f ðuÞ À ðhÞ ðvÞ À ÃC ðxà ' À u À vÞg : The equality (18) has been proved and similarly, by using (11) and (12), we get (19) and (20) g When the set constraint is absent (i.e C ¼ X), we get COROLLARY 4.3 Assume that C ¼ X and that the condition (CC) holds Then & infPị ẳ inf x 2X Proof max ỵ ' fg x Þ À f ðuÞ À ðhÞ ðx À uÞg : à ð, uÞ2S ÂX The conclusion follows directly from Theorem 4.2, (19) Example 4.1 g Consider the following problem (E1): q  inf x21 x2 x21 ỵ x22 E1ị subject to x ẳ x1 , x2 ị R2 , x1 ỵ x2 0: q Let fxị ẳ x21 x2 , gxị ẳ x21 ỵ x22 , X ẳ R2, Z ẳ R, S ẳ Sỵ ẳ Rỵ, CẳX, h(x) ẳ x1 ỵ x2 It is clear that f, g are convex, continuous functions on R2, while h is S-convex and continuous Then (E1) has the form of (P) For each u ¼ (u1, u2) R2, one has that < u1 if u2 ¼ 1, f uị ẳ sup u1 x1 ỵ u2 x2 x21 ỵ x2 ị ẳ : x2R ỵ1 if u2 6ẳ 1: For each v ẳ (v1, v2) R2 and  Sỵ ẳ Rỵ, it comes that & hị vị ẳ sup v1 x1 ỵ v2 x2 x1 x2 ị ẳ þ1 x2R2 if v1 ¼ v2 ¼ , otherwise: For each a ¼ (a1, a2) R2, the formulae for the conjugate of the norm gives that ( à g aị ẳ ỵ1 if a21 ỵ a22 1, if a21 ỵ a22 > 1: Therefore, epi f ỵ [2Sỵ epi h ẳ fu1 ỵ ,  1, u21 ỵ rịj u1 R, , rị R2ỵ g, 552 N Dinh et al which is a closed subset of R3 By Corollary 4.3, we get that infE1ị ẳ inf max a21 ỵa22 a2 ẳ1 ða1 , a2 Þ2R2 !0 ða1 À Þ2 ða1 À a2 1ị2 ẳ inf : 4 a21 þa22 ða1 , a2 Þ2R2 Downloaded by [University of California Santa Barbara] at 07:23 31 July 2013 Note that pffiffiffiqffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ja1 À a2 À 1j ja1 À a2 j þ a21 þ a22 þ 1: pffiffiffi Þ=4 on the right-hand side of the last Therefore, infE1ị ẳ ỵ 1ị2p and the infimum p equality is attained at a1 ẳ 1ị= 2, a2 ẳ 1= 2Þ Remark It is worth observing that the conclusions of Theorems 4.1 and 4.2 (18) were established in [26, (Propositions 4.1–4.3 and Corollary 4.1]) with the assumptions that f is continuous at one point in C and either  ÀintS (Slater constraint qualification), () x C exists such that h(x) or () Rỵ [S ỵ h(dom f \ C)] is a closed subspace Let us remember (see [21]) that if the Slater condition () holds, then  satisfies (CCCQ) Thus, with the fact that f is continuous at one point in C, it comes that (CC) holds The same conclusion holds under the assumption () as shown in the next corollary COROLLARY 4.4 If f is continuous at some point in C and Rỵ [Sỵh(dom f \ C)] is a closed subspace of Z, then (CC) holds Proof Assume that the hypothesis of the Corollary 4.4 holds Let xÃ0 Xà be arbitrary and set g ¼ x0 Since g x0 ị ẳ if x ẳ x0 and g*(x*) ẳ ỵ1 else, it comes from [26, Corollary 4.1], that & ' à à à à infPị ẳ inf max fg x ị À f ðuÞ À ðhÞ ðvÞ À  ðx À u vịg C x 2X 2Sỵ ;u, v2X ẩ ẫ ẳ max f uị hị vị C x0 u vị : ỵ 2S ;u, v2X Observe that infPị ẳ infx2A ffxị x0 , xịg ẳ f ỵ A ị x0 ị Therefore, ẩ ẫ f ỵ A ị x0 ị ẳ f uị ỵ hị vị ỵ C x0 u vị : ỵ 2S ;u, v2X Since the last equality holds for all xÃ0 Xà , (CC) follows from Theorem 3.1 g Corollary 4.4 gives a sufficient condition to (CC) As the following simple example shows, it is not a necessary one Example 4.2 Consider the simple case when X ¼ R, C ¼ [À1, 1], S ẳ Rỵ, f(x) :ẳ x, g(x)  0, and h(x) ¼ max f0, x} Then, for a R and  Rỵ, we have that (C)* (a) ¼ jaj and that & & if a ¼ 1, if a ½0, Š, à à f aị ẳ hị aị ẳ ỵ1 if a 6ẳ 1, þ1 if a = ½0, Š: Optimization 553 Downloaded by [University of California Santa Barbara] at 07:23 31 July 2013 S Then, epi f ỵ 2Rỵ hị ỵ epi C ẳ f1g Rỵ ỵ Rỵ Rỵ ỵ epij:j is a closed subset of R2 while Rỵ [Rỵ þ h(dom f \ C)] ¼ Rþ [Rþ þ h(R)] ẳ [0, ỵ1) is not a closed subspace of R Remark To close this subsection, let us mention that if we take g(x) ¼ (x*, x), x* X*, the results in Theorems 4.1 and 4.2 yield stability of strong Lagrange duality, Fenchel–Lagrange duality for convex problems under linear perturbations (for an example, see [16]) In particular, if g(x) ¼ (x*, x), x* X* andPthe cone-constraint h(x) S is replaced by the semi-definite constraint F0 ỵ m i¼1 xi Fi # 0, where Fi, i ¼ 1, 2, , m, are symmetric (n  n)-matrices and x ¼ (x1, x2, , xm) Rm, then the corresponding results go back to stable strong duality results for convex semi-definite programs introduced in the unpublished manuscript [23] 4.2 Duality for convex programs Let us derive in this section, results about the convex case from the DC one In order to this, consider the convex program: ðQÞ inf fðxÞ subject to x C, hðxÞ ÀS: Then, the problem (Q) is a special case of (P), where g  4.2.1 Lagrange duality Let us recall that the Lagrange dual problem of (Q) is given by & ' ðLDQÞ sup inf f ỵ hịxị : 2Sỵ x2C Then, we find the strong duality for the Lagrange dual problem, as established recently in [13] COROLLARY 4.5 (Lagrange duality) Suppose that (CC) holds Then the Lagrange strong duality holds for (Q), that is, & ' inf f ỵ hịxị 22ị infQị ẳ max ỵ 2S x2C (the Problem (LDQ) is solvable) Proof Note that if g  then g*(0) ¼ and g*(x*) ẳ ỵ1 if x* X*\f0} Since (CC) holds, by Theorem 4.1, applied to the case where g  0, we get infQị ẳ max2Sỵ f f ỵ h ỵ C ị 0ịg & ' ẳ max2Sỵ supffxị hxịg x2C & ' ẳ max2Sỵ inf fxị ỵ hxịị : 23ị x2C The proof is complete g 554 N Dinh et al 4.2.2 Fenchel–Lagrange duality The Fenchel–Lagrange dual problem of (Q) is defined by: È Ã É ðFLDQÞ max Àf ðuÞ À ðhÞà ðvÞ À ÃC u vị : ỵ 2S ;u, v2X As a consequence of Theorem 4.2, we get the following strong Fenchel–Lagrange duality result for (Q) Downloaded by [University of California Santa Barbara] at 07:23 31 July 2013 COROLLARY 4.6 (Fenchel-Lagrange duality) Suppose that (CC) holds Then the strong duality holds between (Q) and (FLDQ), i.e.: È Ã É Ã Ã Àf ðuÞ À ðhÞ ðvÞ À  ðÀu À vÞ : infQị ẳ max C ỵ 2S ;u, v2X Proof Let g  Since g*(0) ¼ and g*(x*) ẳ ỵ1 if x* X*\f0}, the result follows from the first assertion of Theorem 4.2 Let us present in the following corollary, other forms of Fenchel–Lagrange duality results for (Q) COROLLARY 4.7 Then (Fenchel–Lagrange duality) Suppose that (CC) holds infQị ẳ infQị ẳ ẩ ẫ f uị h ỵ C ị uị , ẩ ẫ f þ C Þà ðuÞ À ðhÞà ðÀuÞ : max þ max , uị2Sỵ X , uị2S X Proof This is a direct consequence of Theorem 4.2 (see (19) and (20)) by letting g  g The first strong duality result in Corollary 4.7 was established recently in [8] where X, Z are Banach spaces, f is a continuous convex function and g a S-convex, continuous mapping (thus, (CC) holds) This result was proved in [7] (see also, [6]) In [7] the convex functions involved are not assumed to be l.s.c, X ¼ Rn, h ¼ (h1, h2, , hm), hi : Rn ! R are convex or affine functions (S ẳ Rm ỵ ), and this duality result was established under an interior-type constraint qualification that distinguishes the convex and the affine inequality constraints For other forms of Fenchel–Lagrange duality results for convex and nearly convex problems, see [6–8] Generalized Farkas lemmas Let us use the duality results obtained in the previous section to derive generalized Farkas lemmas in dual forms for systems involving convex and DC functions All the assumptions on the functions f, g, the mapping h, the spaces X, Z, the cone S and the subset C are as in Section THEOREM 5.1 (Farkas lemma) Suppose that (CC) holds and that  R Then, the following statements are equivalent: (i) x C, h(x) À S ¼) f(x) À g(x) ! , (ii) For each x* X*, there exists  Sỵ, u X* such that g x ị f uị h ỵ C Þà ðxà À uÞ ! , Optimization 555 (iii) For each x* X*, there exists  Sỵ, u X* such that gà ðxÃ Þ À ð f þ C Þà ðuÞ À ðhÞà ðxà À uÞ ! , (iv) For each x* X*, there exists  Sỵ, u, v X* such that g x Þ À fà ðuÞ À ðhÞà ðvÞ À ÃC ðxà À u À vÞ ! : Downloaded by [University of California Santa Barbara] at 07:23 31 July 2013 Proof [(i) ) (iv)] Suppose that (i) holds Then inf (P) !  Since (CC) holds, Theorem 4.2 leads to infPị ẳ inf max x 2X 2Sỵ ;u, v2X ẩ ẫ gà ðxÃ Þ À f à ðuÞ À ðhÞà ðvÞ À ÃC ðxà À u À vÞ : ð24Þ Then, for each x* X*, there exists  Sỵ and u, v X* such that gà ðxÃ Þ À f à ðuÞ À ðhÞà ðvÞ À ÃC ðxà À u À vÞ ! infðPÞ ! , which proves (iv) [(iv) ) (i)] Suppose that (iv) holds Then, for each x* X*, there exists  Sỵ and u, v X*, such that gà ðxÃ Þ À f à ðuÞ À ðhÞà ðvÞ À ÃC ðxà À u vị !  This implies that max ỵ 2S ;u, v2Xà À à à Á g ðx Þ À f à ðuÞ À ðhÞà ðvÞ À ÃC ðxà À u À vÞ ! : Therefore, it comes that inf max x 2X 2Sỵ ;u, v2X Á g ðx Þ À f à ðuÞ À ðhÞà ðvÞ À ÃC ðxà À u À vÞ ! : Since (CC) holds, the strong duality holds (Theorem 4.2), we get that inf (P) ! , which means that (i) holds The equivalence between (i), (ii) and (iii) follows from the same argument as in the proof of the equivalence between (i) and (iv), using (19) or (20) instead of (24) g Let us have a look at the convex systems As consequences of Corollaries 4.6 and 4.7, we get the following generalized versions of Farkas lemma for convex systems COROLLARY 5.2 (Farkas lemma for convex systems) Suppose that (CC) holds and  R Then, the following statements are equivalent: (i) x C, h(x) À S ¼) f(x) ! , (ii) There exists  Sỵ, u X* such that f uị h ỵ C ị uị ! , (iii) There exists  Sỵ, u X* such that f ỵ C ị ðuÞ À ðhÞà ðÀuÞ ! , (iv) There exists  Sỵ, u, v X* such that f uị À ðhÞà ðvÞ À ÃC ðÀu À vÞ ! : 556 N Dinh et al Downloaded by [University of California Santa Barbara] at 07:23 31 July 2013 Proof The proof is quite similar to that of Theorem 5.1 Here, Corollaries 4.6 and 4.7 are used instead of Theorem 4.2 g Let us mention that the equivalence between (i) and (ii) in Corollary 5.2 was proved in [7, Theorem 4.1, p 545] under the assumptions: X ¼ Rn,  ¼ 0, h ¼ (h1, h2, , hm) where hi : Rn ! R convex (S ẳ Rnỵ ) and under the Slater constraint qualification condition In [7], a convex problem with X ¼ Rn, gi(x) for all i I (possibly infinite set) was considered as well A version of Farkas lemma was proved It is similar to the equivalence of (i) and (ii) in Corollary 5.2, but without explicitly showing the existence of  Sỵ (Theorem 5.2, p 549) Note that such a problem can be reduced to the model (Q) by letting h : Rn ! RI with h(x) ¼ (hi(x))i I and S ẳ RIỵ [13,14] For more versions of Farkas lemma involving convex and DC functions, see [16] Optimality condition for DC programs with convex constraints In this section, we first establish necessary and sufficient conditions for global optimality for the problem (P) Then, the result is applied to some special mathematical programming problems such as, convex problems, convex maximum problems and DC programs with semi-definite constraints Let us first state the main result of this section THEOREM 6.1 Let x A \ dom f \ dom g and assume that the condition (CC) holds Then,  there exist x is a global solution of (P) if and only if for each  ! and each x* @ g(x),  and  Sỵ and 1, 2, 3 ! such that 1 ỵ 2 ỵ 3 ẳ  ỵ h(x)  ỵ @2 hịxị  ỵ N3 ðC, xÞ:  xà @1 fðxÞ ð25Þ   Sỵ exists such In particular, if x A is a local solution of (P), then, for each x* @g(x),  ỵ @(h)(x)  ỵ N(C, x)  and h(x)  ¼ that x* @f(x) Proof It is clear that Problem (P) can be written in the form: P0 ị inf ẵ f ỵ A ÞðxÞ À gðxފ: x2X By Lemma 2.2, x is a global solution of (P) if and only if for each  ! 0,  & @ f ỵ A ịxị:  @ gðxÞ Since (CC) holds, it follows from Theorem 3.1 that [ [ ẩ ẫ  ẳ  ỵ @2 hxị  ỵ N3 C, xị  : @1 fxị @ f ỵ A ịxị 2Sỵ 1 , 2 , 3 !0  1 ỵ2 ỵ3 ẳỵhxị Thus, x is a global solution of (P) if and only if for each  ! 0, [ [ È É  &  ỵ @2 hxị  ỵ N3 C, xị  , @1 fxị @ gxị 2Sỵ 1 , 2 , 3 !0  1 ỵ2 ỵ3 ẳỵhxị  there exists  Sỵ and 1, 2, 3 ! such which means that for each x* @g(x),  and x* @1 f(x)  ỵ @2 (h)(x)  þ N3 (C, x)  The first assertion that 1 þ 2 þ 3 ¼  þ h(x) is proved 557 Optimization  (as  Sỵ and x A) So, if  Sỵ Now, if x A is a solution of (P), then h(x)  then 1 ¼ 2 ¼ 3 ¼ h(x)  ¼ The last and 1, 2, 3 ! satisfying 1 ỵ 2 ỵ 3 ẳ h(x), assertion follows from the previous observation and the first assertion of the theorem with  ¼ g Example 6.1 Consider the problem E2ị subject to inf ẵx21 ỵ x2 2x22 jx1 j 2x2 x1 ỵ x2 Downloaded by [University of California Santa Barbara] at 07:23 31 July 2013 x ẳ x1 , x2 ị ẵ1; ẵ1; x2 , h(x) ẳ (jx1j 2x2, x1 ỵ x2), X ẳ X* ẳ R2, C ¼ [À1; 1]  Let fðxÞ ¼ [À1; 1], Z ẳ Z* ẳ R , S ẳ S ẳ R2ỵ It is obvious that f, g are continuous convex functions and that h is continuous and S-convex The Problem (E2) has the form of (P) It is easy to see that for  ẳ 1 , 2 ị Sỵ ẳ R2ỵ and a ẳ (a1, a2) R2, we have that x21 , gxị ẳ 2x22 ỵ hị aị ẳ supa1 2 ịx1 1 jx1 j ỵ a2 ỵ 21 2 ịx2 ị x2X & if ja1 À 2 j 1 , a2 ẳ 2 21 , ẳ ỵ1 otherwise: Thus epi (h)* ẳ [2 1; 2 ỵ 1] f2 21} Rỵ and hence, [ Sỵ epi (h)* ¼ f(r, s, t) R3 j r ! s, t ! 0} On the other hand, it is clear that epi C ẳ fr, s, tị R3 j jrj ỵ jsj tg We have that [2Sỵ epi hị ỵ epi C ẳ fr, s, tị R3 j s À r t, t ! 0g ð26Þ is a closed subset of R3 Since f is continuous on R2, epi f ỵ [2Sỵ epi hị ỵ epi ÃC is closed in R3 which means that (CC) holds for the Problem (E2) Let us consider , 1, 2, 3 ! 0,  ẳ 1 , 2 ị R2ỵ , then we have that p p p p @ g0, 0ị ẳ f0g ẵ2 2 1, 2 1, @1 f0, 0ị ẳ ẵ2 1 , 1 f0g @2 hị0, 0ị ẳ ẵ2 1 ; 2 ỵ 1 f2 21 g 8 Sỵ , N3 C, 0, 0ịị ẳ fa1 , a2 ị R2 j ja1 j ỵ ja2 j 3 g: Let  !p ffiffiffiffibe ffiffiffiffiffi a) @g(0, 0), choose 1 ¼ 3 ¼ ð=2Þ, 2 ¼ 0, 1 ¼ ffi arbitrary For eachp(0, 2 Note p that =2ịpỵ 2 paand 2 ẳ 1pỵ  ỵ h(0) ẳ 1 ỵ 2 ỵ 3, 1 ẳ =2ị ỵ 2ị ỵ 2 aị ẳ =2ị 1ị2 þ ð2 2 À À aÞ ! and 2 ! We have that pffiffiffiffiffi pffiffiffiffiffi    0, aị ẳ 2, 0ị ỵ 2, a ỵ 0, 2 p ẳ 1 , 0ị ỵ 2 1 , 2 21 ị þ ð0, 3 Þ & @1 fð0, 0Þ þ @2 hị0, 0ị ỵ N3 C, 0, 0ịị By Theorem 6.1, x ¼ (0, 0) is a global solution of (E2) We are now coming back to the convex problem (Q) in Section By taking g  in Theorem 6.1, we will get an optimality condition for the convex Problem (Q) as shown in the next corollary 558 N Dinh et al COROLLARY 6.2 Suppose that (CC) holds and that x A Then, x is a (global) solution of (Q) if and only if there exists  Sỵ such that  ỵ @hịxị  ỵ NC, xị,  hxị  ẳ 0: @fxị 27ị Proof Necessity Suppose that x is an optimal solution of (Q) Theorem 6.1 with g  and with  ¼ ensures the existence of  Sỵ, such that (27) holds Then, the  ¼ follows from an argument similar to the last part of the proof of condition h(x) Theorem 6.1 Downloaded by [University of California Santa Barbara] at 07:23 31 July 2013 Sufficiency The sufficiency follows by the standard argument as in [25] g The optimality condition given in Corollary 6.2 was established in [24] It was also established in [9,21,25] by assuming that X is a Banach space,  satisfies (CCCQ) and either f is continuous, or C ẳ X, or epi f * ỵ cl K is weak*-closed In [12], this optimality condition was also proved under the assumption that f was continuous at one point in C, h was sequentially l.s.c and Rỵ[S þ h(C)] was a closed subspace In either of the cases, (CC) holds Similar assumptions were imposed in [13], when dealing with the convex infinite problems to get the same kind of optimality condition for this class of problems Now, let us consider the problem of maximizing a convex function under convex constraints ðCMPÞ sup pðxÞ subject to x C, hðxÞ ÀS, where X, C, S, h are as in the previous sections and p : X ! R [ fỵ 1} is a proper, l.s.c and convex function It is clear that if x A is a (global) solution of the problem (CMP), then it is a solution of the problem (CMP1) below ðCMP1Þ inf ÀpðxÞ subject to x C, hðxÞ ÀS: As a direct consequence of Theorem 6.1, we get: COROLLARY 6.3 Suppose that  satisfies (CCCQ) and that x A \ dom p Then x is a  there exist  Sỵ global solution of (CMP) if and only if for each  ! and each x* @ p(x),  and and 1, 2 ! such that 1 ỵ 2 ẳ  ỵ h(x)  ỵ N2 ðC, xÞ:  xà @1 ðhÞðxÞ ð28Þ  there exists  Sỵ In particular, if x A is a local solution (CMP), then for each x* @g(x),  ỵ N(C, x)  and h(x)  ẳ such that x* @(h)(x) Acknowledgements The authors would like to thank the referees for their valuable comments and suggestions which improved the presentation of the paper Parts of the work of N Dinh were realized during his visit at the Laboratory of Applied Mathematics, University of PAU to which he would like to express his sincere thanks for the hospitality he received 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[16] N Dinh, G Vallet, and T.T .A Nghia, Generalized Farkas lemmas for systems involving convex and DC functions and its applications, J Convex Anal 12 (2008) (to appear) [17] J.B Hiriart-Urruty,