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A closedness condition and its applications to DC programs with convex constraints

N Dinh a , T.T.A Nghia b & G Vallet ca

Department of Mathematics, International University, Ho ChiMinh City, Vietnam

b Department of Mathematics and Computer Science, Ho Chi MinhCity University of Pedagogy, Vietnam

c Laboratory of Applied Mathematics, UMR-CNRS 5142 University

of PAU IPRA, BP 1155, 64013 Pau Cedex, FrancePublished 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 MathematicalProgramming and Operations Research, 59:4, 541-560, DOI: 10.1080/02331930801951348

To link to this article: http://dx.doi.org/10.1080/02331930801951348

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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

duality theorems As consequences, various versions of generalized Farkaslemmas in dual forms for systems involving convex and DC functions are derived

Then, we establish optimality conditions for DC problem under convexconstraints Optimality conditions for convex problems and problems ofmaximizing 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 linkbetween several corresponding known ones published recently for DC programsand for convex programs

Keywords: DC programs; closedness conditions; closed-cone constraint cation; Farkas lemmas; Fenchel–Lagrange duality; Toland–Fenchel–Lagrangeduality

*Corresponding author Email: ndinh@hcmiu.edu.vn

ISSN 0233–1934 print/ISSN 1029–4945 online

ß 2010 Taylor & Francis

DOI: 10.1080/02331930801951348

Further assume that S is a closed convex cone of Z (not necessarily with non-emptyinterior) and h : X ! Z is an S-convex mapping, i.e.

8u, v 2 X, 8t 2 ½0, 1, hðtu þ ð1  tÞvÞ  thðuÞ  ð1  tÞhðvÞ 2 S,such that   h is l.s.c for each  2 Sþ

, the dual cone of S, defined by:

Sþ:¼ f 2 Zj ð, sÞ  0, for all s 2 Sg:

This last property, called star S-l.s.c in [23], corresponds to an extension of the notion oflower semi-continuity to vector-valued functions Other related notions already exist in theliterature, e.g in the sense of Penot–Thera in [28] or in a sequential sense in [12] Since any

 2 Sþ

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  2 Z*, the composition of mappings   h would bedenoted by h

We define, by convention, that þ1  (þ1) ¼ þ1

Recall that a function p : X ! R [ fþ1} is called a DC function, if it can bedecomposed as a difference of two convex functions Such a class of functions covers theclasses 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 thismathematical model (see [2–4]) and several numerical methods have been developed forthis 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 constraintqualification is an essential ingredient for the Lagrange multiplier rule and for the dualitytheory The well-known constraint qualifications for convex and DC optimization areoften of Slater-type conditions (see [12,26,27] for instance) However, these conditions areoften not satisfied for many problems in applications In recent years, a constraintqualification called closed-cone constraint qualification (or (CCCQ) for short), has beendeveloped 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 thefeasible 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 andoptimality conditions for the DC problem (P)

Concerning the problem (P), we consider the system

:¼ fx 2 C, hðxÞ 2 Sgand the set of its solution A which is the feasible set of (P),

where ’* stands for the conjugate of the function ’

The system  is said to satisfy the closed clone constraint qualification ((CCCQ), inbrief) 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 inthese mentioned papers In particular, it was shown that the constraint qualification(CCCQ) is strictly weaker than several generalized Slater type ones, and weaker thanthe Robinson-type one stating that Rþ [S þ h(C)] is a closed subspace (see [8,21] formore details)

Let us introduce the following closedness condition:

ðCCÞ epifþKis weak-closed

It involves the function f and the system  and will play a crucial role in the sequel ofthis 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 * þ clK is weak*-closed, wasintroduced in [13] and [14] where this last condition, with the (CCCQ) implies (CC), wereused to establish optimality conditions, duality and stability results, for convex infiniteprograms

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  A) is a closed subspace of X.Indeed, if cone(dom f  A) ¼ cone(dom f  domA) is a closed subspace, by[9, Proposition 3.1], epi fþepi 

Ais weak*-closed and so, thanks to (4),epi fþepi A¼epi fþclK ¼ epi fþKis weak-closed

Note that if the set constraint ‘x 2 C’ is absent, i.e if  :¼ fh(x) 2 S}, the condition(CC) becomes: epif * þS

2S þepi(h)* is weak*-closed

In this article, characterizations for (CC) in dual forms and in terms of approximatesub-differentials will be established These results will serve as main tools to establishduality 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 Fencheland Lagrange dual problems (see [7,8,21]) We establish several duality theorems whichextend Laghdir’s one in [26] In particular, these results would yield the stable strongduality for convex programs under linear perturbations [16,23] Moreover, variousversions of generalized Farkas lemmas in dual forms for systems involving convex and DCfunctions are derived Applied to convex systems or to convex programs, these results givestrong Lagrange duality and Fenchel–Lagrange duality results, and extend known onesestablished, for example in [7,18,13,15,21]

Optimality conditions for DC problem are also obtained Our results are obtainedunder the (CC) condition and by using its characterizations given in Section 3 They serve

as a link between several corresponding known ones published recently for DC programsand for convex programs

This article is organized as follows: in Section 2, we fix some notations and recallresults needed in the sequel of this article Characterizations of the condition (CC) aregiven in Section 3, and corollaries are derived from simple cases Characterizations of(CCCQ) are also proposed In particular, a representation for the approximate normalcones 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 onesgiven in [26] Corollaries for concrete classes of problems, including the classes of convexand 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 forsystems 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 convexproblems, are proved The problems of maximizing a convex function under convexconstraints is also treated as a special case

2 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 (withsuitable 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 0 if x 2 D and D(x) ¼ þ1else Moreover, the support function Dis given by D(u) ¼ supx2Du(x)

Let f : X ! R [ fþ1} be a proper l.s.c convex function Then:

(i) The conjugate function of f, f * : X* ! R[ fþ1}, is defined by

fðvÞ ¼supfvðxÞ  fðxÞ j x 2 dom f g,where the domain of f is given by dom f :¼ fx 2 X j f(x)5þ1},

(ii) If a 2 dom f then, following [20],

@fðaÞ ¼ fv 2 X j fðxÞ  fðaÞ  ðv, x  aÞ  , 8x 2 dom fg:

(iii) If 40 then @f(a) 6¼ ; Moreover, T

40 @f(a) ¼ @f(a), where @f(a) denotes theusual 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 2 Dis defined by (see [17,18])

NðD, xÞ:¼ @DðxÞ ¼ fx  2X j ðx, x  xÞ  , 8x 2 Dg:

When  ¼ 0, N0(D, x) ¼ N(D, x) is the normal cone to D at xin the sense of convex analysis.Following [30] and [9], it is worth noting that for two proper l.s.c convex functions f1

and f2with dom f1\dom f26¼ ;,

epi ðf1þf2Þ¼clðepi f1þepi f2Þ: ð3ÞNote moreover, that a sufficient condition for epi f

1þepi f

2to be weak*-closed, is that atleast one of the functions f or f is continuous at some point of dom f \dom f

Note that, if, as it is assumed, h is a S-convex mapping such that h is l.s.c for each

(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  2 Sþ, provided that X is metrizable (see [12, Proposition 3.7]).Let us conclude this section by recalling some results on duality and optimalityconditions 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 [ fþ1} Assume that G is a proper, convex andl.s.c function and F is an arbitrary function Then

inf

x2XfFðxÞ  GðxÞg ¼ inf

u2X fGðuÞ  FðuÞg:

LEMMA 2.2 [17] Let X be a locally convex Hausdorff topological vector space and F, G :

X ! R[ fþ1} be l.s.c, proper and convex functions Then

(i) A point a 2 dom F \ dom G is a global minimizer of the problem infx2X

fF(x)  G(x)} if and only if for any   0, @G(a)  @F(a)

(ii) If a 2dom F \ dom G is a local minimizer of infx 2 XfF(x)  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 thecondition that G is l.s.c is replaced by ‘G(x) ¼ G**(x) for all x 2 domF ’ The proofs ofthese conclusions are modifications of the original ones given in [31] and [17] and so theyare omitted

3 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 theirindependent interest Then, characterizations of (CCCQ) and of approximate normalcones 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* 2 X*,

Proof [(i) () (ii)] Let  : X X Z X ! R[ fþ1} be the function defined by

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*) ¼ epif *þK Theequivalence 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  2 Sþ, 1, 2, 30,

u 2 @1f( x), v 2 @2(h)( x) and w 2 N3 (C, x), such that  1þ2þ3¼ þ h( x) andx* ¼ u þ v þ w Since

ðx, x  xÞ    ð f þ AÞðxÞ  ð f þ AÞðxÞ, 8x 2 X,which proves that x* 2 @(f þ A)( x) Thus, we have proved

Thanks to (ii), there exist  2 Sþ and u, v, w 2 X*, such that u þ v þ w ¼ x* and that

ðf þ AÞðxÞ ¼fðuÞ þ ðhÞðvÞ þ 

CðwÞ This and (8) gives

 þ ðx, xÞ  fð xÞ  fðuÞ þ ðhÞðvÞ þ 

which implies that u 2 dom f *, v 2 dom(h)* and that w 2 dom 

C Moreover, since (u, f *(u)) 2 epi f *, 1¼f* (u)  (u, x) þ f( x)  0, and by the construction u02@1f( x)

Similarly, it comes that v 2 @2(h)( x) and that w 2 @0

3CðxÞ ¼ N 0

3ðC, xÞwhere 2¼(h)*(v)  (v, x) þ h( x)  0 and 0

3¼

CðwÞ  ðw, xÞ 0 Since x* ¼ u þ v þ w, it follows from (9)that

 þ hð xÞ  ffðuÞ  ðu, xÞ þ fð xÞg þ fðhÞðvÞ  ðv, xÞ þ hð xÞg þ fCðwÞ  ðw, xÞg,which means that  þ hð xÞ  1þ2þ0

3.Let 3¼ þ h( x)  12, then  þ h( x) ¼ 1þ2þ3 and w 2 N0

Thus, (ii) implies (iii)

[(iii) ¼) (i)] Assume (iii) and consider any (x*, r) 2 cl(epi f * þ K) Since

cl epi fð þKÞ ¼epið f þ AÞ(by (3) and (4)), it comes that (x*, r) 2 epi(fþA)* As x 2dom(f þ A), it follows from (2) that

 0 exists such that x* 2 @(fþA)( x) and r ¼ (x*, x)  f( x) þ  Now, from (iii), there exists

 2 Sþ, u, v, w 2 X* and 1, 2, 30 satisfying x* ¼ u þ v þ w, 1þ2þ3¼ þ h( x) with

u 2 @1f( x), v 2 @2h( x) and w 2 N3(C, x)

Set s ¼ (u, x)  f( x) þ 1, t ¼ (v, x)h( x) þ 2 and k ¼ (w, x) þ 3 Once again, thanks

to (2), one gets (u, s) 2 epi f *, (v, t) 2 epi (h)* and ðw, kÞ 2 epi 

C Moreover, one has

s þ t þ k ¼ ðu, xÞ  fð xÞ þ 1þ ðv, xÞ  hð xÞ þ 2þ ðw, xÞ þ 3

¼ ðx, xÞ  fð xÞ  hð xÞ þ 1þ2þ3¼ ðx, xÞ  fð xÞ þ  ¼ r,and hence,

ðx, rÞ ¼ ðu, sÞ þ ðv, tÞ þ ðw, kÞ 2 epi fþ [2Sþepi ðhÞþepi C¼epi fþK,which proves that epi f * þ K is weak*-closed In other words, (CC) holds and the theorem

Proof Let x* 2 X* Since (CC) holds, it follows from Theorem 3.1-(ii) that  2 Sþ and

u, v 2 X* exists such that

ðf þ AÞðxÞ ¼fðuÞ þ ð hÞðvÞ þ Cðxu  vÞ:

Thus, for each x 2 X, one gets

ðf þ AÞðxÞ  ðu, xÞ  fðxÞ þ ðv, xÞ  hðxÞ þ ðxu  v, xÞ  CðxÞ

 ðx, xÞ  ð f þ h þ CÞðxÞ,which implies

ðf þ AÞðxÞ  ðf þ h þ CÞðxÞ: ð13ÞSimilarly, for each x, y 2 X, one gets

ðf þ AÞðxÞ  ðu, xÞ  fðxÞ þ ðv, yÞ  hðyÞ þ ðxu  v, yÞ  CðyÞ

 ðu, xÞ  fðxÞ þ ðxu, yÞ  ð h þ CÞðyÞ,which implies

Note that in the absence of the set constraint ‘x 2 C’ (i.e C ¼ X), characterizations

of (CC) are given in the following corollary, whose proofs are the same as those ofTheorem 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* 2 X*, ð f þ AÞðxÞ ¼ min

þ fðuÞ þ ðhÞðxuÞ

,

(iii) For each   0 and each x 2dom f \ h1(S),

COROLLARY 3.4 The following statements are equivalent:

(i) The system  satisfies (CCCQ),

(ii) For each x* 2 X*,

AðxÞ ¼ min

ð, uÞ2S þ X ðhÞðuÞ þ CðxuÞ

¼min

2S þ½ðhÞCðxÞ,(iii) For any a 2 A and each   0,

Since for each   0, @A(a) ¼ N(A, a), the conclusion follows from Theorem 3.1

4 Toland–Fenchel–Lagrange duality for DC programs with convex constraints

Duality results are useful in the study of the primal problems In particular, for DCprograms, they have been used successfully in building numerical methods for the primalproblems (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 DCconstraints 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 dualityresults of this type for (P) which extend Laghdir’s one in [26] As consequences of theseresults, 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 thesetypes of dual problems to DC programs