DSpace at VNU: Duality and optimality conditions for generalized equilibrium problems involving DC functions

DOI 10.1007/s10898-009-9486-zDuality and optimality conditions for generalized equilibrium problems involving DC functions N.. For this problem we establish two new dual formulations bas

DOI 10.1007/s10898-009-9486-z

Duality and optimality conditions for generalized

equilibrium problems involving DC functions

N Dinh · J J Strodiot · V H Nguyen

Received: 21 June 2009 / Accepted: 27 October 2009 / Published online: 8 November 2009

© Springer Science+Business Media, LLC 2009

Abstract We consider a generalized equilibrium problem involving DC functions which

is called (GEP) For this problem we establish two new dual formulations based on Fenchel-Lagrange duality for DC programming problems The first one allows us to obtain

Toland-a unified duToland-al Toland-anToland-alysis for mToland-any interesting problems So, this duToland-al coincides with the duToland-alproblem proposed by Martinez-Legaz and Sosa (J Glob Optim 25:311–319, 2006) for equi-librium problems in the sense of Blum and Oettli Furthermore it is equivalent to Mosco’sdual problem (Mosco in J Math Anal Appl 40:202–206, 1972) when applied to a varia-tional inequality problem The second dual problem generalizes to our problem another dualscheme that has been recently introduced by Jacinto and Scheimberg (Optimization 57:795–

805, 2008) for convex equilibrium problems Through these schemes, as by products, weobtain new optimality conditions for (GEP) and also, gap functions for (GEP), which coverthe ones in Antangerel et al (J Oper Res 24:353–371, 2007, Pac J Optim 2:667–678, 2006)for variational inequalities and standard convex equilibrium problems These results, in turn,when applied to DC and convex optimization problems with convex constraints (considered

as special cases of (GEP)) lead to Toland-Fenchel-Lagrange duality for DC problems in Dinh

et al (Optimization 1–20, 2008, J Convex Anal 15:235–262, 2008), Fenchel-Lagrange andLagrange dualities for convex problems as in Antangerel et al (Pac J Optim 2:667–678,2006), Bot and Wanka (Nonlinear Anal to appear), Jeyakumar et al (Applied Mathematicsresearch report AMR04/8, 2004) Besides, as consequences of the main results, we obtainsome new optimality conditions for DC and convex problems

This work was completed while the first author was visiting the Department of Mathematics of the University of Namur, Namur, Belgium in July-August 2007 and in August 2008.

N Dinh

Department of Mathematics, International University, Vietnam National University,

Ho Chi Minh City, Vietnam

J J Strodiot (B) · V H Nguyen

Department of Mathematics, University of Namur (FUNDP), Namur, Belgium

e-mail: jean-jacques.strodiot@fundp.ac.be

Find x ∈ K such that

f (x, y) + (y) ≥ (x) for all y ∈ K,

where X is a locally convex Hausdorff topological space, K is a nonempty closed convex subset of X and f : X × X →R∪ {+∞} and  : X →R∪ {+∞} are functions satisfying:

(a) f (x, x) = 0 for all x ∈ K;

(b) f x (·) := f (x, ·) is proper, lower semi-continuous (l.s.c.), and convex for all x ∈ K ;

(c)  = g − h where g, h: X →R∪ {+∞} are two proper, l.s.c., convex functions

Here, by convention, we assume that∞ − ∞ = ∞ + (−∞) = +∞.

This problem is very general in the sense that it includes, as particular cases, many differentproblems as, for example, the problem of minimizing the difference of two convex functions,the mixed variational inequality problem, and when = 0, the Nash equilibrium problem in

noncooperative games, the fixed point problem, the nonlinear complementarity problem andthe vector optimization problem (see, for instance, Blum and Oettli [6] and the referencesquoted therein) The interest of such a general problem is that it unifies all these particularproblems in a convenient way Moreover, many results obtained for one of these problems can

be extended with suitable modifications to the problem(G E P) However, the generalized

equilibrium problem(G E P) is very important in itself Indeed, it covers some important

models in economics as, for example, the Nash-Cournot oligopolistic market equilibriummodel with concave cost functions [26] In this model, the function f (x, y) = F(x), y − x

where F is affine and the function  is a difference of two convex functions (a DC function

in short)

Recently, duality results and optimality conditions have been obtained for equilibriumproblems by Martinez-Legaz and Sosa [23] when = 0 and by Jacinto and Scheimberg

[18] when is convex Our aim in this paper is to obtain similar results but for the case

where is a DC function First, for each x ∈ K , we consider a DC optimization problem (P x ), which allows us to give a fixed point formulation of the solutions of (G E P) Then, we

associate with each DC problem(P x ), a dual problem by using the Toland-Fenchel-Lagrange

duality This is the subject of Sect.3where we develop general duality and optimality resultsfor a DC problem In that section we also introduce a closedness condition, called(CC),

that plays the role of a constraint qualification for these classes of problems In Sect.4weuse the duality for problems(P x ) to construct a first dual problem (DG E P) associated with

problem(G E P) When  = 0, this dual reduces to the dual presented by Martinez-Legaz

and Sosa in [23] This dual problem also reduces to the ones introduced by Bigi, Castellani,and Kassey in [5] and by Mosco in [25] for variational inequality (VI) (see Sect.7) First weprove weak and strong duality properties for these problems under the closedness condition

(CC) which extended the corresponding results in [5,23,25] We then establish necessaryand sufficient optimality conditions for(G E P) These conditions, at the same time, give rise

to the relationships between the solutions of problems(G E P) and (DG E P) In particular,

we prove that if the optimal value of the dual problem is zero (the primal problem(G E P)

might not have any solution), then for any > 0 the problem (G E P) admits -solutions.

In the last part of this section we introduce another dual scheme that generalizes the dualpresented by Jacinto and Scheimberg in [18] when is convex.

In Sect.5we propose gap functions related to the duality developed in the previous sectionswhich extend the ones introduced in [1,2] for variational inequalities and for equilibriumproblems, while in Sects.6and7we show that our dual scheme allows us to find again well-

known results when applied to special cases of problem (G E P) in [5,11,13,14,23,25]

In particular, we develop in Sect.6the case of convex and DC optimization problems andfind again several results established recently in [11,13,14] Sect.7is devoted to the case ofequilibrium problems in the sense of Blum and Oettli First we prove that in the latter casethe dual problem(DG E P) coincides with Martinez-Legaz and Sosa’s dual [23] Then weshow that in the particular case of variational inequality problems the dual problem(DG E P)

is equivalent to the dual introduced by Bigi, Castellani, and Kassey in [5] and by Mosco in[25]

2 Preliminaries

Let us recall some notations and properties useful in this paper Let X be a locally convex Hausdorff topological vector space with its topological dual X∗, endowed with the weak∗-

topology

The indicator function of a set D ⊂ X is defined by δ D (x) = 0 if x ∈ D and δ D (x) = +∞

if x /∈ D Moreover, the support function σ D is defined on X∗and is given by σ D (u) =

supx ∈D u (x) When D∗is a subset of X∗, cl D∗stands for the closure of D∗with respect to

the weak∗topology in X∗.

Let k : X →R∪ {+∞} be a proper l.s.c., and convex function The conjugate function

of k , k∗: X∗→R∪ {+∞}, is defined for all v ∈ X∗by

k∗(v) = sup{v, x − k(x) | x ∈ dom k},

where the domain of k is given by dom k := {x ∈ X | k(x) < +∞}.

If a ∈ dom k, then, following [19], we have

epi k∗=

≥0 {(v, v(a) +  − k(a)) | v ∈ ∂ k (a)} , (1)

where, for a given ≥ 0, the - subdifferential of k at a ∈ domk, ∂  k (a), is defined as the

possibly empty weak∗-closed convex set

∂  k (a) = {v ∈ X∗ | k(x) − k(a) ≥ (v, x − a) −  for all x ∈ dom k}.

If = 0, then ∂  k (a) collapses to ∂k(a), the usual subdifferential of k at a in the sense of

convex analysis (for more details, see [28])

Now let D be a convex subset of X and let ε ≥ 0 The approximate normal cone at a ∈ D

is defined by

N ε (D, a) = { u ∈ X∗| u(x − a) ≤ ε for all x ∈ D }.

Whenε = 0, N ε (D, a) is the classical cone N(D, a) of convex analysis Moreover, it is easy

to see that N (D, a) = ∂δ D (a).

Finally, we recall some results on DC programs which are useful for our study in the nextsections The first one is due to J.B Hiriart-Urruty [17] and the second one to Toland [27].

Lemma 2.1 [17] Let X be a locally convex Hausdorff topological vector space and let

F , G : X →R∪ {+∞} be l.s.c., proper and convex functions Then

(i) A point a ∈ X is a global minimizer of the problem inf x ∈X {F(x) − G(x)} if and only

if for any ε ≥ 0, ∂ ε G (a) ⊂ ∂ ε F (a),

(ii) If a ∈ X is a local minimizer of inf x ∈X {F(x) − G(x)}, then ∂G(a) ⊂ ∂ F(a).

Lemma 2.2 [17,27] Let X be a locally convex Hausdorff topological vector space and

let F , G : X → R∪ {+∞} be two proper and convex functions If F is l.s.c on X and

G∗∗(x) = G(x) for all x ∈ X, then

inf

x ∈X {F(x) − G(x)} = inf

u ∈X∗{G∗(u) − F∗(u)}.

3 Duality of DC optimization problems

In this section we consider a general DC problem of model(Q) below We establish

opti-mality conditions and dual results for(Q) that will be the main tools for the establishment

of the corresponding results for the generalized equilibrium problem(G E P) in the next

sections However the main results of this section may be of their own interest since theyyield the standard Fenchel duality result for convex optimization problem (see Corollary3.2

and also, [7,8]), cover the subdifferential sum rule of convex functions established recently

in [8] (see Corollary3.1), and give rise to a new Farkas’ lemma involving DC inequalities(see Corollary 3.4)

Consider the problem(G E P) defined in Sect.1 For each x ∈ K , we associate with (G E P) the optimization problem

(P x ) p(x) :=



inf f (x, y) + (y)

s.t y ∈ K.

Since f (x, x) = 0 for all x ∈ K , the following result is straightforward from the

defini-tions of problems(G E P) and (P x)

Lemma 3.1 A point ¯x ∈ K is a solution of (G E P) if and only if ¯x is a solution of (P ¯x ) In that case, p ( ¯x) = ( ¯x) and ¯x ∈ Q( ¯x) where Q(x) := arg min (P x ), i.e., ¯x is a fixed point

of the mapping Q.

It is worth mentioning that for each x ∈ K, (P x ) is the problem of minimizing the DC

function (difference of two convex functions) f x (y) + g(y) − h(y) over the convex set K

These problems(P x ) are special cases of the following general DC problem

(Q)



inf F (y) + G(y) − H(y)

s.t y ∈ K,

where X is a locally convex Hausdorff topological space, K is a closed convex subset of X , and F , G, H : X →R∪ {+∞} are proper, l.s.c., and convex functions It is obvious that

for each x ∈ K , (P x) is of the model(Q) where f x , g, and h play the roles of F, G, and H,

respectively We start with the following proposition which plays a key role in the study of

(Q) This proposition may also have its own interest for it recovers the corresponding

Theo-rem 1 in [8] Several parts of its proof are similar to those of Theorem3.1in [13] However,for the completeness of the paper, we give it in details

From now on, the optimal value of problems(P x ) and (Q) are denoted ν(P x ) and ν(Q),

respectively

Proposition 3.1 (Conjugate and approximate subdifferential sum rules involving convex

functions) Assume that U , V, T : X →R∪ {+∞} are proper, l.s.c., and convex functions such that domU

(i) epi U∗+ epi V∗+ epi T∗is weak∗-closed,

(ii) For each x∗∈ X∗,

(U + V + T )∗(x∗) = min

u∗,v∗∈X∗{U∗(u∗) + V∗(v∗) + T∗(x∗− u∗− v∗)} (3)

(the infimum in the right-hand side is attained),

(iii) For any ¯x ∈ domU ∩ domV ∩ domT and each  ≥ 0,

If x∗ ∗, then(U + V + T )∗(x∗) = +∞ and (ii) holds So, for proving

the converse inequality in (3), it is sufficient to assume that x∗∈ dom(U + V + T )∗ Then

we have

On the other hand we observe that (see (2))

epi(U + V + T )∗= cl epi U∗+ epi (V + T )∗

= cl epi U∗+ cl (epi V∗+ epi T∗)

= cl epi U∗+ epi V∗+ epi T∗

So, since(i) holds, we obtain that

188 J Glob Optim (2010) 48:183–208Combining now (7) and (9), we deduce that

(x∗, (U + V + T )∗(x∗)) = (u∗, r) + (v∗, s) + (w∗, t)

for some(u∗, r) ∈ epi U∗, (v∗, s) ∈ epi V∗, and(w∗, t) ∈ epi T∗ Consequently,

(U + V + T )∗(x∗) ≥ U∗(u∗) + V∗(v∗) + T∗(x∗− u∗− v∗),

and(ii) follows Finally, the infimum in (ii) is attained at some u∗, v∗ such that u∗ ∈

domU∗, v∗∈ domV∗, and x∗− u∗− v∗∈ domT∗.

[(ii) ⇒ (iii)] Assume (ii) Let ¯x ∈ domU ∩ domV ∩ domT and let  ≥ 0 We firstly

observe that the inclusion

in(iii) can be easily verified from the definition of approximate differentials in Sect.1 For

the converse inclusion, let x∗∈ ∂  (U + V + T )( ¯x) Since ¯x ∈ dom(U + V + T ), it follows

from (1) that

 + x∗, ¯x − U( ¯x) − V ( ¯x) − T ( ¯x) ≥ (U + V + T )∗(x∗), (10)

which shows that x∗ ∈ dom(U + V + T )∗ Thanks to(ii), there exist u∗, v∗ ∈ X∗, such

that u∗ ∈ domU∗, v∗∈ domV∗, x∗− u∗− v∗∈ domT∗, and that(U + V + T )∗(x∗) =

U∗(u∗) + V∗(v∗) + T∗(x∗− u∗− v∗) So it follows from (10) that

Note that (from (10) 0≤ 1+ 2+ 

3≤  Let 3:=  − 1− 2 Then3≥ 0 and 

3≤ 3,which entails that∂ 

3T ( ¯x) ⊂ ∂ 3T ( ¯x) Combining this and (12), we obtain

[(iii) ⇒ (i)] Assume that (iii) holds Take a ∈ domU ∩ domV ∩ domT and (x∗, r) ∈

cl {epi U∗+ epi V∗+ epi T∗} Then by (8), (x∗, r) ∈ epi (U + V + T )∗, and hence, it

follows from (1) that there exists ≥ 0 such that x∗∈ ∂ (U + V + T )(a) and

By(iii), there exist 1, 2, 3 ≥ 0 and u∗, v∗, w∗∈ X∗such that1+ 2+ 3 = , x∗=

u∗+ v∗+ w∗, and u∗∈ ∂1U (a), v∗∈ ∂2V (a), w∗∈ ∂3T (a) Again, by (1),

(u∗, u∗, a + 1− U(a)) ∈ epi U∗, (v∗, v∗, a + 2− V (a)) ∈ epi V∗, (w∗, w∗, a + 3− T (a)) ∈ epi T∗,

which, in turn, implies that

(x∗, x∗(a) +  − (U + V + T )(a)) ∈ epi U∗+ epi V∗+ epi T∗.

The last inclusion and (13) entail that(x∗, r) ∈ epi U∗+ epi V∗+ epi T∗, which proves(i).



Remark 3.1 Concerning statement (ii), it is worth noting that if x∗∈ dom(U +V +T )∗, then

there exist u∗∈ domU∗, v∗∈ domV∗such that x∗− u∗− v∗∈ domT∗and the infimum in

the right hand side of (3) is attained at u∗, v∗ Otherwise, i.e., when x∗ ∗,

for arbitrary u∗, v∗∈ X∗, one has

U∗(u∗) + V∗(v∗) + T∗(x∗− u∗− v∗) = +∞.

The following Corollary is useful for the study of Problem(P x ) It recovers Corollary 1

in [8] when one of the functions U , V , and T is a zero constant function.

Corollary 3.1 (Subdifferential sum rule involving convex functions) Assume that U , V, T :

X →R∪ {+∞} are proper, l.s.c., and convex functions If epi U∗+ epi V∗+ epi T∗is

weak∗-closed, then for any ¯x ∈ domU ∩ domV ∩ domT ,

∂(U + V + T )( ¯x) = ∂U( ¯x) + ∂V ( ¯x) + ∂T ( ¯x).

Theorem 3.1 (Optimality Condition for (Q)) For Problem (Q), assume that epi F∗+

epi G∗+ epi δ∗

K is weak∗-closed Then

(i) ¯x ∈ K is a global solution of (Q) if and only if for any  ≥ 0

Moreover, if ¯x is a local solution of (Q), then for any x∗ ∈ ∂ H( ¯x), there exist u∗ ∈

domF∗, v∗∈ domV∗such that x∗− u∗− v∗∈ domδ∗

K and

H∗(x∗) − F∗(u∗) − G∗(v∗) − δ∗K (x∗− u∗− v∗) = F( ¯x) + G( ¯x) − H( ¯x) Proof Note that problem (Q) is equivalent to the minimization problem

inf

x ∈X [(F + G + δ K )(x) − H(x)]

Hence by Lemma2.1,¯x ∈ K is a global solution of (Q) if and only if for any ε ≥ 0,

∂ ε H ( ¯x) ⊂ ∂ ε (F + G + δ K )( ¯x).

Thus,(i) follows from this and Proposition3.1

Similarly,(ii) follows from Lemma2.1and Corollary 3.1 

Theorem 3.2 (Duality for (Q)) For Problem (Q), we have

190 J Glob Optim (2010) 48:183–208

(ii) If epi F∗+ epi G∗+ epi δ∗

K is weak∗-closed, then

that H∗∗ = H In fact, for any x∗, u∗, v∗∈ X∗and any y ∈ K , by definition of conjugate

≤ infx∗∈X∗[H∗(x∗) − x∗, y] + F(y) + G(y)

= F(y) + G(y) − H(y),

which proves(i) since the last inequality holds for any y ∈ K

For the proof of(ii), note that

Since epi F∗+ epi G∗+ epi δ∗

K is weak∗-closed, it follows from Proposition3.1that

Now the general Fenchel duality result for convex problems in infinite dimensional spaces(see [7,8] and the references quoted therein) follows from the previous proposition as shown

in the next corollary

Corollary 3.2 [8] (Fenchel Duality for the sum of convex functions) Assume that F , G :

X→R∪{+∞} are proper, l.s.c., and convex functions Assume further that epi F∗+epi G∗

is weak∗-closed Then

The conclusion now follows directly from Theorem3.2where H ≡ 0 and K = X since

epi F∗+ epi G∗+ epi δ∗K = epiF∗+ epiG∗+ {0} × [0, ∞)

= epiF∗+ epiG∗.



The following corollary is a direct consequence of Theorem 3.2.

Corollary 3.3 (Fenchel Duality for convex problems) For problem (Q), assume that H ≡ 0 and G ≡ 0 Assume further that epi F∗+ epi δ∗

K is weak∗-closed Then

-closed Then the following statements are equivalent:

(i) For all y ∈ K, F(y) + G(y) − H(y) ≥ α,

(ii) For each x∗∈ X∗, there exist u∗, v∗∈ X∗such that

H∗(x∗) − F∗(u∗) − G∗(v∗) − δ∗

K (x∗− u∗− v∗) ≥ α.

Proof It is clear that (i) is equivalent to

inf

y ∈K [F (y) + G(y) − H(y)] ≥ α.

By Theorem3.2, the last inequality is equivalent to

inf

x∗∈X∗ max

u∗,v∗∈X∗{H(x∗) − F∗(u∗) − G∗(v∗) − δ∗

K (x∗− u∗− v∗)} ≥ α,

We now come back to problem(P x ) associated with (G E P) The duality results for the

general problem(Q) give rise to the corresponding ones for (P x ) as shown below But first,

we introduce the definition of a constraint qualification called closedness property for(P x ).

Definition 3.1 (Closedness Condition) Let x ∈ K If the set

x + epi g∗+ epi δ∗

K

is weak∗-closed in the dual space of X×R, then problem(P x ) is said to satisfy the

closed-ness condition,((CC) in short), or equivalently, that problem (G E P) satisfies the closedness

condition(CC) at x.

It is worth observing that the dual form constraint qualification of the type(CC) seems

to be used for the first time in [8,20] This condition is weaker than several qualificationconditions known in the literature such as generalized Slater conditions and interior-typeconditions It was successfully used for establishing optimality conditions, duality, stability

of convex programming problems [8,20], convex infinite programs [10,15], DC problemswith convex constraints [13,14], and DC infinite programs with parameters [12] It was alsoused to study the variational inequalities and equilibrium problems (see [1,2]) In this paperthis condition plays a key role in the study of duality and other topics in the next sections

For x ∈ K, x∗, u∗, v∗∈ X∗, set

L (x, x∗, u∗, v∗) = h∗(x∗) − f∗

x (u∗) − g∗(v∗) − δ∗

K (x∗− u∗− v∗). (16)

As a consequence of Theorem3.2, we obtain the following corollary

Corollary 3.5 (Duality for(P x )) Let x ∈ K Then

(the “sup” in the right hand-side is attained).

Proof This is a direct consequence of Theorem3.2where f x , g, and h play the roles of F, G,

4 Duality and optimality conditions for(G E P)

In this section we introduce a dual problem(DG E P) associated with problem (G E P).

We give weak and strong duality results and optimality conditions for(G E P) as well The

latter, at the same time, shows the relationships between the solutions of(G E P) and those

of(DG E P) Besides, it is shown that if the primal problem (G E P) possesses an optimal

solution then the value of the dual problem(DG E P) is zero, i.e., ν(DG E P) = 0 However,

the converse is not true In such a situation, i.e., when the optimal value of the dual problem

(DG E P) is zero, we prove that for any  > 0, the problem (G E P) possesses at least an

-solution As it is proven in Sect.7, when = 0, the dual problem (DG E P) coincides

with the dual problem defined by Martinez-Legaz and Sosa in [23], and when h ≡ 0 and

f (x, y) = F(x), y − x where F is an operator from X to X∗, the dual problem(DG E P)

is equivalent to the dual problem for variational inequality problems in the sense of Mosco[25] Finally, to end this section, we generalize to our problem another dual scheme thatcovers the one recently introduced by Jacinto and Scheimberg in [18] for problem(G E P)

for the case where h≡ 0

First let us recall that for problem(G E P), each x ∈ K is associated to an optimization

problem(P x ):

p (x) = inf

y ∈K [ f x (y) + (y)].

From this definition and from Lemma3.1, we can conclude that

• p(x) ≤ (x) for all x ∈ K , and

• ¯x ∈ K is a solution of (G E P) if and only if p( ¯x) = ( ¯x).

Definition 4.1 (Local solutions of (G E P)) A point ¯x ∈ K is called a local solution of (G E P) if there exists a neighborhood U of ¯x such that

f ( ¯x, y) + (y) ≥ ( ¯x) for all y ∈ U ∩ K.

It is obvious that ¯x ∈ K is a local solution of (G E P) if and only if it is a local solution of (P¯x) Furthermore, any global solution of (G P E) is also a local solution of this problem.

So the problem of finding (local/global) solutions of(G E P) reduces to the one of finding

(local/global) solutions of the optimization problem

x ∈K [p(x) − (x)],

or, equivalently, to the problem

is called the dual problem of the generalized equilibrium problem(G E P).

It is worth mentioning that this dual problem collapses to the one introduced by Legaz and Sosa in [23] when h ≡ 0 and g ≡ 0 When h ≡ 0 and f (x, y) = F(x), y − x

Martinez-where F is an operator from X to X∗, the dual problem(DG E P) is equivalent to the dual

problem for variational inequality problems in the sense of Mosco [25] and that of Bigi,Castellani, and Kassey in [5]

Definition 4.2 (Solution of (DG E P)) A solution of the dual problem is a point ¯x ∈ K such

that for any x∗∈ dom h∗, there exist u∗, v∗∈ X∗such that the following equality holds

The point ¯x is said to be a weak solution of the dual if for any x∗ ∈ ∂h( ¯x), there exist

u∗, v∗∈ X∗such that the equality (19) holds.

Remark 4.1 Note that the inequality

L ( ¯x, x∗, u∗, v∗) ≤ g( ¯x) − h( ¯x)

is always true Indeed, using the same argument as in the first part of the proof of Theorem3.2,

we get for all y ∈ K ,

L ( ¯x, x∗, u∗, v∗) = h∗(x∗) − f∗

¯x (u∗) − g∗(v∗) − δ∗

K (x∗− u∗− v∗)

≤ h∗(x∗) − x∗, y + g(y) + f ¯x (y)

≤ −h(y) + g(y) + f ¯x (y).

Taking y = ¯x, we obtain the desired inequality Therefore, the equality in the definition (19)

is equivalent to

L ( ¯x, x∗, u∗, v∗) ≥ g( ¯x) − h( ¯x).

Furthermore, it is easy to see thatν(DG E P) ≤ 0.

The weak and strong duality results are given in the next theorem

Theorem 4.1 (Weak and strong duality for(G E P)) For problem (G E P), the following properties hold:

(ii) If for each x ∈ K , the closedness condition (CC) holds, then

ν(P) = ν(DG E P).

194 J Glob Optim (2010) 48:183–208

Proof (i) is obvious To prove (ii), note that if (CC) holds, then problem (P x) enjoys the

strong duality property for each x, i.e.,

p (x) = inf

x∗∈X∗ max

u∗,v∗∈X∗L (x, x∗, u∗, v∗).

Remark 4.2 Note that if the problem (G E P) has a solution then ν(P) = 0 In Theorem4.1,

the equality (ii) holds provided that (CC) holds for every x ∈ K But even in this case, the

values of each side in this equality might not be zero if(G E P) has no solution as shown in

the following simple example

Example 4.1 [26] Consider the generalized equilibrium problem(P1) of finding x ∈ K := [−1, 1] such that

x, y − x − y2≥ −x2 for all y ∈ K.

This problem is of the model(G E P) where (x) = h(x) = −x2is a concave function,

g ≡ 0, and f (x, y) = x, y − x It is easy to see that (CC) holds for all x ∈ [−1, 1] and

emphasize that(P1) has no solution.

We now establish the relationship between the solutions of (G E P) and those of (DG E P),

and we derive at the same time, optimality conditions for(G E P) First we consider the local

solutions of (G E P).

Theorem 4.2 Let ¯x ∈ K For the problem (G E P), assume that the closedness condition

there exist u∗∈ dom f∗

¯x , v∗∈ domg∗such that

In particular, ¯x is a weak solution of (DG E P).

Proof Let ¯x be a local solution of (G E P) Then ¯x is a local solution of the DC program

inf

y ∈K [ f ¯x(y) + g(y) − h(y)].

Since(CC) holds for (P x ), it follows from [27] and the subdifferential sum rule, lary3.1, that

Corol-∂h( ¯x) ⊂ ∂( f ¯x + g + δ K )( ¯x)

⊂ ∂ f ¯x( ¯x) + ∂g( ¯x) + N K ( ¯x). (21)

Let x∗∈ ∂h( ¯x) (this set is non-empty by assumption) By (21), there exist u∗∈ ∂ f ¯x( ¯x), v∗∈

∂g( ¯x), and w∗∈ N K ( ¯x) = ∂δ K ( ¯x) such that x∗= u∗+ v∗+ w∗, which give rise to

Now we consider the global solutions of (G E P).

Theorem 4.3 Let ¯x ∈ K For problem (G E P), assume that the closedness condition (CC) holds at ¯x If ¯x is a global solution of (G E P), then for each x∗∈ X∗, there exist u∗, v∗∈ X∗

satisfying

In this case, ¯x is a solution of the dual problem (DG E P) and ν(DG E P) = 0.

Proof Assume that ¯x is a solution of (G E P) Then by Lemma3.1, ¯x solves (P ¯x), which

means thatν(P ¯x ) = g( ¯x) − h( ¯x) It now follows from Corollary3.5(ii) that for each

then ¯x is a global solution of (G E P).

Proof Let x∗∈ dom h∗ By assumption, there exist u∗∈ dom f∗

¯x , v∗∈ dom g∗such that

(23) holds Using the same argument as in the first part of the proof of Theorem3.2, we

obtain, for all y ∈ K , that