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Abstract and Applied Analysis
Volume 2013 (2013), Article ID 176363, 8 pages
Strong Duality and Optimality Conditions for Generalized Equilibrium Problems
1College of Mathematics and Statistics, Jishou University, Jishou 416000, China
2Department of Applied Mathematics, National Sun Yat-sen University, Kaohsiung 80424, Taiwan
3School of Mathematics, Physics and Information Science, Zhejiang Ocean University, Zhoushan 316004, China
Received 29 August 2013; Accepted 4 September 2013
Academic Editor: Jen-Chih Yao
Copyright © 2013 D. H. Fang and J. F. Bao. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
We consider a generalized equilibrium problem involving DC functions. By using the properties of the epigraph of the conjugate functions, some sufficient and/or necessary conditions for the weak and strong duality results and optimality conditions for generalized equilibrium problems are provided.
Consider the following generalized equilibrium problem: where is a locally convex Hausdorff topological space, is a nonempty convex subset of , , and are proper functions satisfying the following.(a) for all .(b) is proper convex for all .(c), where are two proper convex functions. Here and throughout the whole paper, following [1, page 39], we adapt the convention that .
As mentioned in , equilibrium problems theory provides us with a unified, natural, innovative, and general framework to study a wide class of problems arising in finance, economics, network analysis, transportation, elasticity, and optimization. This theory has witnessed an explosive growth in theoretical advances and applications across all disciplines of pure and applied sciences. Equilibrium problems have been studied extensively, and many problems such as optimization problems, Nash equilibria problems, complementarity problems, fixed point problems, variational inequality problems, and convex vector optimization problems can be recast into the form (GEP); see, for example, [3–10] and the references therein.
Duality for equilibrium problems was first studied in . The schemes proposed in that paper are extensions of a classical duality theory for variational inequalities. In spirit of convex optimization, duality results and optimality conditions have been obtained for equilibrium problems by Martínez-Legaz and Sosa  when and by Jacinto and Scheimberg  when is convex, which extended the classical convex duality results. Recently, the authors in  considered the generalized equilibrium problems in the case where is a DC function. Under the assumptions that is closed and functions , , are lower semicontinuous (lsc in brief), they gave some weak and strong duality results and optimality conditions for (GEP) via a closedness qualification condition.
Inspired by the works mentioned above, we continue to study the generalized equilibrium problems. Our main aim in the present paper is to give some new regularity conditions which characterize the weak duality, the strong duality, and optimality conditions for (GEP). In general, we do not impose any topological assumption on or on , , and ; that is, is not necessarily closed, and , , and are not necessarily lsc. Most of results obtained in the present paper are proper extensions of the early results (e.g., [5, 14–16]). In particular, even in the special case when the closedness of and lower semicontinuity of , , and are satisfied, our results provide improved versions of [5, Theorems 4.2, 4.3 and 4.4].
The paper is organized as follows. Section 2 contains some necessary notations and preliminary results. In Section 3, we develop general duality and optimality results for a DC optimization problem. The weak and strong duality results and optimality conditions for generalized equilibrium problems are given in Section 4.
2. Notations and Preliminaries
The notation used in the present paper is standard (cf. ). In particular, we assume throughout the whole paper that is a real locally convex Hausdorff topological vector space, and denotes the dual space, endowed with the -topology . By we will denote the value of the functional at ; that is, . Let be a set in . The closure of is denoted by . If , then denotes the -closure of . For the whole paper, we endow with the product topology of and the usual Euclidean topology.
The indicator function of a nonempty set is defined by if and if . Let be a proper function. The conjugate function and the epigraph of are denoted by and , respectively; they are defined by where the effective domain . It is well known and easy to verify that is -closed. The lsc hull of , denoted by , is defined by Then, by [1, Theorem 2.3.1], and by [1, Theorem 2.3.4], if is proper and convex, then . Let . The subdifferential of at is defined by if , and otherwise. Clearly, the following equivalence holds: By definition, the Young-Fenchel inequality below holds: Moreover, by [1, Theorem 2.4.2(iii)], (the equality in (7) is usually referred to as Young's equality). If , are proper, then Furthermore, for each and , Moreover, if is convex and lsc on , then, by [17, Lemma 2.3],
Lemma 1. Let be proper convex functions satisfying .(i)If , are lsc, then (ii)If either or is continuous at some point of , then
3. Duality and Optimality Conditions for DC Optimization Problem
Let . Consider the following DC optimization problem: where is a convex subset of (not necessarily closed) and are proper convex functions (not necessaily lsc). Following , we define the dual problem of by where is defined by Let and denote the optimal values of problems and , respectively. Especially, in the case when , we write , and for , , and , respectively. One of the main aims in this section is devoted to the study of the weak duality and the strong duality between and , which are defined as follows.
Definition 2. We say that(a)the weak duality holds (between and ) if ;(b)the strong duality holds (between and ) if and for each , there exists such that ;(c)the stable weak duality (resp., the stable strong duality) holds if the weak duality (resp., the strong duality) holds between and for each .
If is lsc, then by [5, Theorem 3.2(i)], the weak duality holds. However, the weak duality does not necessarily hold in general as will be shown in the following example.
Example 3. Let and . Let be defined by , , and Then, , , and are proper convex functions and . Clearly, , , and . Hence, . This implies that . Consequently, the weak duality does not hold.
To consider the weak duality, the strong duality, and optimality conditions for problem , we introduced the following conditions. For simplicity, we denote where we adapt the convention .
Definition 4. The family is said to satisfy(i)the weak closure condition at 0 () if (ii)the closure condition at 0 () if (iii)the weak closure condition () if (iv)the closure condition () if (v)the Moreau-Rockafellar formula (MRF) at if (vi) (MRF) if it satisfies (MRF) at each point in .
The following proposition describes the relationship between the (resp., the ) and the (resp., the ).
Proposition 6. The family satisfies the (resp., the ) if and only if for each , satisfies the (resp., the ).
Proof. Let , and let be the set defined by Then, by (9), the following equality is clear: Hence, we have that Moreover, using (9), we conclude that Thus, the conclusion holds by definitions and the proof is complete.
Under the assumption that the authors in  introduced the following closure condition to consider the strong duality and optimality conditions for DC optimization problem (14). The following proposition describes the relationships among the , the , and (30).
Proposition 7. The following implication holds: Furthermore, if (29) holds, then the following implications hold:
Proof. Suppose that the holds. Let , and let . Then, by (7),
thanks to the . Hence, for each ,
Let . Then, there exist , , and such that
Below we show that , , and . To do this, note by the definition that
Moreover, since , it follows from (7) that
Hence, by (35)–(37) and the Young-Fenchel inequality (6), we have that
This implies that by (7). Using the same argument, we have that and . Hence, , and
since is arbitrary. Therefore, the holds.
Furthermore, suppose that (29) holds. To show that (32), we only need to show the implication (30) holds. To do this, we assume that (30) holds. Since is lsc, it follows from (10) that Note that , are lsc and is closed; by Lemma 1(i), one has that This together with (30) and (41) implies that the holds. The proof is complete.
To study the weak duality and the strong duality, we need the following lemma.
Lemma 8. Let . Then, the following assertions hold:(i) if and only if .(ii) if and only if and for each , there exist and such that
Proof. (i) By the definition of the conjugate function, one has
Hence, the result is clear.
(ii) Let and let . Then Thus, there exist , , and such that Since it follows from (46) that This together with the definition of implies that and , satisfy (43).
Conversely, suppose that and for each , there exist and satisfying (43). Let . Then, there exist and such that (43) holds. Then This means that Therefore, Noting that is arbitrary, we have that Thus, we complete the proof.
Our first theorem of this section shows that the is a sufficient and necessary condition for the weak duality to hold.
Theorem 9. (i) The weak duality holds if and only if the family satisfies the .
(ii) The stable weak duality holds if and only if the family satisfies the .
Proof. As assertion (ii) is a global version of assertion (i). Hence, by Proposition 6, it suffices to prove assertion (i). Suppose that the weak duality holds. Let . Then, by Lemma 8(ii), we have and hence , which implies that , thanks to Lemma 8(i). Hence, (19) holds; that is, the holds.
Conversely, suppose that the family satisfies the . To show that , suppose on the contrary that . Then, there exists such that . Thus, by the definition of , we have that for each , there exist and such that (43) holds. Hence, by Lemma 8(ii), and by the . This together with Lemma 8(i) implies that , which contradicts to . Consequently, we have and complete the proof.
Theorem 10. (i) The strong duality holds if and only if the family satisfies the .
(ii) The stable strong duality holds if and only if the family satisfies the .
Proof. As before, it is sufficient to prove assertion (i). Suppose that the strong duality holds. Let . Then, and there exist and such that . By Theorem 9(i), holds, and so, we only need to verify the following inclusion:
To do this, let . Then, by Lemma 8(i), we have . Hence, and , , satisfying (43). This together with Lemma 8(ii) implies that as is arbitrary. Hence, (53) holds and so does the .
Conversely, suppose that the holds. Then, the family satisfies , and so by Theorem 9(i). Thus, to prove the strong duality, it suffices to show that and that for each there exist and satisfying . Note that the conclusion holds trivially if . Below we only consider the case when . By Lemma 8(i), , and so , thanks to the . Then, by Lemma 8(ii) and the definition of , we have that and for each there exist and satisfying . Hence, the strong duality holds. The proof is complete.
Theorem 11. Let be a solution of . Suppose that the family satisfies the at . Then Furthermore, if , then for each , there exist and such that
Proof. Since is a solution of , it follows that
Then, by the at , one has that
which is equivalent to (54) holds.
Furthermore, assume that . Then by (54), there exist , , and such that Since , , , and , it follows from (7) that Hence, which completes the proof.
Remark 12. In the case when (29) holds, Dinh et al. established the weak duality and strong duality in [5, Theorem 3.2] and the optimality condition in [5, Theorem 3.1] under the assumption that (30) holds. Clearly, by Proposition 7, Theorems 9 and 10 extend and improve [5, Theorem 3.2] and Theorem 11 extends and improves [5, Theorem 3.1].
4. Optimality Conditions and Dualities for Equilibrium Problem
Recall the optimization problem is defined as in Section 1. Let and consider the DC optimization problem Then, by the definitions of and , a point is a solution of if and only if is a solution of (cf. [5, Lemma 3.1]). Moreover, by the definition, we can find that for each , and, is a solution of if and only if . Hence, the problem of finding solutions of can be reduced to the one of finding solutions of the following optimization problem: Following , we defined the dual problem of by where is defined by
Let and denote the optimal values of problems and , respectively. Unlike , the weak duality (i.e., ) does not necessarily holds in general. Recall from Definition 4 that for each , the family satisfies the if and it satisfies the if Then, we have the following theorem.
Theorem 13. (i) Suppose that for each , the family satisfies the . Then, .
(ii) Suppose that for each , the family satisfies the . Then, .
Proof. (i) Since for each , the family satisfies the , it follows from Theorem 9(i) that
Hence, by the definitions of and , we see that .
(ii) Since for each , the family satisfies the , it follows from Theorem 10(i) that Thus, the result is seen to hold.
The following theorems establish the relationships between the solutions of and those of . First, we recall that a point is said to be a solution of if and it is said to be a solution of the dual problem if for each , there exist such that partially, if for each , there exist such that (70) holds, then is said to be a weak solution of problem .
Remark 14. (a) Obviously, is a solution of if and only if is a solution of .
(b) Let . If is lsc at , then for each , there exist and such that Consequently, is a solution of the dual problem if and only if for each , there exist such that In fact, let , , and . Then by the Young-Fenchel inequality (6), it is easy to see that Hence, (71) holds. However, (71) does not necessarily hold in general as will be showen in the following example.
Example 15. Let and . Let and be defined by , , and Let . Then, , and is not lsc at . Clearly, , , and . Then, , , and . Let . Then, for each and , one has that this implies that (71) does not hold.
By Theorem 11, we obtain the the following theorem straightforwardly, which was established in [5, Theorem 4.2] under the assumptions that (29) holds and Thus, by Proposition 7, our Theorem 16 improves the corresponding result in [5, Theorem 4.2].
Theorem 16. Let such that . Suppose that the family satisfies the at . If is a solution of , then is a weak solution of .
Theorem 17. Let . Suppose that is lsc and that the family satisfies the . Then, is a solution of if and only if is a solution of the dual problem .
Proof. Suppose that is a solution of . Then, is a solution of . Hence,
Since the family satisfies the , it follows from Theorem 10 that
which implies that for each , there exist and such that
Moreover, since is lsc at , it follows from Remark 14(b) that
Hence, (70) holds. This means that is a solution of the dual problem .
Conversely, suppose that is a solution of the dual problem . Let . Then, there exist and such that (70) holds. Note by the Young-Fenchel inequality (6) that for each , Then by (70), one has that for each , Since the above inequality holds for each , it follows that Consequently, by the lower semicontinuity of , one has that for each , and hence . This implies that since holds automatically. Thus, is a solution of .
Remark 18. Theorem 17 was established in [5, Theorems 4.3 and 4.4] under the assumptions that (29) and (76) hold. Thus, our Theorem 16 extends the corresponding result in [5, Theorems 4.3 and 4.4] to suit the case when , are not lsc and is not closed.
Below we will give a upper estimate for the Fréchet subdifferential of the function defined in (61). We first recall from  or [19, page 90] that the Fréchet subdifferential of at a point with is defined by where is an extended real-valued function.
Theorem 19. Let be a solution of . Suppose that the family satisfies the at ; that is, Then for each , one has that where denotes the unit ball in .
Proof. Let . If , then (87) holds trivially. Below let and . Let . Then, by the definition of of Fréchet subdifferential of at , there exists such that where denotes the unit ball in . Since is a solution of , it follows that . Note that holds for each . Then where the last inequality holds because . Define by Then, is a proper convex function on , and is a solution to the following convex programming: Hence, , and Note that is an interior point of the set ; then the function is continuous at . Hence, This together with (86) and (92) implies that that is, Which implies that (87) holds by the arbitrariness of . The proof is complete.
Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.
D. H. Fang was supported in part by the National Natural Science Foundation of China (Grant no. 11101186) and supported in part by the Scientific Research Fund of Hunan Provincial Education Department (Grant no. 13B095).
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