Strong Duality and Optimality Conditions for Generalized Equilibrium Problems
Abstract
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.
1. Introduction
- (a)
f(x, x) = 0 for all x ∈ C.
- (b)
fx(·)≔f(x, ·) is proper convex for all x ∈ C.
- (c)
ψ≔g − h, where are two proper convex functions.
As mentioned in [2], 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 [11]. 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 [12] when ψ = 0 and by Jacinto and Scheimberg [13] when ψ is convex, which extended the classical convex duality results. Recently, the authors in [5] considered the generalized equilibrium problems in the case where ψ is a DC function. Under the assumptions that C is closed and functions f(·, x), g, h 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 C or on fx(·), g, and h; that is, C is not necessarily closed, and fx(·), g, and h 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 C and lower semicontinuity of fx(·), g, and h 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. [1]). In particular, we assume throughout the whole paper that X is a real locally convex Hausdorff topological vector space, and X* denotes the dual space, endowed with the weak*-topology w*(X*, X). By 〈x*, x〉 we will denote the value of the functional x* ∈ X* at x ∈ X; that is, 〈x*, x〉 = x*(x). Let Z be a set in X. The closure of Z is denoted by cl Z. If W⊆X*, then cl W denotes the weak*-closure of W. For the whole paper, we endow X* × ℝ with the product topology of w*(X*, X) and the usual Euclidean topology.
The following lemma is known in [1, 6] (cf. [6, Lemma 2.1] for (11) and (12) and [1, Theorem 2.8.7] for (13)).
Lemma 1. Let be proper convex functions satisfying dom g∩dom h ≠ ∅.
- (i)
If g, h are lsc, then
() - (ii)
If either g or h is continuous at some point of dom g∩dom h, then
()()
3. Duality and Optimality Conditions for DC Optimization Problem
Definition 2. We say that
- (a)
the weak duality holds (between (P) and (D)) if v(D) ≤ v(P);
- (b)
the strong duality holds (between (P) and (D)) if v(P) = v(D) and for each x* ∈ dom F*, there exists (u*, v*) ∈ dom F* × dom G* such that L(x*, u*, v*) ≥ v(D);
- (c)
the stable weak duality (resp., the stable strong duality) holds if the weak duality (resp., the strong duality) holds between (Pp) and (Dp) for each p ∈ X*.
If H 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 X≔ℝ and C≔[0, +∞). Let be defined by F≔0, G≔δ[0,+∞), and
Definition 4. The family {F, G, H, δC} is said to satisfy
- (i)
the weak closure condition at 0 ((WCC) 0) if
() - (ii)
the closure condition at 0 ((CC) 0) if
() - (iii)
the weak closure condition ((WCC)) if
() - (iv)
the closure condition ((CC)) if
() - (v)
the Moreau-Rockafellar formula (MRF) at x ∈ dom (F + G − H)∩C if
() - (vi)
(MRF) if it satisfies (MRF) at each point in dom (F + G − H)∩C.
Remark 5. If H is lsc, then by (10), we have that
The following proposition describes the relationship between the (CC) (resp., the (WCC)) and the (CC) 0 (resp., the (WCC) 0).
Proposition 6. The family {F, G, H, δC} satisfies the (CC) (resp., the (WCC)) if and only if for each p ∈ X*, {F − p, G, H, δC} satisfies the (CC) 0 (resp., the (WCC) 0).
Proof. Let p ∈ X*, and let K(p) be the set defined by
Proposition 7. The following implication holds:
Proof. Suppose that the (CC) holds. Let x0 ∈ A, and let p ∈ ∂(F + G − H + δC)(x0). Then, by (7),
Furthermore, suppose that (29) holds. To show that (32), we only need to show the implication (30)⇒(CC) holds. To do this, we assume that (30) holds. Since H is lsc, it follows from (10) that
To study the weak duality and the strong duality, we need the following lemma.
Lemma 8. Let r ∈ ℝ. Then, the following assertions hold:
- (i)
(0, r) ∈ epi (F + G − H + δC) * if and only if v(P)≥−r.
- (ii)
(0, r) ∈ K if and only if v(D)≥−r and for each x* ∈ dom H*, there exist u* ∈ dom F* and v* ∈ dom G* such that
()
Proof. (i) By the definition of the conjugate function, one has
(ii) Let (0, r) ∈ K and let x* ∈ dom H*. Then
Conversely, suppose that v(D)≥−r and for each x* ∈ dom H*, there exist u* ∈ dom F* and v* ∈ dom G* satisfying (43). Let x* ∈ dom H*. Then, there exist u* ∈ dom F* and v* ∈ dom G* such that (43) holds. Then
Our first theorem of this section shows that the (WCC) 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 {F, G, H, δC} satisfies the (WCC) 0.
(ii) The stable weak duality holds if and only if the family {F, G, H, δC} satisfies the (WCC).
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 (0, r) ∈ K. Then, by Lemma 8(ii), we have v(D)≥−r and hence v(P)≥−r, which implies that (0, r)∈(F + G − H + δC) *, thanks to Lemma 8(i). Hence, (19) holds; that is, the (WCC) 0 holds.
Conversely, suppose that the family {F, G, H, δC} satisfies the (WCC) 0. To show that v(D) ≤ v(P), suppose on the contrary that v(P) < v(D). Then, there exists r ∈ ℝ such that v(P)<−r ≤ v(D). Thus, by the definition of v(D), we have that for each x* ∈ dom H*, there exist u* ∈ dom F* and v* ∈ dom G* such that (43) holds. Hence, (0, r) ∈ K by Lemma 8(ii), and (0, r) ∈ epi (F + G − H + δC) * by the (WCC) 0. This together with Lemma 8(i) implies that v(P)≥−r, which contradicts to v(P)<−r. Consequently, we have v(D) ≤ v(P) and complete the proof.
Theorem 10. (i) The strong duality holds if and only if the family {F, G, H, δC} satisfies the (CC) 0.
(ii) The stable strong duality holds if and only if the family {F, G, H, δC} satisfies the (CC).
Proof. As before, it is sufficient to prove assertion (i). Suppose that the strong duality holds. Let x* ∈ dom H*. Then, v(P) = v(D) and there exist u* ∈ dom F* and v* ∈ dom G* such that L(x*, u*, v*) ≥ v(D). By Theorem 9(i), (WCC) 0 holds, and so, we only need to verify the following inclusion:
Conversely, suppose that the (CC) 0 holds. Then, the family {f, g, δC; ft, gt : t ∈ T} satisfies (WCC) 0, and so v(D) ≤ v(P) by Theorem 9(i). Thus, to prove the strong duality, it suffices to show that v(D) ≥ v(P) and that for each x* ∈ dom H* there exist u* ∈ dom F* and v* ∈ dom G* satisfying L(x*, u*, v*) ≥ v(D). Note that the conclusion holds trivially if v(P) = −∞. Below we only consider the case when −r≔v(P) ∈ ℝ. By Lemma 8(i), (0, r) ∈ epi (F + G − H + δC) *, and so (0, r) ∈ K, thanks to the (CC) 0. Then, by Lemma 8(ii) and the definition of v(D), we have that v(D)≥−r and for each x* ∈ dom H* there exist u* ∈ dom F* and v* ∈ dom G* satisfying L(x*, u*, v*)≥−r. Hence, the strong duality holds. The proof is complete.
Theorem 11. Let x0 be a solution of (P). Suppose that the family {F, G, H, δC} satisfies the (MRF) at x0. Then
Proof. Since x0 is a solution of (P), it follows that
Furthermore, assume that x* ∈ ∂H(x0). Then by (54), there exist u* ∈ ∂F(x0), v* ∈ ∂G(x0), and w* ∈ NC(x0) such that
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
Theorem 13. (i) Suppose that for each x ∈ C, the family {fx, g, h, δC} satisfies the (WCC) 0. Then, v(𝒟) ≤ v(𝒫).
(ii) Suppose that for each x ∈ C, the family {fx, g, h, δC} satisfies the (CC) 0. Then, v(𝒟) = v(𝒫).
Proof. (i) Since for each x ∈ C, the family {fx, g, h, δC} satisfies the (WCC) 0, it follows from Theorem 9(i) that
(ii) Since for each x ∈ C, the family {fx, g, h, δC} satisfies the (CC) 0, it follows from Theorem 10(i) that
Remark 14. (a) Obviously, x0 ∈ C is a solution of (GEP ) if and only if x0 is a solution of .
(b) Let x0 ∈ C. If h is lsc at x0, then for each x* ∈ dom h*, there exist and v* ∈ dom g* such that
Example 15. Let X≔ℝ and C≔(−∞, 0]. Let and be defined by f≔0, g≔δ[0,+∞), and
Theorem 16. Let x0 ∈ C such that ∂H(x0) ≠ ∅. Suppose that the family satisfies the (MRF) at x0. If x0 is a solution of (GEP), then x0 is a weak solution of (𝒟).
Theorem 17. Let x0 ∈ C. Suppose that h is lsc and that the family satisfies the (CC) 0. Then, x0 is a solution of (GEP) if and only if x0 is a solution of the dual problem (𝒟).
Proof. Suppose that x0 is a solution of (GEP ). Then, x0 is a solution of . Hence,
Conversely, suppose that x0 is a solution of the dual problem (𝒟). Let x* ∈ dom h*. Then, there exist and v* ∈ dom g* such that (70) holds. Note by the Young-Fenchel inequality (6) that for each x ∈ C,
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 g, h are not lsc and C is not closed.
Theorem 19. Let x0 ∈ C be a solution of (GEP). Suppose that the family {g, δC} satisfies the (MRF) at x0; that is,
Proof. Let x0 ∈ C. If ∂H(x0) = ∅, then (87) holds trivially. Below let and x* ∈ ∂h(x0). Let γ > 0. Then, by the definition of of Fréchet subdifferential of p at x0, there exists η > 0 such that
Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.
Acknowledgments
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).
