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

Consider the following generalized equilibrium problem: ()𝑓𝑦(𝑥)GEPFind𝑥∈𝐶suchthat𝑥,𝑦+𝜓≥𝜓foreach𝑦∈𝐶, (1) where 𝑋 is a locally convex Hausdorff topological space, 𝐶 is a nonempty convex subset of 𝑋, 𝑓∶𝑋×𝑋→ℝ∶=ℝ∪{+∞}, and 𝜓∶𝑋→ℝ are proper functions satisfying the following.(a)𝑓(𝑥,𝑥)=0 for all 𝑥∈𝐶.(b)𝑓_𝑥(⋅)∶=𝑓(𝑥,⋅) is proper convex for all 𝑥∈𝐶.(c)𝜓∶=𝑔−ℎ, where 𝑔,ℎ∶𝑋→ℝ are two proper convex functions. Here and throughout the whole paper, following [1, page 39], we adapt the convention that (+∞)+(−∞)=(+∞)−(+∞)=+∞.

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 𝑓_𝑥(⋅), 𝑔, and ℎ; that is, 𝐶 is not necessarily closed, and 𝑓_𝑥(⋅), 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 𝑓_𝑥(⋅), 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 cl𝑍. If 𝑊⊆𝑋^∗, then cl𝑊 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 indicator function 𝛿_𝑍 of a nonempty set 𝑍 is defined by 𝛿_𝑍(𝑥)=0 if 𝑥∈𝑍 and 𝛿_𝑍(𝑥)=+∞ if 𝑥∉𝑍. Let 𝑓∶𝑋→ℝ be a proper function. The conjugate function and the epigraph of 𝑓 are denoted by 𝑓^∗ and epi𝑓, respectively; they are defined by 𝑓^∗𝑥^∗∶=sup_{𝑥∈dom𝑓}𝑥^∗,𝑥−𝑓(𝑥)foreach𝑥^∗∈𝑋^∗,epi𝑓∶=(𝑥,𝑟)∈𝑋×ℝ∶𝑓(𝑥)≤𝑟, (2) where the effective domain dom𝑓∶={𝑥∈𝑋∶𝑓(𝑥)<+∞}. It is well known and easy to verify that epi𝑓^∗ is weak^∗-closed. The lsc hull of 𝑓, denoted by cl𝑓, is defined by epicl𝑓=clepi𝑓. (3) Then, by [1, Theorem 2.3.1], 𝑓^∗=(cl𝑓)^∗ and by [1, Theorem 2.3.4], if cl𝑓 is proper and convex, then 𝑓^∗∗=cl𝑓. Let 𝑥∈𝑋. The subdifferential of 𝑓 at 𝑥 is defined by 𝜕𝑓(𝑥)∶=𝑥^∗∈𝑋^∗∶𝑓(𝑥)+𝑥^∗,𝑦−𝑥≤𝑓𝑦foreach𝑦∈𝑋 (4) if 𝑥∈dom𝑓, and 𝜕𝑓(𝑥)∶=∅ otherwise. Clearly, the following equivalence holds: 𝑥₀isaminimizerof𝑓iff0∈𝜕𝑓𝑥₀. (5) By definition, the Young-Fenchel inequality below holds: 𝑓(𝑥)+𝑓^∗𝑥^∗≥𝑥,𝑥^∗foreachpair𝑥,𝑥^∗∈𝑋×𝑋^∗. (6) Moreover, by [1, Theorem 2.4.2(iii)], 𝑓(𝑥)+𝑓^∗𝑥^∗=⟨𝑥^∗,𝑥⟩iff𝑥^∗∈𝜕𝑓(𝑥) (7) (the equality in (7) is usually referred to as Young's equality). If 𝑔, ℎ are proper, then epi𝑔^∗+epiℎ^∗⊆epi𝑔+ℎ^∗,𝑔≤ℎ⟹𝑔^∗≥ℎ^∗⟺epi𝑔^∗⊆epiℎ^∗,𝜕𝑔(𝑎)+𝜕ℎ(𝑎)⊆𝜕𝑔+ℎ(𝑎)foreach𝑎∈dom𝑔∩domℎ. (8) Furthermore, for each 𝑝∈𝑋^∗ and 𝑎∈ℝ, epiℎ+𝑝+𝑎^∗=epiℎ^∗+𝑝,−𝑎. (9) Moreover, if 𝑔 is convex and lsc on domℎ, then, by [17, Lemma 2.3], epiℎ−𝑔^∗=_{𝑥∗∈dom𝑔∗}epiℎ^∗−𝑥^∗,𝑔^∗𝑥^∗. (10)

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 𝑔,ℎ∶𝑋→ℝ be proper convex functions satisfying dom𝑔∩domℎ≠∅.(i)If 𝑔, ℎ are lsc, then epi𝑔+ℎ^∗=clepi𝑔^∗+epiℎ^∗. (11)(ii)If either 𝑔 or ℎ is continuous at some point of dom𝑔∩domℎ, then epi𝑔+ℎ^∗=epi𝑔^∗+epiℎ^∗,𝜕(𝑥)(𝑥)(𝑥)𝑔+ℎ=𝜕𝑔+𝜕ℎ𝑓𝑜𝑟𝑒𝑎𝑐ℎ𝑥∈dom𝑔∩domℎ. (12)(13)

3. Duality and Optimality Conditions for DC Optimization Problem

Let 𝑝∈𝑋^∗. Consider the following DC optimization problem: 𝑃_𝑝inf_𝑦∈𝐶𝐹𝑦+𝐺𝑦−𝐻𝑦−𝑝,𝑦, (14) where 𝐶 is a convex subset of 𝑋 (not necessarily closed) and 𝐹,𝐺,𝐻∶𝑋→ℝ are proper convex functions (not necessaily lsc). Following [5], we define the dual problem of (𝑃_𝑝) by 𝐷_𝑝inf_{𝑥∗∈dom𝐻∗}sup_{𝑢∗∈dom𝐹∗,𝑣∗∈dom𝐺∗}𝐿_𝑝𝑥^∗,𝑢^∗,𝑣^∗, (15) where 𝐿_𝑝∶dom𝐻^∗×dom𝐹^∗×dom𝐺^∗→ℝ is defined by 𝐿_𝑝𝑥^∗,𝑢^∗,𝑣^∗∶=𝐻^∗𝑥^∗−𝐹^∗𝑝+𝑢^∗−𝐺^∗𝑣^∗−𝛿^∗_𝐶𝑥^∗−𝑢^∗−𝑣^∗. (16) Let 𝑣(𝑃_𝑝) and 𝑣(𝐷_𝑝) denote the optimal values of problems (𝑃_𝑝) and (𝐷_𝑝), respectively. Especially, in the case when 𝑝=0, we write (𝑃), (𝐷) and 𝐿(⋅,⋅,⋅) for (𝑃₀), (𝐷₀), and 𝐿₀(⋅,⋅,⋅), respectively. One of the main aims in this section is devoted to the study of the weak duality and the strong duality between (𝑃) and (𝐷), which are defined as follows.

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^∗∈dom𝐹^∗, there exists (𝑢^∗,𝑣^∗)∈dom𝐹^∗×dom𝐺^∗ 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 (𝑃_𝑝) and (𝐷_𝑝) 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 𝐹,𝐺,𝐻∶ℝ→ℝ be defined by 𝐹∶=0, 𝐺∶=𝛿_[0,+∞), and 𝐻𝑦∶=0𝑦<0,1𝑦=0,+∞𝑦>0. (17) Then, 𝐹, 𝐺, and 𝐻 are proper convex functions and 𝑣(𝑃)=−1. Clearly, 𝐹^∗=𝛿_{0}, 𝐺^∗=𝛿^∗_𝐶=𝛿_(−∞,0], and 𝐻^∗=𝛿_[0,+∞). Hence, 𝑣(𝐷)=0. This implies that 𝑣(𝐷)>𝑣(𝑃). Consequently, the weak duality does not hold.

To consider the weak duality, the strong duality, and optimality conditions for problem (𝑃), we introduced the following conditions. For simplicity, we denote 𝐾∶=_{𝑥∗∈dom𝐻∗}epi𝐹^∗+epi𝐺^∗+epi𝛿^∗_𝐶−𝑥^∗,𝐻^∗𝑥^∗, (18) where we adapt the convention ∩_𝑡∈∅𝑆_𝑡=𝑋.

Definition 4. The family {𝐹,𝐺,𝐻,𝛿_𝐶} is said to satisfy(i)the weak closure condition at 0 ((WCC)₀) if 𝐾∩({0}×ℝ)⊆epi𝐹+𝐺−𝐻+𝛿_𝐶^∗∩({0}×ℝ); (19)(ii)the closure condition at 0 ((CC)₀) if 𝐾∩({0}×ℝ)=epi𝐹+𝐺−𝐻+𝛿_𝐶^∗∩({0}×ℝ); (20)(iii)the weak closure condition ((WCC)) if 𝐾⊆epi𝐹+𝐺−𝐻+𝛿_𝐶^∗; (21)(iv)the closure condition ((CC)) if 𝐾=epi𝐹+𝐺−𝐻+𝛿_𝐶^∗; (22)(v)the Moreau-Rockafellar formula (MRF) at 𝑥∈dom(𝐹+𝐺−𝐻)∩𝐶 if 𝜕𝐹+𝐺−𝐻+𝛿_𝐶(𝑥)⊆_{𝑥∗(𝑥)∈𝜕𝐻}𝜕𝐹(𝑥)+𝜕𝐺(𝑥)+𝑁_𝐶(𝑥)−𝑥^∗; (23)(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 𝐾=epi(𝐹−𝐻)^∗+epi𝐺^∗+epi𝛿^∗_𝐶 (24) and, by (8), (21) holds; that is, the (WCC) holds.

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,𝛿_𝐶} satisfies the (𝐶𝐶) (resp., the (𝑊𝐶𝐶)) if and only if for each 𝑝∈𝑋^∗, {𝐹−𝑝,𝐺,𝐻,𝛿_𝐶} satisfies the (𝐶𝐶)₀ (resp., the (𝑊𝐶𝐶)₀).

Proof. Let 𝑝∈𝑋^∗, and let 𝐾(𝑝) be the set defined by 𝐾𝑝∶=_{𝑥∗∈dom𝐻∗}epi𝐹−𝑝^∗+epi𝐺^∗+epi𝛿^∗_𝐶−𝑥^∗,𝐻^∗𝑥^∗. (25) Then, by (9), the following equality is clear: 𝐾𝑝=𝐾+−𝑝,0. (26) Hence, we have that 𝐾𝑝∩({0}×ℝ)=𝐾∩𝑝×ℝ+−𝑝,0. (27) Moreover, using (9), we conclude that epi𝐹−𝑝+𝐺−𝐻+𝛿_𝐶^∗∩({0}×ℝ)=epi𝐹+𝐺−𝐻+𝛿_𝐶^∗∩𝑝×ℝ+−𝑝,0. (28) Thus, the conclusion holds by definitions and the proof is complete.

Under the assumption that 𝐹,𝐺,𝐻arelscand𝐶isclosed, (29) the authors in [5] introduced the following closure condition epi𝐹^∗+epi𝐺^∗+epi𝛿^∗_𝐶isweak^∗-closed (30) to consider the strong duality and optimality conditions for DC optimization problem (14). The following proposition describes the relationships among the (CC), the (MRF), and (30).

Proposition 7. The following implication holds: (𝐶𝐶)⟹(𝑀𝑅𝐹). (31) Furthermore, if (29) holds, then the following implications hold: (30)⟹(𝐶𝐶)⟹(𝑀𝑅𝐹). (32)

Proof. Suppose that the (CC) holds. Let 𝑥₀∈𝐴, and let 𝑝∈𝜕(𝐹+𝐺−𝐻+𝛿_𝐶)(𝑥₀). Then, by (7), 𝑝,𝑝,𝑥₀−𝐹+𝐺−𝐻+𝛿_𝐶𝑥₀∈epi𝐹+𝐺−𝐻+𝛿_𝐶^∗=𝐾, (33) thanks to the (CC). Hence, for each 𝑥^∗∈𝜕𝐻(𝑥₀), 𝑝,𝑝,𝑥₀−𝐹+𝐺−𝐻+𝛿_𝐶𝑥₀∈epi𝐹^∗+epi𝐺^∗+epi𝛿^∗_𝐶−𝑥^∗,𝐻^∗𝑥^∗. (34) Let 𝑥^∗∈𝜕𝐻(𝑥₀). Then, there exist (𝑢^∗,𝑟₁)∈epi𝐹^∗, (𝑣^∗,𝑟₂)∈epi𝐺^∗, and (𝑤^∗,𝑟₃)∈epi𝛿^∗_𝐶 such that 𝑢^∗+𝑣^∗+𝑤^∗−𝑥^∗𝑟=𝑝,₁+𝑟₂+𝑟₃−𝐻^∗𝑥^∗=𝑝,𝑥₀−𝐹+𝐺−𝐻+𝛿_𝐶𝑥₀. (35) Below we show that 𝑢^∗∈𝜕𝐹(𝑥₀), 𝑣^∗∈𝜕𝐺(𝑥₀), and 𝑤^∗∈𝑁_𝐶(𝑥₀). To do this, note by the definition that 𝐹^∗𝑢^∗≤𝑟₁,𝐺^∗𝑣^∗≤𝑟₂,𝛿^∗_𝐶𝑤^∗≤𝑟₃. (36) Moreover, since 𝑥^∗∈𝜕𝐻(𝑥₀), it follows from (7) that 𝐻𝑥₀+𝐻^∗𝑥^∗=𝑥^∗,𝑥₀. (37) Hence, by (35)–(37) and the Young-Fenchel inequality (6), we have that 0≤𝐹^∗𝑢^∗+𝐹𝑥₀−𝑢^∗,𝑥₀≤𝐹^∗𝑢^∗+𝐹𝑥₀−𝑥^∗+𝑝−𝑣^∗−𝑤^∗,𝑥₀≤𝐹^∗𝑢^∗−𝑟₁+𝑣^∗,𝑥₀−𝐺𝑥₀−𝑟₂+𝑤^∗,𝑥₀−𝛿_𝐶𝑥₀−𝑟₃+𝐻𝑥₀+𝐻^∗𝑥^∗−𝑥^∗,𝑥₀≤𝐹^∗𝑢^∗−𝑟₁+𝐺^∗𝑣^∗−𝑟₂+𝛿^∗_𝐶𝑤^∗−𝑟₃≤0. (38) Thus, 𝐹^∗𝑢^∗+𝐹𝑥₀−𝑢^∗,𝑥₀=0. (39) This implies that 𝑢^∗∈𝜕𝐹(𝑥₀) by (7). Using the same argument, we have that 𝑣^∗∈𝜕𝐺(𝑥₀) and 𝑤^∗∈𝑁_𝐶(𝑥₀). Hence, 𝑝∈𝜕𝐹(𝑥₀)+𝜕𝐺(𝑥₀)+𝑁_𝐶(𝑥₀)−𝑥^∗, and 𝑝∈_{𝑥∗𝑥∈𝜕𝐻0}𝜕𝐹𝑥₀+𝜕𝐺𝑥₀+𝑁_𝐶𝑥₀−𝑥^∗ (40) since 𝑥^∗∈𝜕𝐻(𝑥₀) is arbitrary. Therefore, the (MRF) holds.
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 epi𝐹+𝐺+𝛿_𝐶−𝐻^∗=_{𝑥∗∈dom𝐻∗}epi𝐹+𝐺+𝛿_𝐶^∗−𝑥^∗,𝐻^∗𝑥^∗. (41) Note that 𝐹, 𝐺 are lsc and 𝐶 is closed; by Lemma 1(i), one has that epi𝐹+𝐺+𝛿_𝐶^∗=clepi𝐹^∗+epi𝐺^∗+epi𝛿^∗_𝐶. (42) This together with (30) and (41) implies that the (CC) holds. The proof is complete.

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+𝛿_𝐶)^∗ if and only if 𝑣(𝑃)≥−𝑟.(ii)(0,𝑟)∈𝐾 if and only if 𝑣(𝐷)≥−𝑟 and for each 𝑥^∗∈dom𝐻^∗, there exist 𝑢^∗∈dom𝐹^∗ and 𝑣^∗∈dom𝐺^∗ such that 𝐿𝑥^∗,𝑢^∗,𝑣^∗≥−𝑟. (43)

Proof. (i) By the definition of the conjugate function, one has 𝑣(𝑃)=−𝐹+𝐺−𝐻+𝛿_𝐶^∗(0). (44) Hence, the result is clear.
(ii) Let (0,𝑟)∈𝐾 and let 𝑥^∗∈dom𝐻^∗. Then (0,𝑟)∈epi𝐹^∗+epi𝐺^∗+epi𝛿^∗_𝐶−𝑥^∗,𝐻^∗𝑥^∗. (45) Thus, there exist (𝑢^∗,𝑟₁)∈epi𝐹^∗, (𝑣^∗,𝑟₂)∈epi𝐺^∗, and (𝑤^∗,𝑟₃)∈epi𝛿^∗_𝐶 such that 𝑢^∗+𝑣^∗+𝑤^∗−𝑥^∗𝑟=0,₁+𝑟₂+𝑟₃−𝐻^∗𝑥^∗=𝑟. (46) Since 𝐹^∗𝑢^∗≤𝑟₁,𝐺^∗𝑣^∗≤𝑟₂,𝛿^∗_𝐶𝑤^∗≤𝑟₃, (47) it follows from (46) that 𝐿𝑥^∗,𝑢^∗,𝑣^∗=𝐻^∗𝑥^∗−𝐹^∗𝑢^∗−𝐺^∗𝑣^∗−𝛿^∗_𝐶𝑥^∗−𝑢^∗−𝑣^∗≥𝐻^∗𝑥^∗−𝑟₁−𝑟₂−𝑟₃=−𝑟. (48) This together with the definition of 𝑣(𝐷) implies that 𝑣(𝐷)≥−𝑟 and 𝑢^∗, 𝑣^∗ satisfy (43).
Conversely, suppose that 𝑣(𝐷)≥−𝑟 and for each 𝑥^∗∈dom𝐻^∗, there exist 𝑢^∗∈dom𝐹^∗ and 𝑣^∗∈dom𝐺^∗ satisfying (43). Let 𝑥^∗∈dom𝐻^∗. Then, there exist 𝑢^∗∈dom𝐹^∗ and 𝑣^∗∈dom𝐺^∗ such that (43) holds. Then 𝛿^∗_𝐶𝑥^∗−𝑢^∗−𝑣^∗≤𝑟+𝐻^∗𝑥^∗−𝐹^∗𝑢^∗−𝐺^∗𝑣^∗. (49) This means that 𝑥^∗−𝑢^∗−𝑣^∗,𝑟+𝐻^∗𝑥^∗−𝐹^∗𝑢^∗−𝐺^∗𝑣^∗∈epi𝛿^∗_𝐶. (50) Therefore, (0,𝑟)=𝑢^∗,𝐹^∗𝑢^∗+𝑣^∗,𝐺^∗𝑣^∗−𝑥^∗,𝐻^∗𝑥^∗+𝑥^∗−𝑢^∗−𝑣^∗,𝑟+𝐻^∗𝑥^∗−𝐹^∗𝑢^∗−𝐺^∗𝑣^∗∈epi𝐹^∗+epi𝐺+epi𝛿^∗_𝐶−𝑥^∗,𝐻^∗𝑥^∗. (51) Noting that 𝑥^∗∈dom𝐻^∗ is arbitrary, we have that (0,𝑟)∈_{𝑥∗∈dom𝐻∗}epi𝐹^∗+epi𝐺+epi𝛿^∗_𝐶−𝑥^∗,𝐻^∗𝑥^∗=𝐾. (52) Thus, we complete the proof.

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,𝛿_𝐶} satisfies the (𝑊𝐶𝐶)₀.
(ii) The stable weak duality holds if and only if the family {𝐹,𝐺,𝐻,𝛿_𝐶} 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+𝛿_𝐶)^∗, thanks to Lemma 8(i). Hence, (19) holds; that is, the (WCC)₀ holds.
Conversely, suppose that the family {𝐹,𝐺,𝐻,𝛿_𝐶} 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^∗∈dom𝐻^∗, there exist 𝑢^∗∈dom𝐹^∗ and 𝑣^∗∈dom𝐺^∗ such that (43) holds. Hence, (0,𝑟)∈𝐾 by Lemma 8(ii), and (0,𝑟)∈epi(𝐹+𝐺−𝐻+𝛿_𝐶)^∗ 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,𝛿_𝐶} satisfies the (𝐶𝐶)₀.
(ii) The stable strong duality holds if and only if the family {𝐹,𝐺,𝐻,𝛿_𝐶} satisfies the (𝐶𝐶).

Proof. As before, it is sufficient to prove assertion (i). Suppose that the strong duality holds. Let 𝑥^∗∈dom𝐻^∗. Then, 𝑣(𝑃)=𝑣(𝐷) and there exist 𝑢^∗∈dom𝐹^∗ and 𝑣^∗∈dom𝐺^∗ 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: epi𝐹+𝐺−𝐻+𝛿_𝐶^∗∩({0}×ℝ)⊆𝐾∩({0}×ℝ). (53) To do this, let (0,𝑟)∈epi(𝐹+𝐺−𝐻+𝛿_𝐶)^∗. Then, by Lemma 8(i), we have 𝑣(𝑃)≥−𝑟. Hence, 𝑣(𝐷)=𝑣(𝑃)≥−𝑟 and 𝑥^∗∈dom𝐻^∗, 𝑢^∗∈dom𝐹^∗, 𝑣^∗∈dom𝐺^∗ satisfying (43). This together with Lemma 8(ii) implies that (0,𝑟)∈𝐾 as 𝑥^∗∈dom𝐻^∗ is arbitrary. Hence, (53) holds and so does the (CC)₀.
Conversely, suppose that the (CC)₀ holds. Then, the family {𝑓,𝑔,𝛿_𝐶;𝑓_𝑡,𝑔_𝑡∶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^∗∈dom𝐻^∗ there exist 𝑢^∗∈dom𝐹^∗ and 𝑣^∗∈dom𝐺^∗ 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+𝛿_𝐶)^∗, 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^∗∈dom𝐻^∗ there exist 𝑢^∗∈dom𝐹^∗ and 𝑣^∗∈dom𝐺^∗ 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,𝛿_𝐶} satisfies the (𝑀𝑅𝐹) at 𝑥₀. Then 𝜕𝐻𝑥₀⊆𝜕𝐹𝑥₀+𝜕𝐺𝑥₀+𝑁_𝐶𝑥₀. (54) Furthermore, if 𝜕𝐻(𝑥₀)≠∅, then for each 𝑥^∗∈𝜕𝐻(𝑥₀), there exist 𝑢^∗∈dom𝐹^∗ and 𝑣^∗∈dom𝐺^∗ such that 𝑣(𝑃)=𝐿𝑥^∗,𝑢^∗,𝑣^∗. (55)

Proof. Since 𝑥₀ is a solution of (𝑃), it follows that 0∈𝜕𝐹+𝐺−𝐻+𝛿_𝐶𝑥₀. (56) Then, by the (MRF) at 𝑥₀, one has that 0∈_{𝑥∗𝑥∈𝜕𝐻0}𝜕𝐹𝑥₀+𝜕𝐺𝑥₀+𝑁_𝐶𝑥₀−𝑥^∗, (57) which is equivalent to (54) holds.
Furthermore, assume that 𝑥^∗∈𝜕𝐻(𝑥₀). Then by (54), there exist 𝑢^∗∈𝜕𝐹(𝑥₀), 𝑣^∗∈𝜕𝐺(𝑥₀), and 𝑤^∗∈𝑁_𝐶(𝑥₀) such that 𝑥^∗=𝑢^∗+𝑣^∗+𝑤^∗. (58) Since 𝑥^∗∈𝜕𝐻(𝑥₀), 𝑢^∗∈𝜕𝐹(𝑥₀), 𝑣^∗∈𝜕𝐺(𝑥₀), and 𝑤^∗∈𝑁_𝐶(𝑥₀), it follows from (7) that 𝑥^∗,𝑥₀=𝐻^∗𝑥^∗+𝐻𝑥₀,𝑢^∗,𝑥₀=𝐹^∗𝑢^∗+𝐹𝑥₀,𝑣^∗,𝑥₀=𝐺^∗𝑣^∗+𝐺𝑥₀,𝑤^∗,𝑥₀=𝛿^∗_𝐶𝑢^∗. (59) Hence, 𝐿𝑥^∗,𝑢^∗,𝑣^∗=𝐹𝑥₀+𝐺𝑥₀−𝐻𝑥₀=𝑣(𝑃), (60) 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 (GEP) is defined as in Section 1. Let 𝑥∈𝐶 and consider the DC optimization problem 𝒫_𝑥𝑝(𝑥)∶=inf_𝑦∈𝐶𝑓𝑥,𝑦+𝜓𝑦. (61) Then, by the definitions of (GEP) and (𝒫_𝑥), a point 𝑥₀∈𝐶 is a solution of (GEP) if and only if 𝑥₀ is a solution of (𝒫_𝑥0) (cf. [5, Lemma 3.1]). Moreover, by the definition, we can find that 𝑝(𝑥)≤𝜓(𝑥) for each 𝑥∈𝐶, and, 𝑥₀ is a solution of (GEP) if and only if 𝑝(𝑥₀)=𝜓(𝑥₀). Hence, the problem of finding solutions of (GEP) can be reduced to the one of finding solutions of the following optimization problem: (𝒫)max_𝑥∈𝐶inf_𝑦∈𝐶𝑓_𝑥𝑦+𝜓𝑦−𝜓(𝑥). (62) Following [5], we defined the dual problem of (𝒫) by (𝒟)max_𝑥∈𝐶inf_{𝑥∗∈domℎ∗}max_{𝑢∗∈dom𝑓∗𝑥,𝑣∗∈dom𝑔∗}𝐿𝑥,𝑥^∗,𝑢^∗,𝑣^∗−𝜓(𝑥), (63) where 𝐿∶𝐶×𝑋^∗×𝑋^∗×𝑋^∗→ℝ is defined by 𝐿𝑥,𝑥^∗,𝑢^∗,𝑣^∗∶=ℎ^∗𝑥^∗−𝑓^∗_𝑥𝑢^∗−𝑔^∗𝑣^∗−𝛿^∗_𝐶𝑥^∗−𝑢^∗−𝑣^∗. (64)

Let 𝑣(𝒫) and 𝑣(𝒟) denote the optimal values of problems (𝒫) and (𝒟), respectively. Unlike [5], the weak duality (i.e., 𝑣(𝒟)≤𝑣(𝒫)) does not necessarily holds in general. Recall from Definition 4 that for each 𝑥∈𝐶, the family {𝑓_𝑥,𝑔,ℎ,𝛿_𝐶} satisfies the (WCC)₀ if _{𝑥∗∈domℎ∗}epi𝑓^∗_𝑥+epi𝑔^∗+epi𝛿^∗_𝐶−𝑥^∗,ℎ^∗𝑥^∗∩({0}×ℝ)⊆epi𝑓_𝑥+𝑔−ℎ+𝛿_𝐶^∗∩({0}×ℝ) (65) and it satisfies the (CC)₀ if _{𝑥∗∈domℎ∗}epi𝑓^∗_𝑥+epi𝑔^∗+epi𝛿^∗_𝐶−𝑥^∗,ℎ^∗𝑥^∗∩({0}×ℝ)=epi𝑓_𝑥+𝑔−ℎ+𝛿_𝐶^∗∩({0}×ℝ). (66) Then, we have the following theorem.

Theorem 13. (i) Suppose that for each 𝑥∈𝐶, the family {𝑓_𝑥,𝑔,ℎ,𝛿_𝐶} satisfies the (𝑊𝐶𝐶)₀. Then, 𝑣(𝒟)≤𝑣(𝒫).
(ii) Suppose that for each 𝑥∈𝐶, the family {𝑓_𝑥,𝑔,ℎ,𝛿_𝐶} satisfies the (𝐶𝐶)₀. Then, 𝑣(𝒟)=𝑣(𝒫).

Proof. (i) Since for each 𝑥∈𝐶, the family {𝑓_𝑥,𝑔,ℎ,𝛿_𝐶} satisfies the (WCC)₀, it follows from Theorem 9(i) that 𝑝(𝑥)≥inf_{𝑥∗∈domℎ∗}max_{𝑢∗∈dom𝑓∗𝑥,𝑣∗∈dom𝑔∗}𝐿𝑥,𝑥^∗,𝑢^∗,𝑣^∗. (67) Hence, by the definitions of 𝑣(𝒫) and 𝑣(𝒟), we see that 𝑣(𝒟)≤𝑣(𝒫).
(ii) Since for each 𝑥∈𝐶, the family {𝑓_𝑥,𝑔,ℎ,𝛿_𝐶} satisfies the (CC)₀, it follows from Theorem 10(i) that 𝑝(𝑥)=inf_{𝑥∗∈domℎ∗}max_{𝑢∗∈dom𝑓∗𝑥,𝑣∗∈dom𝑔∗}𝐿𝑥,𝑥^∗,𝑢^∗,𝑣^∗. (68) Thus, the result is seen to hold.

The following theorems establish the relationships between the solutions of (GEP) and those of (𝒟). First, we recall that a point 𝑥₀∈𝐶 is said to be a solution of (GEP) if 𝑓𝑥₀,𝑦+𝜓𝑦≥𝜓𝑥₀foreach𝑦∈𝐶, (69) and it is said to be a solution of the dual problem (𝒟) if for each 𝑥^∗∈domℎ^∗, there exist 𝑢^∗,𝑣^∗∈𝑋^∗ such that 𝐿𝑥₀,𝑥^∗,𝑢^∗,𝑣^∗=𝜓𝑥₀=𝑔𝑥₀−ℎ𝑥₀. (70) partially, if for each 𝑥^∗∈𝜕ℎ(𝑥₀), there exist 𝑢^∗,𝑣^∗∈𝑋^∗ such that (70) holds, then 𝑥₀ is said to be a weak solution of problem (𝒟).

Remark 14. (a) Obviously, 𝑥₀∈𝐶 is a solution of (GEP) if and only if 𝑥₀ is a solution of (𝒫_𝑥0).
(b) Let 𝑥₀∈𝐶. If ℎ is lsc at 𝑥₀, then for each 𝑥^∗∈domℎ^∗, there exist 𝑢^∗∈dom𝑓^∗_𝑥0 and 𝑣^∗∈dom𝑔^∗ such that 𝐿𝑥₀,𝑥^∗,𝑢^∗,𝑣^∗≤𝑔𝑥₀−ℎ𝑥₀. (71) Consequently, 𝑥₀ is a solution of the dual problem (𝒟) if and only if for each 𝑥^∗∈domℎ^∗, there exist 𝑢^∗,𝑣^∗∈𝑋^∗ such that 𝐿𝑥₀,𝑥^∗,𝑢^∗,𝑣^∗≥𝑔𝑥₀−ℎ𝑥₀. (72) In fact, let 𝑥^∗∈domℎ^∗, 𝑢^∗∈dom𝑓^∗_𝑥0, and 𝑣^∗∈dom𝑔^∗. Then by the Young-Fenchel inequality (6), it is easy to see that 𝐿𝑥₀,𝑥^∗,𝑢^∗,𝑣^∗≤ℎ^∗𝑥^∗−𝑥^∗,𝑥₀+𝑔𝑥₀+𝑓_𝑥0𝑥₀≤inf_{𝑥∗∈domℎ∗}ℎ^∗𝑥^∗−𝑥^∗,𝑥₀+𝑔𝑥₀+𝑓_𝑥0𝑥₀=−(clℎ)𝑥₀+𝑔𝑥₀=𝑔𝑥₀−ℎ𝑥₀. (73) Hence, (71) holds. However, (71) does not necessarily hold in general as will be showen in the following example.

Example 15. Let 𝑋∶=ℝ and 𝐶∶=(−∞,0]. Let 𝑓∶ℝ×ℝ→ℝ and 𝑔,ℎ∶ℝ→ℝ be defined by 𝑓∶=0, 𝑔∶=𝛿_[0,+∞), and ℎ(𝑥)∶=0𝑥<0,1𝑥=0,+∞𝑥>0. (74) Let 𝑥₀=0. Then, 𝑔(𝑥₀)−ℎ(𝑥₀)=−1, and ℎ is not lsc at 𝑥₀. Clearly, 𝑓^∗_𝑥0=𝛿_{0}, 𝑔^∗=𝛿_(−∞,0], and ℎ^∗=𝛿^∗_𝐶=𝛿_[0,+∞). Then, dom𝑓^∗_𝑥0={0}, dom𝑔^∗=(−∞,0], and domℎ^∗=dom𝛿^∗_𝐶=[0,+∞). Let 𝑥^∗=0. Then, for each 𝑢^∗∈dom𝑓^∗_𝑥0 and 𝑣^∗∈dom𝑔^∗, one has that 𝐿𝑥₀,𝑥^∗,𝑢^∗,𝑣^∗=−𝑔^∗𝑣^∗−𝛿^∗_𝐶−𝑣^∗=0; (75) 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 epi𝑓^∗_𝑥+epi𝑔^∗+epi𝛿^∗_𝐶isweak^∗-closed. (76) Thus, by Proposition 7, our Theorem 16 improves the corresponding result in [5, Theorem 4.2].

Theorem 16. Let 𝑥₀∈𝐶 such that 𝜕𝐻(𝑥₀)≠∅. Suppose that the family {𝑓_𝑥0,𝑔,ℎ,𝛿_𝐶} 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 {𝑓_𝑥0,𝑔,ℎ,𝛿_𝐶} 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 (𝒫_𝑥0). Hence, 𝑣𝒫_𝑥0=𝑓_𝑥0𝑥₀+𝑔𝑥₀−ℎ𝑥₀=𝑔𝑥₀−ℎ𝑥₀. (77) Since the family {𝑓_𝑥0,𝑔,ℎ,𝛿_𝐶} satisfies the (CC)₀, it follows from Theorem 10 that 𝑣𝒫_𝑥0=inf_{𝑥∗∈domℎ∗}max_{𝑢∗∈dom𝑓∗𝑥0,𝑣∗∈dom𝑔∗}𝐿𝑥₀,𝑥^∗,𝑢^∗,𝑣^∗, (78) which implies that for each 𝑥^∗∈domℎ^∗, there exist 𝑢^∗∈dom𝑓^∗_𝑥0 and 𝑣^∗∈dom𝑔^∗ such that 𝐿𝑥₀,𝑥^∗,𝑢^∗,𝑣^∗≥𝑣𝑃_𝑥0. (79) Moreover, since ℎ is lsc at 𝑥₀, it follows from Remark 14(b) that 𝐿𝑥₀,𝑥^∗,𝑢^∗,𝑣^∗≤𝑔𝑥₀−ℎ𝑥₀=𝑣𝑃_𝑥0. (80) Hence, (70) holds. This means that 𝑥₀ is a solution of the dual problem (𝒟).
Conversely, suppose that 𝑥₀ is a solution of the dual problem (𝒟). Let 𝑥^∗∈domℎ^∗. Then, there exist 𝑢^∗∈dom𝑓^∗_𝑥0 and 𝑣^∗∈dom𝑔^∗ such that (70) holds. Note by the Young-Fenchel inequality (6) that for each 𝑥∈𝐶, 𝐿𝑥₀,𝑥^∗,𝑢^∗,𝑣^∗≤ℎ^∗𝑥^∗−𝑥^∗,𝑥+𝑔(𝑥)+𝑓_𝑥0(𝑥). (81) Then by (70), one has that for each 𝑥∈𝐶, 𝑔𝑥₀−ℎ𝑥₀≤ℎ^∗𝑥^∗−𝑥^∗,𝑥+𝑔(𝑥)+𝑓_𝑥0(𝑥). (82) Since the above inequality holds for each 𝑥^∗∈domℎ^∗, it follows that 𝑔𝑥₀−ℎ𝑥₀≤inf_{𝑥∗∈domℎ∗}ℎ^∗𝑥^∗−𝑥^∗(𝑥),𝑦+𝑔+𝑓_𝑥0(𝑥)=−(clℎ)(𝑥)+𝑔(𝑥)+𝑓_𝑥0(𝑥). (83) Consequently, by the lower semicontinuity of ℎ, one has that for each 𝑥∈𝐶, 𝑔𝑥₀−ℎ𝑥₀≤𝑓_𝑥0(𝑥)+𝑔(𝑥)−ℎ(𝑥), (84) and hence 𝑔(𝑥₀)−ℎ(𝑥₀)≤𝑝(𝑥₀). This implies that 𝑔(𝑥₀)−ℎ(𝑥₀)=𝑝(𝑥₀) since 𝑔(𝑥₀)−ℎ(𝑥₀)≥𝑝(𝑥₀) holds automatically. Thus, 𝑥₀ is a solution of (GEP).

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.

Below we will give a upper estimate for the Fréchet subdifferential of the function 𝑝 defined in (61). We first recall from [18] or [19, page 90] that the Fréchet subdifferential of 𝜙 at a point 𝑥₀∈dom𝜙 with |𝜙(𝑥₀)|<∞ is defined by 𝜕𝜙𝑥₀∶=𝑥^∗∈𝑋^∗∶liminf_𝑥→𝑥0𝜙(𝑥)−𝜙𝑥₀−𝑥^∗,𝑥−𝑥₀𝑥−𝑥₀≥0, (85) where 𝜙∶𝑋→ℝ∪{±∞} is an extended real-valued function.

Theorem 19. Let 𝑥₀∈𝐶 be a solution of (𝐺𝐸𝑃). Suppose that the family {𝑔,𝛿_𝐶} satisfies the (𝑀𝑅𝐹) at 𝑥₀; that is, 𝜕𝑔+𝛿_𝐶𝑥₀=𝜕𝑔𝑥₀+𝑁_𝐶𝑥₀. (86) Then for each 𝛾>0, one has that 𝑝𝑥₀⊆_{𝑥∗𝑥∈𝜕ℎ0}𝜕𝑔𝑥₀+𝑁_𝐶𝑥₀−𝑥^∗+𝛾𝔹^∗, (87) where 𝔹^∗ denotes the unit ball in 𝑋^∗.

Proof. Let 𝑥₀∈𝐶. If 𝜕𝐻(𝑥₀)=∅, then (87) holds trivially. Below let 𝑢^∗∈𝜕𝑝(𝑥₀) 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 𝑝(𝑥)−𝑝𝑥₀−𝑢^∗,𝑥−𝑥₀+𝛾𝑥−𝑥₀≥0foreach𝑥∈𝑥₀+𝜂𝔹, (88) where 𝔹 denotes the unit ball in 𝑋. Since 𝑥₀ is a solution of (GEP), it follows that 𝑝(𝑥₀)=𝑔(𝑥₀)−ℎ(𝑥₀). Note that 𝑝(𝑥)≤𝑔(𝑥)−ℎ(𝑥) holds for each 𝑥∈𝐶. Then 0≤𝑔(𝑥)−ℎ(𝑥)−𝑔𝑥₀+ℎ𝑥₀−𝑢^∗,𝑥−𝑥₀+𝛾𝑥−𝑥₀≤𝑔(𝑥)−𝑔𝑥₀−𝑢^∗+𝑥^∗,𝑥−𝑥₀+𝛾𝑥−𝑥₀, (89) where the last inequality holds because 𝑥^∗∈𝜕ℎ(𝑥₀). Define 𝜙∶𝑋→ℝ by 𝜙(𝑥)∶=𝑔(𝑥)−𝑔𝑥₀−𝑢^∗+𝑥^∗,𝑥−𝑥₀+𝛾𝑥−𝑥₀. (90) Then, 𝜙 is a proper convex function on 𝑋, and 𝑥₀ is a solution to the following convex programming: (𝑥),𝑥Min𝜙s.t.𝑥∈𝐶∩₀+𝜂𝔹. (91) Hence, 0∈𝜕(𝜙+𝛿_{𝐶∩(𝑥0+𝜂𝔹)})(𝑥₀), and 0∈𝜕𝑔+𝛿_{𝐶∩(𝑥0+𝜂𝔹)}𝑥₀−𝑢^∗+𝑥^∗+𝛾𝔹^∗. (92) Note that 𝑥₀ is an interior point of the set 𝑥₀+𝜂𝔹; then the function 𝛿_{𝑥0+𝜂𝔹} is continuous at 𝑥₀. Hence, 𝜕𝑔+𝛿_{𝐶∩(𝑥0+𝜂𝔹)}𝑥₀=𝜕𝑔+𝛿_𝐶𝑥₀. (93) This together with (86) and (92) implies that 0∈𝜕𝑔𝑥₀+𝑁_𝐶𝑥₀−𝑢^∗+𝑥^∗+𝛾𝔹^∗; (94) that is, 𝑢^∗∈𝜕𝑔𝑥₀+𝑁_𝐶𝑥₀−𝑥^∗+𝛾𝔹^∗, (95) Which implies that (87) holds by the arbitrariness of 𝑥^∗. The proof is complete.

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).