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Abstract and Applied Analysis
Volume 2011, Article ID 432963, 16 pages
http://dx.doi.org/10.1155/2011/432963
Research Article

Second-Order Optimality Conditions for Set-Valued Optimization Problems Under Benson Proper Efficiency

1College of Sciences, Chongqing Jiaotong University, Chongqing 400074, China
2Research Institute of Information and System Computation Science, Beifang University of Nationalities, Yinchuan 750021, China

Received 4 October 2011; Accepted 18 November 2011

Academic Editor: D. Anderson

Copyright © 2011 Qilin Wang and Guolin Yu. 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.

Abstract

Some new properties are obtained for generalized second-order contingent (adjacent) epiderivatives of set-valued maps. By employing the generalized second-order adjacent epiderivatives, necessary and sufficient conditions of Benson proper efficient solutions are given for set-valued optimization problems. The results obtained improve the corresponding results in the literature.

1. Introduction

The investigation of the optimality conditions is one of the most attractive topics of optimization theory since many optimization problems encountered in economics, engineering, and other fields involve vector-valued maps (or set-valued maps) as constraints and objectives (see [13]). With the concept of contingent derivative for a set-valued map (see [3]), Corley [4] investigated optimality conditions for general set-valued optimization problems. But it turns out that the optimality conditions do not coincide under standard assumptions. Jahn and Rauh [5] introduced the contingent epiderivative of a set-valued map and then obtained unified necessary and sufficient optimality conditions. The essential differences between the definitions of the contingent derivative and the contingent epiderivative are that the graph is replaced by the epigraph and the derivative is single-valued. But the existence of the contingent epiderivative of a set-valued map in a general setting is an open question. To overcome the difficulty, Chen and Jahn [6] introduced a generalized contingent epiderivative of a set-valued map which is a set-valued map. They proved that the generalized contingent epiderivative exists under standard assumptions and obtained a unified necessary and sufficient condition in terms of the generalized contingent epiderivative. As to other concepts of epiderivatives for set-valued maps and applications to optimality conditions, one can refer to [711].

Since higher-order tangent sets introduced in [3], in general, are not cones and convex sets, there are some difficulties in studying higher-order optimality conditions and duality for general set-valued optimization problems. Until now, there are only a few papers to deal with higher-order optimality conditions and duality of set-valued optimization problems by virtue of the higher-order derivatives or epiderivatives introduced by the higher-order tangent sets. Jahn et al. [12] introduced second-order contingent epiderivative and generalized contingent epiderivative for a set-valued map and obtained some second-order optimality conditions based on these concepts. In [13], Li et al. studied some properties of higher-order tangent sets and higher-order derivatives introduced in [3] and then obtained higher-order Fritz John type necessary and sufficient optimality conditions for set-valued optimization problems in terms of the higher-order derivatives. By using these concepts, they also discussed higher-order Mond-Weir duality for a set-valued optimization problem in [14]. In general, since the epigraph of a set-valued map has nicer properties than the graph of a set-valued map, it is advantageous to employ the epiderivatives in set-valued optimization. Li and Chen [15] introduced the definitions of higher-order generalized contingent epiderivative and higher-order generalized adjacent epiderivative and obtained higher-order optimality conditions for Henig properly efficient solutions of a set-valued optimization problem with constraints based on the higher-order generalized adjacent epiderivative and contingent epiderivative. Chen et al. [16] introduced the concepts of higher-order weak contingent epiderivative and higher-order weak adjacent epiderivative for set-valued maps and obtained higher order Mond-Weir type duality, higher-order Wolfe type duality, and higher-order Kuhn-Tucker type optimality conditions to a constrained set-valued optimization problem based on the higher-order weak adjacent (contingent) epiderivatives and Henig efficiency. Since the existence of the higher-order contingent (adjacent) derivative and the higher-order generalized contingent (adjacent) epiderivative of a set-valued map in a general setting may not exist, Wang and Li [17] introduced the generalized higher-order contingent (adjacent) epiderivative of a set-valued map and obtained the optimality conditions for Henig efficient solutions to set-valued optimization problems. Wang et al. [18] introduced the generalized higher-order contingent (adjacent) derivatives of set-valued maps, and necessary and sufficient optimality conditions are obtained for weakly efficient solutions of set-valued optimization problems under no convexity assumptions.

To the best of our knowledge, by virtue of the second-order derivatives or epiderivatives introduced by the higher-order tangent sets, the second-order optimality conditions have not been studied for Benson proper efficient solutions of set-valued optimization problems until now. This paper aims to investigate the second-order optimality conditions for Benson proper efficient solutions of set-valued optimization problems by virtue of the generalized second-order epiderivatives.

The rest of the paper is organized as follows. In Section 2, we collect some of the concepts and introduce two kinds of set-valued optimization problem models. In Section 3, we recall the concepts and properties of generalized second-order contingent (adjacent) epiderivatives of set-valued maps and discuss some new properties of them. In Section 4, we establish second-order necessary and sufficient conditions for Benson proper efficient solutions to a set-valued optimization problem, whose constraint set is determined by a fixed set. In Section 5, we establish second-order Kuhn-Tuck type optimality conditions for Benson proper efficient solutions to a set-valued optimization problem, whose constraint set is determined by a set-valued map.

2. Preliminaries

Throughout this paper, let 𝑋,𝑌, and 𝑍 be three real normed spaces, where the spaces 𝑌 and 𝑍 are partially ordered by nontrivial closed convex pointed cones 𝐶𝑌 and 𝐷𝑍 with int𝐷, respectively. Denote by 𝑌 the topological dual space of 𝑌 and by 0𝑌 the zero element in 𝑌. The dual cone of 𝐶 is defined as𝐶+=𝑓𝑌𝑓(𝑐)0,𝑐𝐶.(2.1) The quasi-interior 𝐶+𝑖 of 𝐶+ is the set𝐶+𝑖=𝑓𝑌0𝑓(𝑐)>0,𝑐𝐶𝑌.(2.2) Recall that a base of a cone 𝐶 is a convex subset 𝐵 of 𝐶 such that0𝑌cl𝐵,𝐶=cone𝐵.(2.3) Of course, 𝐶 is pointed whenever 𝐶 has a base. Furthermore, if 𝐶 is a nonempty closed convex pointed cone in 𝑌, then 𝐶+𝑖 if and only if 𝐶 has a base.

Let 𝐹𝐸2𝑌 and 𝐺𝐸2𝑍 be set-valued maps. The effective domain, the graph, and the epigraph of 𝐹 are defined by dom(𝐹)={𝑥𝐸𝐹(𝑥)},gph(𝐹)={(𝑥,𝑦)𝐸×𝑌𝑥𝐸,𝑦𝐹(𝑥)} and epi(𝐹)={(𝑥,𝑦)𝐸×𝑌𝑥𝐸𝑦𝐹(𝑥)+𝐶}, respectively. Denote 𝐹(𝐸)=𝑥𝐸𝐹(𝑥). Let (𝑥0,𝑦0)gph(𝐹),𝑧0𝐺(𝑥0)(𝐷).

Definition 2.1 (see [19]). 𝐹 is called 𝐶-convex on a nonempty convex subset 𝐸, if, for any 𝑥1,𝑥2𝐸 and 𝜆(0,1), such that 𝜆𝐹(𝑥1)+(1𝜆)𝐹(𝑥2)𝐹(𝜆𝑥1+(1𝜆)𝑥2)+𝐶.

Definition 2.2 (see [20]). Let 𝐹𝐸2𝑌 be a set-valued map, 𝑥0𝐸,(𝑥0,𝑦0)gph(𝐹). 𝐹 is said to be generalized 𝐶-convex at (𝑥0,𝑦0) on 𝐸, if cone(epi𝐹{(𝑥0,𝑦0)}) is convex.

Remark 2.3 (see [20]). If 𝐹 is 𝐶-convex on convex set 𝐸, then 𝐹 is generalized 𝐶-convex at (𝑥0,𝑦0)gph(𝐹) on 𝐸. But the converse may not hold.

Definition 2.4 (see [6]). Let 𝑀 be a subset of 𝑌. 𝑦0𝑀 is said to be a Benson proper efficient point of 𝑀 if (𝐶)clcone(𝑀+𝐶{𝑦0})={0𝑌}. The set of all Benson proper efficient points of 𝑀 is denoted by 𝑃min[𝑀,𝐶].

Definition 2.5 (see [21]). Let 𝑌 be a real normed space, 𝐵 be a base of 𝐶, and 𝑀𝑌 be a nonempty subset. 𝑦𝑀 is said to be a Henig efficient point of 𝑀, written as 𝑦𝐻𝐸(𝑀,𝐵), if for some 𝜖(0,𝛿), (𝑀{𝑦})int(cone(𝜖𝑈+𝐵))=,(2.4) where 𝛿=inf{𝑏𝑏𝐵} and 𝑈 is the closed unit ball of 𝑌.

Lemma 2.6 (see [22]). Let 𝑃 and 𝐶 be two closed convex cones in a locally convex vector space, and let 𝐶 be pointed and have a compact base. If 𝑃(𝐶)={0𝑌}, then there exists 𝐶+𝑖 such that 𝑃+.

Definition 2.7 (see [6]). A set-valued map 𝐻𝑋2𝑌 is said to be (i)strictly positive homogeneous if 𝐻(𝛼𝑥)=𝛼𝐻(𝑥),𝛼>0,𝑥𝑋,(2.5)(ii)subadditive if 𝐻𝑥1𝑥+𝐻2𝑥𝐻1+𝑥2+𝐶.(2.6)

In this paper, consider the following set-valued optimization problem:min𝐹(𝑥),s.t.𝑥𝐸.(OP)

that is, to find a pair (𝑥0,𝑦0) with 𝑥0𝐾 and 𝑦0𝐹(𝑥0) is called a Benson proper efficient element of problem (OP) if 𝑦0𝑃min[𝐹(𝐸),𝐶]. We also consider the following constraint set-valued optimization problem:min𝐹(𝑥),s.t.𝐺(𝑥)(𝐷),𝑥𝐸.(SOP)

Set 𝐾={𝑥𝐸𝐺(𝑥)(𝐷)}. A pair (𝑥0,𝑦0) with 𝑥0𝐾 and 𝑦0𝐹(𝑥0) is called a Benson proper efficient element of problem (SOP) if 𝑦0𝑃min[𝐹(𝐾),𝐶].

3. Properties of Generalized Second-Order Contingent (Adjacent) Epiderivatives

In this section, we discuss some new properties of generalized second-order contingent and adjacent epiderivatives. Moreover, we give an example to show these properties. Let 𝐹 be a set-valued map from 𝐸𝑋 to 𝑌.

Definition 3.1 (see [17]). Let 𝐹 be a set-valued map from 𝐸𝑋 to 𝑌, (𝑥0,𝑦0)gph(𝐻),𝑢𝑋,𝑣𝑌.(i)The generalized second-order contingent epiderivative 𝐺-𝐷(2)𝐹(𝑥0,𝑦0,𝑢,𝑣) of 𝐹 at (𝑥0,𝑦0) for vectors (𝑢,𝑣) is the set-valued map from 𝑋 to 𝑌 defined by gph𝐺-𝐷(2)𝐹𝑥0,𝑦0,𝑢,𝑣=𝐺-𝑇(2)epi(𝐹)𝑥0,𝑦0.,𝑢,𝑣(3.1)(ii)The generalized second-order adjacent epiderivative 𝐺-𝐷(2)𝐹(𝑥0,𝑦0,𝑢,𝑣) of 𝐹 at (𝑥0,𝑦0) for vectors (𝑢,𝑣) is the set-valued map from 𝑋 to 𝑌 defined by gph𝐺-𝐷(2)𝐹𝑥0,𝑦0,𝑢,𝑣=𝐺-𝑇(2)epi(𝐹)𝑥0,𝑦0,𝑢,𝑣.(3.2)

Remark 3.2. If (𝑢,𝑣)=(0𝑋,0𝑌), then 𝐺-𝐷(2)𝐹𝑥0,𝑦0𝑥,𝑢,𝑣=𝐺-𝐷𝐹0,𝑦0,𝐺-𝐷(2)𝐹𝑥0,𝑦0,𝑢,𝑣=𝐺-𝐷𝐹𝑥0,𝑦0.(3.3)

From [17, Proposition 3.9 and Corollary 3.11], we have the following result.

Proposition 3.3. Let 𝑥,𝑥0𝐸, 𝑦0𝐹(𝑥0), (𝑢,𝑣){0𝑋}×𝐶. Then, (i)𝐹(𝑥)+𝐶{𝑦0}𝐺-𝐷(2)𝐹(𝑥0,𝑦0,𝑢,𝑣)(𝑥𝑥0); (ii)𝐹(𝑥)+𝐶{𝑦0}𝐺-𝐷(2)𝐹(𝑥0,𝑦0,𝑢,𝑣)(𝑥x0).

Proposition 3.4. Let 𝐸 be a nonempty subset of 𝑋, 𝑥0𝐸, 𝑦0𝐹(𝑥0). Let 𝐹 be generalized 𝐶-convex at (𝑥0,𝑦0) on 𝐸, 𝑢𝐸,𝑣𝐹(𝑢)+𝐶. Then, 𝑦𝐹(𝑥)+𝐶0𝐺-𝐷(2)𝐹𝑥0,𝑦0,𝑢𝑥0,𝑣y0𝑥𝑥0,forany𝑥𝐸.(3.4)

Proof. Take a arbitrary sequence {𝑛} with 𝑛0+, any 𝑐𝐶, 𝑥𝐸 and 𝑦𝐹(𝑥). Since 𝐻 is generalized 𝐶-convex at (𝑥0,𝑦0) on 𝐸, cone(epi(𝐹){(𝑥0,𝑦0)}) is convex, and then 𝑛𝑢𝑥0,𝑣𝑦0𝑥coneepi𝐹0,𝑦0.(3.5) It follows from 𝑛>0, and cone(epi𝐹{(𝑥0,𝑦0)}) is a convex cone that 𝑥𝑛,𝑦𝑛=𝑛𝑢𝑥0,𝑣𝑦0+2𝑛𝑥𝑥0,𝑦+𝑐𝑦0𝑥coneepi𝐹0,𝑦0.(3.6) Then, 𝑥𝑥0,𝑦+𝑐𝑦0=𝑥𝑛,𝑦𝑛𝑛𝑢𝑥0,𝑣𝑦02𝑛,(3.7) Thus, 𝑦𝐹(𝑥)+𝐶0𝐺-𝐷(2)𝐹𝑥0,𝑦0,𝑢𝑥0,𝑣𝑦0𝑥𝑥0,(3.8) and the proof of the proposition is complete.

Corollary 3.5. Let 𝐸 be a nonempty subset of 𝑋, 𝑥0𝐸, 𝑦0𝐹(𝑥0). If 𝐹 is generalized 𝐶-convex at (𝑥0,𝑦0) on 𝐸, 𝑢𝐸,𝑣𝐹(𝑢)+𝐶, then 𝑦𝐹(𝑥)+𝐶0𝐺-𝐷(2)𝐹𝑥0,𝑦0,𝑢𝑥0,𝑣𝑦0𝑥𝑥0,forany𝑥𝐸.(3.9)

Proposition 3.6. Let (𝑥0,𝑦0)gph(𝐹),(𝑢,𝑣)𝑋×𝑌. Then, (i)𝐺-𝐷(2)𝐹(𝑥0,𝑦0,𝑢,𝑣) is strictly positive homogeneous. Moreover, if 𝐹 is generalized 𝐶-convex at (𝑥0,𝑦0) on 𝐸, then (ii)𝐺-𝐷(2)𝐹(𝑥0,𝑦0,𝑢,𝑣) is subadditive.

Proof. (i) Since 𝐺-𝑇(2)epi(𝐹)𝑥0,𝑦0,𝑢,𝑣=𝑇(2)cone(epi(𝐹){(𝑥0,𝑦0)})0𝑋,0𝑌,𝑢,𝑣=𝑇cone(epi(𝐹){(𝑥0,𝑦0)})(𝑢,𝑣),(3.10)𝐺-𝐷(2)𝐹(𝑥0,𝑦0,𝑢,𝑣) is strictly positive homogeneous.
(ii) Let 𝑥1,𝑥2𝑋, 𝑦1𝐺-𝐷(2)𝐹(𝑥0,𝑦0,𝑢,𝑣)(𝑥1), 𝑦2𝐺-𝐷(2)𝐹(𝑥0,𝑦0,𝑢,𝑣)(𝑥2). Then one has (𝑥1,𝑦1),(𝑥2,𝑦2)𝐺-𝑇(2)epi(𝐹)(𝑥0,𝑦0,𝑢,𝑣). Since 𝐹 is generalized 𝐶-convex at (𝑥0,𝑦0) on 𝐸, cone(epi𝐹{(𝑥0,𝑦0)}) is convex, and then it follows from the proof of [13, Proposition 3.2], that 𝐺-𝑇(2)epi(𝐹)(𝑥0,𝑦0,𝑢,𝑣) is convex. Thus, we have 12𝑥1,𝑦1+12𝑥2,𝑦2=12𝑥1+𝑥2,𝑦1+𝑦2𝐺-𝑇(2)epi(𝐹)𝑥0,𝑦0,,𝑢,𝑣(3.11) and then it follows from (i) that 𝑦1+𝑦2𝐺-𝐷(2)𝐹𝑥0,𝑦0𝑥,𝑢,𝑣1+𝑥2,(3.12) which implies 𝐺-𝐷(2)𝐹𝑥0,𝑦0𝑥,𝑢,𝑣1+𝐺-𝐷(2)𝐹𝑥0,𝑦0𝑥,𝑢,𝑣2𝐺-𝐷(2)𝐹𝑥0,𝑦0𝑥,𝑢,𝑣1+𝑥2+𝐶.(3.13) The proof of the proposition is complete.

For the sake of comparison, we recall some notions in [13, 15, 16].

Definition 3.7 (see [13]). The second-order adjacent derivative 𝐷(2)𝐹(𝑥0,𝑦0,𝑢,𝑣) of 𝐹 at (𝑥0,𝑦0) for vector (𝑢,𝑣) is the set-valued map from 𝑋 to 𝑌 defined by 𝐷gph(2)𝐹𝑥0,𝑦0,𝑢,𝑣=𝑇(2)Graph(𝐹)𝑥0,𝑦0,𝑢,𝑣.(3.14)

Definition 3.8 (see [15]). The second-order generalized adjacent epiderivative 𝐷𝑔(2)𝐹(𝑥0,𝑦0,𝑢,𝑣) of 𝐹 at (𝑥0,𝑦0) for vectors (𝑢,𝑣) is the set-valued map from 𝑋 to 𝑌 defined by 𝐷𝑔(2)𝐹𝑥0,𝑦0,𝑢,𝑣(𝑥)=Min𝐶𝑦𝑌(𝑥,𝑦)𝑇(2)epi(𝐹)𝑥0,𝑦0,𝑢,𝑣.(3.15)

Definition 3.9 (see [16]). The second-order weak adjacent epiderivative 𝐷𝑔(2)𝐹(𝑥0,𝑦0,𝑢,𝑣) of 𝐹 at (𝑥0,𝑦0) for vector (𝑢,𝑣) is the set-valued map from 𝑋 to 𝑌 defined by 𝐷𝑤(2)𝐹𝑥0,𝑦0,𝑢,𝑣(𝑥)=𝑊Min𝐶𝑦𝑌(𝑥,𝑦)𝑇(2)epi(𝐹)𝑥0,𝑦0,𝑢,𝑣.(3.16)

Remark 3.10. Notice that in Proposition 3.6, we establish a special property of 𝐺-𝐷(2)𝐹(𝑥0,𝑦0,𝑢,𝑣), which is similar to the corresponding property of the generalized contingent epiderivative in [6, Theorem 1]. But we cannot obtain the similar property for 𝐷(2)𝐹(𝑥0,𝑦0,𝑢,𝑣), 𝐷𝑔(2)𝐹(𝑥0,𝑦0,𝑢,𝑣), and 𝐷𝑤(2)𝐹(𝑥0,𝑦0,𝑢,𝑣), even though 𝐹 is a 𝐶-convex map. The following example explains the case.

Example 3.11. Let 𝑋=𝑅,𝑌=𝑅2,𝐶=𝑅2+, 𝐹(𝑥)={(𝑦1,𝑦2)𝑅2𝑦1𝑥,𝑦2𝑥2},𝑥𝑋. Take (𝑥0,𝑦0)=(0,(0,0))gph(𝐹) and (𝑢,𝑣)=(1,(1,0)). Then, 𝑇(2)gph(𝐹)𝑥0,𝑦0,𝑢,𝑣=𝑇(2)epi(𝐹)𝑥0,𝑦0=,𝑢,𝑣𝑥𝑅[[({𝑥}×𝑥,+)×1,+)),𝐺-𝑇(2)epi(𝐹)𝑥0,𝑦0=,𝑢,𝑣𝑥𝑅{[𝑥}×𝑥,+)×𝑅+.(3.17) Therefore, for any 𝑥𝑋, one has 𝐷(2)𝐹𝑥0,𝑦0[[𝐷,𝑢,𝑣(𝑥)=𝑥,+)×1,+),𝑔(2)𝐹𝑥0,𝑦0𝐷,𝑢,𝑣(𝑥)={(𝑥,1)},𝑤(2)𝐹𝑥0,𝑦0,𝑢,𝑣(𝑥)=𝑥,𝑦2𝑦2𝑦11,1𝑦1,𝑥𝐺-𝐷(2)𝐹𝑥0,𝑦0[,𝑢,𝑣(𝑥)=𝑥,+)×𝑅+.(3.18) Naturally, 𝐷(2)𝐹(𝑥0,𝑦0,𝑢,𝑣), 𝐷𝑔(2)𝐹(𝑥0,𝑦0,𝑢,𝑣), and 𝐷𝑤(2)𝐹(𝑥0,𝑦0,𝑢,𝑣) are not strictly positive homogeneous, but 𝐺-𝐷(2)𝐹(𝑥0,𝑦0,𝑢,𝑣) is strictly positive homogeneous here.

It follows from [13, Corollary 3.1], and the proof of Proposition 3.6 that the following result holds.

Corollary 3.12. Let (𝑥0,𝑦0)gph(𝐹),(𝑢,𝑣)𝑋×𝑌. Then (i)𝐺-𝐷(2)𝐹(𝑥0,𝑦0,𝑢,𝑣) is strictly positive homogeneous. Moreover, if 𝐹 is generalized 𝐶-convex at (𝑥0,𝑦0) on 𝐸, and (𝑢,𝑣)cone(epi(𝐻){(𝑥0,𝑦0)}), then (ii)𝐺-𝐷(2)𝐹(𝑥0,𝑦0,𝑢,𝑣) is subadditive.

Remark 3.13. Under the assumption of cone-convex maps, Chen and Jahn [6, Theorem 1] obtained the subadditivity of 𝐷𝑔𝐹(𝑥0,𝑦0). However, it is under the assumption of generalized cone-convex maps that we obtain the subadditivity of 𝐺-𝐷(2)𝐹(𝑥0,𝑦0,𝑢,𝑣) in Corollary 3.12.

By Propositions 3.33.6 and Corollary 3.12, we obtain the following results.

Corollary 3.14. Let (𝑥0,𝑦0)gph(𝐹),(𝑢,𝑣)𝑋×𝑌. Then, 0𝑋,0𝑌gph𝐺-𝐷(2)𝐹𝑥0,𝑦0,0,𝑢,𝑣𝑋,0𝑌gph𝐺-𝐷(2)𝐹𝑥0,𝑦0.,𝑢,𝑣(3.19)

Corollary 3.15. Let (𝑥0,𝑦0)gph(𝐹) and (𝑢,𝑣)𝑋×𝑌. If 𝐹 is generalized 𝐶-convex at (𝑥0,𝑦0) on 𝐸, then(i)gph(𝐺-𝐷(2)𝐹(𝑥0,𝑦0,𝑢,𝑣)) is a closed convex cone. Moreover, if (𝑢,𝑣)𝑐𝑜𝑛𝑒(𝑒𝑝𝑖(𝐻){(𝑥0,𝑦0)}), then (ii)gph(𝐺-𝐷(2)𝐹(𝑥0,𝑦0,𝑢,𝑣)) is a closed convex cone.

4. Second-Order Optimality Conditions of (OP)

In this section, by employing the generalized second-order adjacent epiderivative, we will discuss the second-order optimality conditions for Benson proper efficient solutions of (OP).

Theorem 4.1. Let 𝑥0𝐸,𝑦0𝐹(𝑥0), and (𝑢,𝑣)𝑋×(𝐶). If (𝑥0,𝑦0) is a Benson proper efficient element of (OP), then, for every 𝑥Ω=dom[𝐺-𝐷(2)𝐹(𝑥0,𝑦0,𝑢,𝑣)], one has 𝐺-𝐷(2)𝐹𝑥0,𝑦00,𝑢,𝑣(𝑥)𝑌𝐶𝑌.(4.1)

Proof. Since (𝑥0,𝑦0) is a Benson proper efficient element of (OP), 𝐹𝑦clcone(𝐸)+𝐶00(𝐶)=𝑌.(4.2)
Assume that there exists an 𝑥Ω such that (4.1) does not hold, that is, there exists 𝑦𝐺-𝐷(2)𝐹𝑥0,𝑦0,𝑢,𝑣𝑥0𝐶𝑌,(4.3) and then, by the definition of generalized second-order adjacent epiderivatives, for arbitrary sequence {𝑛} with 𝑛0+, there exists a sequence 𝑥𝑛,𝑦𝑛𝑥coneepi𝐹0,𝑦0(4.4) such that 𝑥𝑛,𝑦𝑛𝑛(𝑢,𝑣)2𝑛𝑥,𝑦.(4.5)
Since 𝑣𝐶,𝑛>0 and 𝐶 is a convex cone, 𝑛𝑣𝐶. Then, by (4.4), we get 𝑦𝑛𝑛𝐹𝑦𝑣cone(𝐸)+𝐶0.(4.6) Thus, from (4.5) and (4.3), one obtains 𝐹𝑦𝑦clcone(𝐸)+𝐶00𝐶𝑌,(4.7) which contradicts (4.2). So (4.1) holds and the proof is complete.

Theorem 4.2. Let (𝑢,𝑣)𝑋×(𝐶), 𝑥0𝐸,𝑦0𝐹(𝑥0) and let 𝐶 be locally compact. Suppose that the following conditions are satisfied: (i)𝐹 is generalized 𝐶-convex at (𝑥0,𝑦0) on 𝐸; (ii)the pair (𝑥0,𝑦0) is a Benson proper efficient element of (OP).
Then, there exists 𝜙𝐶+𝑖 such that inf𝜙(𝑦)𝑦𝑥Ω𝐺-𝐷(2)𝐹𝑥0,𝑦0(,𝑢,𝑣𝑥)=0,(4.8) where Ω=dom[𝐺𝐷(2)𝐹(𝑥0,𝑦0,𝑢,𝑣)].

Proof. Define 𝑀=𝑥Ω𝐺-𝐷(2)𝐹𝑥0,𝑦0,𝑢,𝑣(𝑥).(4.9)
By the similar line of proof for convexity of 𝑀 in [15], Theorem 5.1, we obtain that 𝑀 is a convex set. It follows from Theorem 4.1, that 0𝑀(𝐶)=𝑌.(4.10)
Thus, by Lemma 2.6, there exists 𝜙𝐶+𝑖 such that 𝜙𝑀+. So, we have 𝜙(𝑦)0,𝑦𝑀.(4.11) It follows from Corollary 3.15 that 0𝑌𝑀, so inf𝜙(𝑦)𝑦𝑥Ω𝐺-𝐷(2)𝐹𝑥0,𝑦0(,𝑢,𝑣𝑥)=0,(4.12) and the proof of the theorem is complete.

Theorem 4.3. Let 𝑥0𝐸,𝑦0𝐹(𝑥0) and (𝑢,𝑣){0𝑋}×𝐶. If there exists 𝜙𝐶+𝑖 such that 𝜙(𝑦)0,𝑦𝐺-𝐷(2)𝐹𝑥0,𝑦0,𝑢,𝑣𝑥𝑥0,𝑥𝐸,(4.13) then the pair (𝑥0,𝑦0) is a Benson proper efficient solution of (OP).

Proof. It follows from Proposition 3.3 that 𝑦𝐹(𝑥)+𝐶0𝐺-𝐷(2)𝐹𝑥0,𝑦0,𝑢,𝑣𝑥𝑥0,(4.14) for all 𝑥𝐸,𝑦𝐹(𝑥). Then, by (4.13), (4.14), and 𝜙𝐶+𝑖, we get 𝜙𝑦(𝑦)𝜙0,𝑦𝐹(𝐸).(4.15)
Thus, by the sufficient condition of [23, Theorem 4.1], we get that (𝑥0,𝑦0) is a Benson proper efficient solution of (OP), and the proof of the theorem is complete.

Remark 4.4. In Theorem 4.3, no assumption of generalized convexity is imposed.

Remark 4.5. If we use 𝐷(2)𝐹(𝑥0,𝑦0,𝑢,𝑣) instead of 𝐺-𝐷(2)𝐹(𝑥0,𝑦0,𝑢,𝑣) in Theorem 4.3, then the corresponding result for 𝐷(2)𝐹(𝑥0,𝑦0,𝑢,𝑣) may not hold. The following example explains the case.

Example 4.6. Suppose that 𝑋=𝑌=𝐸=𝑅, 𝐶=𝑅+. Let 𝐹𝐸2𝑌 be a set-valued map with 1𝐹(𝑥)=𝑦𝑅𝑦|𝑥|+1,𝑥2,1𝑦𝑅𝑦𝑥+1,𝑥>2.(4.16) Consider the following constrained set-valued optimization problem (4.1): min𝐹(𝑥),s.t.𝑥𝐸.(4.17) Take (𝑥0,𝑦0)=(0,1)gph(𝐹),(𝑢,𝑣)=(0,0){0}×𝐶. We have 𝐷(2)𝐹𝑥0,𝑦0,𝑢,𝑣𝑥𝑥0={𝑦𝑅𝑦|𝑥|},𝑥𝐸.(4.18) Then, for any 𝜙𝐶+𝑖, we have 𝜙(𝑦)0,𝑦𝐷(2)𝐹𝑥0,𝑦0,𝑢,𝑣𝑥𝑥0,𝑥𝐸.(4.19) Naturally, the pair (𝑥0,𝑦0) is not a Benson proper efficient solution of (4.1).

From Proposition 3.4, we know that the following theorem holds.

Theorem 4.7. Let 𝑥0𝐸,𝑦0𝐹(𝑥0). Suppose that the following conditions are satisfied: (i)𝑢𝐸,𝑣𝐹(𝑢)+𝐶, (ii)𝐹 is generalized 𝐶-convex at (𝑥0,𝑦0) on 𝐸, (iii)there exists 𝜙𝐶+𝑖 such that 𝜙(𝑦)0,𝑦𝐺-𝐷(2)𝐹𝑥0,𝑦0,𝑢𝑥0,𝑣𝑦0𝑥𝑥0,𝑥𝐸.(4.20)
Then the pair (𝑥0,𝑦0) is a Benson proper efficient solution of (OP).

5. Second-Order Kuhn-Tuck Type Optimality Conditions of (SOP)

In this section, by employing the generalized second-order adjacent epiderivative, we will discuss the second-order Kuhn-Tuck type optimality conditions for Benson proper efficient solutions of (SOP). The notation (𝐹,𝐺)(𝑥) is used to denote 𝐹(𝑥)×𝐺(𝑥). Let 𝑥0𝐾,𝑦0𝐹(𝑥0). Firstly, we recall a result in [24].

The interior tangent cone of 𝐾 at 𝑥0 is defined as IT𝐾𝑥0=𝑢𝑋𝜆>0,𝑡(0,𝜆),𝑢𝐵𝑋(𝑢,𝜆),𝑥0+𝑡𝑢𝐾,(5.1) where 𝐵𝑋(𝑢,𝜆) stands for the closed ball centered at 𝑢𝑋 and of radius 𝜆.

Lemma 5.1 (see [24]). If 𝐾𝑋 is convex, 𝑥0𝐾 and int𝐾, then ITint𝐾𝑥0𝑥=intcone𝐾0.(5.2)

Theorem 5.2. Let (𝑢,𝑣,𝑤)𝑋×(𝐶)×(𝐷), 𝑧0𝐺(𝑥0)(𝐷). If (𝑥0,𝑦0) is a Benson proper efficient element of (𝐶𝑃), then 𝐺-𝐷(2)𝑥(𝐹,𝐺)0,𝑦0,𝑧0,𝑢,𝑣,𝑤1+𝑧00(𝑥)(𝑌×𝑍)𝐶𝑌×int𝐷+𝑧0,(5.3) for all 𝑥Ω=dom[𝐺-𝐷(2)(𝐹,𝐺)(𝑥0,𝑦0,𝑧0,𝑢,𝑣,𝑤1+𝑧0)].

Proof. Since (𝑥0,𝑦0) is a Benson proper efficient element of (𝐶𝑃), 𝐹𝑦clcone(𝐾)+𝐶00𝐶=𝑌.(5.4)
Suppose that there exists some 𝑥Ω such that (5.3) does not hold, that is, there exists some 𝑦,𝑧𝐺-𝐷(2)𝑥(𝐹,𝐺)0,𝑦0,𝑧0,𝑢,𝑣,𝑤+𝑧0𝑥,(5.5)𝑦,𝑧0𝐶𝑌×𝑧int𝐷+0.(5.6) It follows from (5.5) and the definition of generalized second-order contingent epiderivatives that there exist sequences {𝑛} with 𝑛0+ and 𝑥𝑛,𝑦𝑛,𝑧𝑛𝑥coneepi(𝐹,𝐺)0,𝑦0,𝑧0(5.7) such that 𝑥𝑛,𝑦𝑛,𝑧𝑛𝑛𝑢,𝑣,𝑤+𝑧02𝑛𝑥,𝑦,𝑧.(5.8)
From (5.6) and (5.8), there exists a sufficiently large natural number 𝑁1 such that 𝑧𝑛𝑧=𝑛𝑛𝑤+𝑧02𝑛=1𝑛𝑧𝑛𝑛𝑤𝑛𝑧0int𝐷+𝑧0𝑧intcone𝐷+0,𝑛>𝑁1.(5.9)
According to (5.9) and Lemma 5.1, we obtain 𝑧ITint𝐷(𝑧0). Then, it follows from the definitions of ITint𝐷(𝑧0) that 𝜆>0, 𝑡(0,𝜆),𝑢𝐵𝑍(𝑧,𝜆),𝑧0+𝑡𝑢int𝐷. Since 𝑛0+ and (5.9), there exists a sufficiently large natural number 𝑁2 such that 𝑛(0,𝜆),for 𝑛>𝑁2 and 𝑧0+𝑛(𝑧𝑛)int𝐷, for 𝑛>𝑁2, that is, 𝑧𝑛𝑛𝑤𝑛int𝐷,for𝑛>𝑁2.(5.10) It follows from 𝑛>0,𝑤𝐷 and 𝐷 is a convex cone that 𝑧𝑛int𝐷,for𝑛>𝑁2.(5.11) Then from (5.7), 𝑧𝑛cone(𝐺(𝑥𝑛)+𝐷{𝑧0}), there exist 𝜆𝑛>0, ̃𝑥𝑛𝐸, ̃𝑧𝑛𝐺(̃𝑥𝑛), ̃𝑦𝑛𝐹(̃𝑥𝑛), 𝑐𝑛𝐶 and 𝑑𝑛𝐷 such that 𝑦𝑛=𝜆𝑛(̃𝑦𝑛+𝑐𝑛𝑦0)) and 𝑧𝑛=𝜆𝑛(̃𝑧𝑛+𝑑𝑛𝑧0)). It follows from (5.11) that ̃𝑧𝑛𝐺(̃𝑥𝑛)(𝐷), for 𝑛>𝑁2, and then ̃𝑥𝑛𝐾,for𝑛>𝑁2.(5.12)
Since 𝑣𝐶,𝑛>0 and 𝐶 is a convex cone, 𝑛𝑣𝐶. Then by (5.7), (5.12), and (5.8), we get 𝐹𝑦𝑦clcone(𝐾)+𝐶0𝑁,for𝑛>max1,𝑁2.(5.13) Then it follows from (5.6) that 𝐹𝑦𝑦clcone(𝐾)+𝐶00𝐶𝑌𝑁,for𝑛>max1,𝑁2,(5.14) which contradicts (5.4). Thus, (5.3) holds and the proof of the theorem is complete.

Theorem 5.3. Let (𝑢,𝑣,𝑤)𝑋×(𝐶)×(𝐷), 𝑥0𝐸,(𝑦0,𝑧0)(𝐹,𝐺)(𝑥0). Suppose that the following conditions are satisfied: (i)𝐶 has a compact base; (ii)(𝐹,𝐺) is generalized 𝐶×𝐷-convex at (𝑥0,𝑦0,𝑧0) on 𝐸; (iii)the pair (𝑥0,𝑦0) is a Benson proper efficient element of (SOP).
Then, there exist 𝜙𝐶+𝑖 and 𝜓𝐷 such that 𝑧inf{𝜙(𝑦)+𝜓(𝑧)(𝑦,𝑧)Δ}=0,𝜓0=0,(5.15) where Δ=𝑥Ω𝐺-𝐷(2)(𝐹,𝐺)(𝑥0,𝑦0,𝑧0,𝑢,𝑣,𝑤+𝑧0)(𝑥) and Ω=dom[𝐺-𝐷(2)(𝐹,𝐺)(𝑥0,𝑦0,𝑧0,𝑢,𝑣,𝑤+𝑧0)].

Proof. Define 𝑀=𝑥Ω𝐺-𝐷(2)𝑥(𝐹,𝐺)0,𝑦0,𝑧0,𝑢,𝑣,𝑤+𝑧00(𝑥)+𝑌,𝑧0.(5.16)
By the similar line of proof for convexity of 𝑀 in [15, Theorem 5.1], we obtain that 𝑀 is a convex set. It follows from Theorem 5.2 that 0𝑀𝐶𝑌×int𝐷=.(5.17) Thus, by employing Corollary 3.15, it follows from a standard separation theorem of convex sets and the similar proof method of [8, Theorem 1] that there exist 𝜙𝐶+𝑖 and 𝜓𝐷 such that 𝑧inf{𝜙(𝑦)+𝜓(𝑧)(𝑦,𝑧)Δ}=0,𝜓0=0.(5.18) The proof is complete.

Remark 5.4. It follows from Remarks 2.3 and 3.2 that the necessary optimality condition in Theorem 5.3 is obtained under weaker assumptions than those assumed of [8, Theorem 1].

Now we give an example to illustrate the necessary optimality conditions for generalized second-order contingent epiderivatives.

Example 5.5. Suppose that 𝑋=𝑌=𝑍=𝐸=𝑅, 𝐶=𝐷=𝑅+. Let 𝐹𝐸2𝑌 be a set-valued map with 𝐹(𝑥)=𝑦𝑅𝑦𝑥2/3,𝑥𝐸,(5.19) and 𝐺𝐸𝑍 be a set-valued map with 𝐺(𝑥)={𝑧𝑅𝑧𝑥},𝑥𝐸.(5.20)
Consider the following constrained set-valued optimization problem (4.2): min𝐹(𝑥),s.t.𝑥𝐸,𝐺(𝑥)(𝐷).(5.21) Take (𝑥0,𝑦0,𝑧0)=(0,0,0)gph(𝐹,𝐺) and (𝑢,𝑣,𝑤)=(1,0,1)𝑋×(𝐶)×(𝐷). Naturally, (𝐹,𝐺) is generalized 𝐶×𝐷-convex at (𝑥0,𝑦0,𝑧0) on 𝐸, and (𝑥0,𝑦0) is a Benson proper efficient solution of (4.2). By directly calculation, we have 𝐺-𝐷(2)𝑥(𝐹,𝐺)0,𝑦0,𝑧0,𝑢1,𝑣1,𝑤1+𝑧0(𝑥)=(𝑦,𝑧)𝑅2𝑦0,𝑧𝑥,𝑥𝐸.(5.22) Take 𝜙=1𝐶+𝑖 and 𝜑=0𝐷. Naturally, necessary optimality condition of Theorem 5.3 holds here.

Theorem 5.6. Let 𝑥0𝐸,(𝑦0,𝑧0)(𝐹,𝐺)(𝑥0). Suppose that the following conditions are satisfied: (i)(𝑢,𝑣,𝑤){0𝑋}×𝐶×𝐷, (ii) there exist 𝜙𝐶+𝑖 and 𝜓𝐷 such that 𝑧inf{𝜙(𝑦)+𝜓(𝑧)(𝑦,𝑧)Θ}=0,𝜓0=0,(5.23) where Θ=𝑥𝐾𝐺-𝐷(2)(𝐹,𝐺)(𝑥0,𝑦0,𝑧0,𝑢,𝑣,𝑤)(𝑥𝑥0).
Then, the pair (𝑥0,𝑦0) is a Benson proper efficient solution of (SOP).

Proof. It follows from Proposition 3.3 that 𝑦𝑦0,𝑧𝑧0𝐺-𝐷(2)𝑥(𝐹,𝐺)0,𝑦0,𝑧0,𝑢,𝑣,𝑤𝑥𝑥0,(5.24) for all 𝑦𝐹(𝑥),𝑧𝐺(𝑥),𝑥𝐾. Then, by assumption (ii), we have 𝜙𝑦𝑦0+𝜓𝑧𝑧00,𝑦𝐹(𝐾),𝑧𝐺(𝐾).(5.25) Thus, there exists a 𝑧𝐺(𝐾) with 𝑧𝐷 such that 𝜓(𝑧)0. It follows from 𝜓(𝑧0)=0 and (5.25) that 𝜙𝑦(𝑦)𝜙0,𝑦𝐹(𝐾).(5.26) It follows from the sufficient condition of [23, Theorem 4.1], that (𝑥0,𝑦0) is a Benson proper efficient solution of (SOP) and the proof of the theorem is complete.

Remark 5.7. Since Theorem 5.6 does not involve the assumption of convexity, it improves and generalizes [8, Theorem 2].

From Proposition 3.4 and the proof of Theorem 5.6, we easily obtain that the following Theorem holds.

Theorem 5.8. Let 𝑥0𝐸,(𝑦0,𝑧0)(𝐹,𝐺)(𝑥0). Suppose that the following conditions are satisfied: (i)𝑢𝐾,𝑣𝐹(𝑢)+𝐶,𝑤𝐺(𝑢)+𝐷, (ii)(𝐹,𝐺) is generalized 𝐶×𝐷-convex at (𝑥0,𝑦0,𝑧0) on 𝐸;(iii) there exist 𝜙𝐶+𝑖 and 𝜓𝐷 such that inf(𝑦,𝑧)Θ𝑧{𝜙(𝑦)+𝜓(𝑧)}=0,𝜓0=0,(5.27) where Θ=𝑥𝐾𝐺-𝐷(2)(𝐹,𝐺)(𝑥0,𝑦0,𝑧0,𝑢𝑥0,𝑣𝑦0,𝑤𝑧0)(𝑥𝑥0).
Then, the pair (𝑥0,𝑦0) is a Benson proper efficient solution of (SOP).

Acknowledgments

The authors would like to thank anonymous referees for their valuable comments and suggestions, which helped to improve the paper. This research was partially supported by the National Natural Science Foundation of China (10901004, 11171362, and 11071267), Natural Science Foundation Project of CQ CSTC (cstcjjA00019), and Science and Technology Research Project of Chong Qing Municipal Education Commission (KJ100419).

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