- Research
- Open Access
Approximate solutions for nonconvex set-valued optimization and vector variational inequality
- Guolin Yu^{1}Email author and
- Xiangyu Kong^{1}
https://doi.org/10.1186/s13660-015-0839-5
© Yu and Kong 2015
- Received: 10 June 2015
- Accepted: 23 September 2015
- Published: 9 October 2015
Abstract
This note deals with approximate solutions in vector optimization involving a generalized cone-invex set-valued mapping. First, a new class of generalized cone-invex set-valued maps, called cone-subinvex set-valued maps, is introduced. Then the sufficient optimality condition and two types dual theorems are established for weakly approximate minimizers under the assumption of cone-subinvexity. Finally, it also reveals the closed relationships between a weakly approximate minimizer of a cone-subinvex set-valued optimization problem and a weakly approximate solution of a kind of vector variational inequality.
Keywords
- generalized invex set-valued mappings
- optimality conditions
- contingent epiderivative
- approximate solutions
MSC
- 90C29
- 90C46
- 26B25
1 Introduction
Recently, there has been an increasing interest in the extension of vector optimization to set-valued optimization. As a bridge between different areas of optimization, the theory of set-valued optimization problems has wide applications in differential inclusion, variational inequality, optimal control, game theory, economic equilibrium problem, decision making, etc. For more details of set-valued optimization theory and applications, the reader can refer to the excellent books [1–4].
The derivative of set-valued maps is most important for the formulation of optimality conditions. Aubin and Frankowsa [1] introduced the notion of a contingent derivative of a set-valued map as an extension of the concept of Fréchet differentiability. From then on, various approaches have been followed in defining the concept of derivative for set-valued maps. Among these notions, a meaningful and useful concept is the contingent epiderivative, which was given by Jahn and Rauh [5]. It is important to note that the contingent epiderivative is a single-valued map. Recently, much attention has been paid to characterizing optimality conditions for set-valued optimization and related problems by utilizing contingent epiderivatives; for example, see [6–12]. On the other hand, convex analysis is a powerful tool for the investigation of optimal solutions of vector optimization problems. Various notions of generalized convexity have been introduced to weaken convexity. One of such generalizations is invexity, which was firstly introduced by Hanson [13] for nonlinear programming. Based upon this concept, some scholars developed further generalizations of invexity to vector optimization involving set-valued maps. For example, Luc and Malivert [14] extended the concept of invexity to set-valued optimization and investigated the necessary and sufficient optimality conditions; Sach and Craven [15] proved duality theorems for set-valued optimization problems under invexity assumptions. In [16–19], optimality conditions and a characterization of solution sets of set-valued optimization problems involving generalized invexity are investigated.
Since duality assertions allow us to study a minimization problem through a maximization problem and to know what one can expect in the best case. At the same time, duality has resulted in many applications within optimization, and it has provided many unifying conceptual insights into economics and management science. So, it is not surprising that duality is one of the important topics in set-valued optimization. There are many papers dedicated to duality theory of set-valued optimization, for instance, [20–25] cited in this paper are closely related to the present work.
On the other hand, since it has been introduced by Giannessi [26], the theory of vector variational inequalities has shown many applications in vector optimization problems and traffic equilibrium problems. In fact, some recent work has shown that optimality conditions of some vector optimization problems can be characterized by vector variational inequalities. For example, Al-Homidan and Ansari [27] dealt with different kinds of generalized vector variational-like inequality problems and a vector optimization problem. Some relationships between the solutions of generalized vector variational-like inequality problem and an efficient solution of a vector optimization problem have been established; Ansari et al. [28] worked on the generalized vector variational-like inequalities involving the Dini subdifferential, and some relations among these inequalities and vector optimization problems are presented. In the literature [29, 30], the authors focused on the exponential type vector variational-like inequalities and vector optimization problems with exponential type invexities. Observing the above mentioned papers, we found that the invexity plays exactly the same role in variational-like inequalities as the classical convexity plays in variational inequalities. Motivated by this work, this paper will extend the partial results to the setting of a set-valued mapping under the weaker invexity assumption.
In addition, approximate solutions of optimization problems are very important from both the theoretical and the practical points of view because they exist under very mild hypotheses and a lot of solution methods propose this kind of solutions. Thus, it is meaningful to consider various concepts of approximate solutions to set-valued optimization problems. Recently, approximate solutions for set-valued optimization have caught many scholars’ attention; for example, see [31–34] and the references therein.
Based upon the above observation, the purpose of this paper is two aspects: first, to introduce a new class of generalized set-valued cone-invex maps and establish sufficient optimal conditions and dual theorems of approximate solutions for set-valued optimization problems under these generalized convexities; second, to study the optimality conditions of weakly approximate minimizer in vector optimization involving generalized cone-invex set-valued mappings by using the notions of vector variational inequality.
This paper is structured as follows: In Section 2, some well-known definitions and results used in the sequel are recalled; a new class of generalized set-valued cone-invex maps, named cone-subinvex set-valued maps, is introduced. In Section 3 and Section 4, we give the sufficient optimality conditions and two types dual theorems of weakly approximate minimizers, respectively. Section 5 is devoted to revealing the closed relation between weakly approximate solutions of vector optimization and variational inequality involving set-valued cone-subinvex mappings.
2 Preliminaries
Definition 2.1
(see [5])
Next, we begin with recalling the concepts of a convex set and an invex set. It is well known that a subset S of X is a convex set, if for any \(x,z\in S\), \(t\in[0,1]\), we have \(tz+(1-t)x\in S \).
Definition 2.2
(see [14])
A subset \(S\subset X\) is said to be an η-invex set, if, for every \(x\in S\), \(z\in S\), there exists a map \(\eta: X\times X\rightarrow X\) such that \(z+t\eta(x,z)\in S\), for all \(t\in[0,1]\).
Example 2.3
Let \(S=(0, +\infty)\subset\mathbb{R}=X\). Then S is invex with respect to \(\eta: X\times X\rightarrow X\), \(\eta (x,y)=x+y\).
In Definition 2.2, when \(\eta(x,z)=x-z\), then S is a convex set. In general, the opposite is not true.
Example 2.4
Definition 2.5
(see [14])
Example 2.6
Definition 2.7
Obviously, if F is a D-η-invex map at \((\bar{x},\bar{y})\in \operatorname{graph}(F) \), then F is a D-η-subinvex set-valued mapping at \((\bar{x},\bar{y})\) with respect to \(\varepsilon\cdot e\). However, the inverse proposition is not necessarily true, as is illustrated in the following example.
Example 2.8
3 Sufficient optimality conditions
A point \((\bar{x},\bar{y})\in X\times Y\) is said to be a feasible point of the problem (SOP-I) if \(\bar{x}\in X\), \(\bar{y}\in F(\bar{x})\), and \(G(\bar{x})\cap(-E)\neq\emptyset\). Let \(\Omega=\{x\in X: (x,y) \mbox{ is a feasible point of the problem (SOP-I)}\}\). Then the weakly approximate minimizer for the set-valued optimization problem (SOP-I) is defined in the following way.
Definition 3.1
(see [21])
Theorem 3.2
(Sufficient optimality condition)
Proof
4 Duality theorems
4.1 Mond-Weir type duality
A point \((x',y',z',y^{*},z^{*})\) satisfying all the constraints of the problem (MWD) is called a feasible point of the problem (MWD). Let \(K_{1}=\{y': (x',y',z',y^{*},z^{*}) \mbox{ is a feasible point of} \mbox{ }\mbox{(MWD)}\}\).
Definition 4.1
Theorem 4.2
(Weak duality)
Proof
Theorem 4.3
(Strong duality)
Let \(e\in\operatorname{int}(D)\), \(\varepsilon\geq0\), and \((\bar{x},\bar{y})\) be a weak \(\varepsilon\cdot e\)-minimizer of (SOP-I). Suppose that, for some \((y^{*},z^{*})\in (D^{*}\backslash\{0_{Y^{*}}\} )\times E^{*}\) and \(\bar {z}\in G(\bar{x})\cap(-E)\), (3.1) and (3.2) are satisfied. Then \((\bar{x},\bar{y},\bar{z},y^{*},z^{*})\) is a feasible point for (MWD). Furthermore, if the weak duality theorem, Theorem 4.2, between (SOP-I) and (MWD) holds, then \((\bar{x},\bar{y},\bar{z},y^{*},z^{*})\) is a weak \(\varepsilon\cdot e\)-maximizer of (MWD).
Proof
Theorem 4.4
(Converse duality)
Let \(e\in\operatorname{int}(D)\), \(\varepsilon\geq0\), and \((x',y',z',y^{*},z^{*})\) be a feasible point of the problem (MWD) and \(z'\in G(x')\cap(-E)\). Suppose that F is D-η-subinvex at \((x',y')\) with respect to \(\varepsilon\cdot e\) and G is E-η-invex at \((x',z')\) with respect to the same η. Then \((x',y')\) is a weak \(\varepsilon\cdot e\)-minimizer of the problem (SOP-I).
Proof
4.2 Wolfe type duality
A point \((x',y',z',y^{*},z^{*})\) satisfying all the constraints of problem (WD) is called a feasible point of the problem (WD). Let \(K_{2}=\{ y'+z^{*T}z'\cdot d_{0}: (x',y',z',y^{*},z^{*}) \mbox{ is a feasible point}\mbox{ }\mbox{of (WD)}\}\).
Definition 4.5
Theorem 4.6
(Weak duality)
Proof
Theorem 4.7
(Strong duality)
Let \(e\in\operatorname {int}(D)\), \(\varepsilon\geq0\), and \((\bar{x},\bar{y})\) be a weak \(\varepsilon\cdot e\)-minimizer of (SOP-I). Suppose that, for some \((y^{*},z^{*})\in (D^{*}\backslash\{0_{Y^{*}}\} )\times E^{*}\) with \(y^{*T}\cdot d_{0}=1\), (3.1) and (3.2) are satisfied for some \(\bar{z}\in G(\bar{x})\cap(-E)\). Then \((\bar{x},\bar{y},\bar{z},y^{*},z^{*})\) is a feasible point for (WD). Furthermore, if the weak duality theorem, Theorem 4.6, between (SOP-I) and (WD) holds, then \((\bar{x},\bar {y},\bar{z},y^{*},z^{*})\) is a weak \(\varepsilon\cdot e\)-maximizer of (WD).
Proof
Theorem 4.8
(Converse duality)
Let \(e\in\operatorname {int}(D)\), \(\varepsilon\geq0\), and \((x',y',z',y^{*},z^{*})\) be a feasible point of the problem (WD) with \(z'\in G(x')\cap(-E)\) and \(z^{*T}z'=0\). Suppose that F is D-η-subinvex at \((x',y')\) with respect to \(\varepsilon\cdot e\) and G is E-η-invex at \((x',z')\) with respect to the same η. Then \((x',y')\) is a weak \(\varepsilon\cdot e\)-minimizer of the problem (SOP-I).
Proof
5 Vector optimization and variational inequality
Let \(\bar{x}\in S\) , \(\bar{y}\in F(\bar{x})\) and \(\eta: X\times X \rightarrow X\) be a map. In the following, it is assumed that \(DF(\bar {x},\bar{y})\) exists, and \(\eta(S,\bar{x}):=\{\eta(x,\bar{x}): x\in S\}\) belongs to the domain of \(DF(\bar{x},\bar{y})\) .
When \(\eta(x,\bar{x})=x-\bar{x}\), the vector variational inequality problem (VVIP) was investigated by Liu and Gong [35].
Definition 5.1
Theorem 5.2
Let \(e\in\operatorname{int}(D)\) and \(\varepsilon \geq0\). If a pair \((\bar{x},\bar{y})\) is a weak \(\varepsilon\cdot e\)-minimizer of problem (SOP-II), then \((\bar{x},\bar{y})\) is a weak \(\varepsilon\cdot e\)-efficient solution of \((\mathrm{VVIP})_{\eta}\).
Proof
For the problem (SOP-II), since every weak minimizer is a weak \(\varepsilon\cdot e\)-minimizer, we can immediately derive Corollary 5.3.
Corollary 5.3
Let \(e\in\operatorname{int}(D)\) and \(\varepsilon \geq0\). If a pair \((\bar{x},\bar{y})\) is a weak minimizer of problem (SOP-II), then \((\bar{x},\bar{y})\) is a weak \(\varepsilon\cdot e\)-minimizer of \((\mathrm{VVIP})_{\eta}\).
Theorem 5.4
Let S be an invex set with respect to η and the set-valued mapping \(F: S\rightarrow2^{Y}\) be D-η-subinvex on S with respect to \(\varepsilon\cdot e\). If \((\bar{x},\bar{y})\) is a weak efficient solution of \((\mathrm{VVIP})_{\eta}\), then \((\bar{x},\bar{y})\) is a weak \(\varepsilon\cdot e\)-minimizer of the problem (SOP-II).
Proof
Remark 5.5
Remark 5.6
This note only presents the relationships between a kind of generalized variational-like inequalities and set-valued optimization problem. However, we do not discuss the relationships of other kinds of variational-like inequalities and set-valued optimization, and the existence problems of variational-like inequalities are not involved. For more details related to these problems, we refer the reader to [27, 28, 36].
6 Conclusions and remarks
In this paper, we focus on the approximate solutions in set-valued optimization. We present the notion of cone-subinvex set-valued maps and investigate its properties. A sufficient optimality condition and two types dual theorems are established for weakly approximate minimizers under the assumption of cone-subinvexity. We also discuss the relationships between a kind of vector variational inequality and set-valued optimization. Under the assumption cone-subinvexity, it shows that the weakly approximate minimizers of set-valued optimization are characterized by the weakly approximate solution of a kind of vector variational inequality.
It is worthy underlining that Ansari and Jahn [36] defined the \(\mathbb {T}\)-epiderivative of a set-valued map, which includes the contingent epiderivative as its special case. They provided necessary and sufficient conditions for a solution of a set-valued problems and some existence results for solutions of set-valued optimization problems and a generalized vector \(\mathbb{T}\)-inequality problem under the assumption of cone-convexity for set-valued maps. It is possible to extend the notion of cone-subinvexity in the setting of \(\mathbb {T}\)-epiderivative and to deal with similar problems for approximate solutions in set-valued optimization, such as optimality conditions and duality. This must be an interesting and meaningful work.
Declarations
Acknowledgements
This research was supported by Natural Science Foundation of China under Grant No. 11361001; Ministry of Education Science and technology key projects under Grant No. 212204; Natural Science Foundation of Ningxia under Grant No. NZ14101. The authors are grateful to the anonymous referees who have contributed to improve the quality of the paper.
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
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