Approximate solutions for nonconvex set-valued optimization and vector variational inequality

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.

In this paper, X, Y, and Z are assumed to be Banach spaces with topological dual \(X^{*}\), \(Y^{*}\), and \(Z^{*}\), respectively. For any \(x\in X\) and \(x^{*}\in X^{*}\), the canonical form between X and \(X^{*}\) is denoted by \(x^{*T}x\). Let \(D\subset Y\) and \(E\subset Z\) be pointed closed convex cones with \(\operatorname{int} (D)\neq\emptyset\). We write and similarly for \(E^{*}\). Let \(F: X\rightarrow2^{Y}\) be a set-valued mapping. The set is called the domain of F. The set is called the graph of the map F. The set is called the epigraph of F.

Let \((\bar{x},\bar{y})\in\operatorname{graph}(F)\). The contingent cone (see [5]) to \(\operatorname{epi}(F)\) at \((\bar{x},\bar{y})\) denoted by \(T(\operatorname{epi}(F),(\bar{x},\bar{y}))\), which consists of all tangent vectors where \((\bar{x},\bar{y})=\lim_{n\rightarrow \infty}(x_{n},y_{n})\), \((x_{n},y_{n})\in\operatorname{epi}(F)\), \(\lambda_{n} >0\), for all \(n\in\mathbb{N}\).

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 \(tz+(1-t)x\in S \).

In Definition 2.2, when \(\eta(x,z)=x-z\), then S is a convex set. In general, the opposite is not true.

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.

Let \(F: X\rightarrow2^{Y}\) and \(G: X\rightarrow2^{Z}\) be two set-valued maps with \(\operatorname{dom}(F)=\operatorname{dom}(G)=X\). In Section 3 and Section 4, we consider the following set-valued optimization problem:

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.

This section is devoted to a discussion of the relationship between approximate solutions of set-valued optimization and that of a kind of vector variational inequality. Let S be a nonempty invex subset of X and \(F: X\rightarrow2^{Y}\) be a set-valued mapping. Considering the following set-valued optimization problem (SOP-II):

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 \(DF(\bar{x},\bar{y})\) .

Now, we consider the vector variational inequality problem \((\mathrm{VVIP})_{\eta }\), that is, to find \(\bar{x}\in S\), \(\bar{y}\in F(\bar{x})\) such that

When \(\eta(x,\bar{x})=x-\bar{x}\), the vector variational inequality problem (VVIP) was investigated by Liu and Gong [35].

For the problem (SOP-II), since every weak minimizer is a weak \(\varepsilon\cdot e\)-minimizer, we can immediately derive Corollary 5.3.

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.