Abstract
This note deals with approximate solutions in vector optimization involving a generalized coneinvex setvalued mapping. First, a new class of generalized coneinvex setvalued maps, called conesubinvex setvalued maps, is introduced. Then the sufficient optimality condition and two types dual theorems are established for weakly approximate minimizers under the assumption of conesubinvexity. Finally, it also reveals the closed relationships between a weakly approximate minimizer of a conesubinvex setvalued optimization problem and a weakly approximate solution of a kind of vector variational inequality.
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1 Introduction
Recently, there has been an increasing interest in the extension of vector optimization to setvalued optimization. As a bridge between different areas of optimization, the theory of setvalued optimization problems has wide applications in differential inclusion, variational inequality, optimal control, game theory, economic equilibrium problem, decision making, etc. For more details of setvalued optimization theory and applications, the reader can refer to the excellent books [1–4].
The derivative of setvalued maps is most important for the formulation of optimality conditions. Aubin and Frankowsa [1] introduced the notion of a contingent derivative of a setvalued 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 setvalued 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 singlevalued map. Recently, much attention has been paid to characterizing optimality conditions for setvalued 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 setvalued maps. For example, Luc and Malivert [14] extended the concept of invexity to setvalued optimization and investigated the necessary and sufficient optimality conditions; Sach and Craven [15] proved duality theorems for setvalued optimization problems under invexity assumptions. In [16–19], optimality conditions and a characterization of solution sets of setvalued 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 setvalued optimization. There are many papers dedicated to duality theory of setvalued 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, AlHomidan and Ansari [27] dealt with different kinds of generalized vector variationallike inequality problems and a vector optimization problem. Some relationships between the solutions of generalized vector variationallike inequality problem and an efficient solution of a vector optimization problem have been established; Ansari et al. [28] worked on the generalized vector variationallike 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 variationallike 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 variationallike 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 setvalued 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 setvalued optimization problems. Recently, approximate solutions for setvalued 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 setvalued coneinvex maps and establish sufficient optimal conditions and dual theorems of approximate solutions for setvalued optimization problems under these generalized convexities; second, to study the optimality conditions of weakly approximate minimizer in vector optimization involving generalized coneinvex setvalued mappings by using the notions of vector variational inequality.
This paper is structured as follows: In Section 2, some wellknown definitions and results used in the sequel are recalled; a new class of generalized setvalued coneinvex maps, named conesubinvex setvalued 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 setvalued conesubinvex mappings.
2 Preliminaries
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 setvalued 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 \(n\in\mathbb{N}\).
Definition 2.1
(see [5])
Let a pair \((\bar{x},\bar {y})\in\operatorname{graph}(F)\) be given. A singlevalued map \(D F(\bar{x},\bar{y}): X\rightarrow Y\) whose epigraph equals the contingent cone to the epigraph of F at \((\bar{x},\bar{y})\), that is,
is called contingent epiderivative of F at \((\bar{x},\bar{y})\). If the contingent epiderivative of F at any point in \(\operatorname{graph}(F)\) exists, then we say F is a contingent epiderivable setvalued map.
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 \(tz+(1t)x\in S \).
Definition 2.2
(see [14])
A subset \(S\subset X\) is said to be an ηinvex 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 \(\eta(x,z)=xz\), then S is a convex set. In general, the opposite is not true.
Example 2.4
Suppose that \(S=[5,2]\cup[2,10]\), then S is an invex with respect to \(\eta(x,y)\), defined by
Obviously, S is not a convex set.
Definition 2.5
(see [14])
Let \(\eta: X\times X\rightarrow X\) be a map and \(F: X\rightarrow2^{Y}\) be a contingent epiderivable setvalued map at a point \((\bar{x},\bar{y})\in\operatorname {graph}(F)\) with \(\operatorname{dom} (DF(\bar{x},\bar{y}) )=X\). Then F is said to be a Dηinvex map at \((\bar{x},\bar{y})\) if
F is said to be a Dηinvex map at x̄, if (2.1) holds for any \(\bar{y}\in F(\bar{x})\).
Example 2.6
Let \(X=\mathbb{R}_{+}\), \(Y=\mathbb{R}^{2}\), \(D=\mathbb{R}^{2}_{+} \), and \(F: \mathbb{R}_{+}\rightarrow2^{\mathbb {R}{^{2}}}\) be defined by
and the epigraph of F
Let \(0=\bar{x}\in\mathbb{R}_{+}\) and \((0,0)=\bar{y}=(\bar{y_{1}},\bar {y_{2}})\in F(\bar{x})\). By calculating, we have
Hence, for any \(x\in\mathbb{R}_{+}\),
Therefore, \(DF(0,(0,0))\) exists for any \(x\in\mathbb{R}_{+}\) and for any map \(\eta: \mathbb{R}_{+}\times\mathbb{R}_{+}\rightarrow\mathbb{R}_{+}\), one has
Hence,
So, F is a Dηinvex map at \((\bar{x},\bar{y})= (0,(0,0) )\) with respect to any mapping η.
Definition 2.7
Suppose that \(\eta: X\times X\rightarrow X\), \(e\in\operatorname{int} (D)\), \(\varepsilon\geq0\), and \(F: X\rightarrow2^{Y}\) is a contingent epiderivable setvalued map at a point \((\bar{x},\bar{y})\in\operatorname{graph}(F)\) with \(\operatorname{dom} (DF(\bar{x},\bar{y}) )=X\). Then F is said to be a Dηsubinvex setvalued mapping at \((\bar{x},\bar{y})\) with respect to \(\varepsilon\cdot e\), if
F is said to be a Dηsubinvex map at x̄ with respect to \(\varepsilon\cdot e\), if (2.2) holds for any \(\bar{y}\in F(\bar {x})\).
Obviously, if F is a Dηinvex map at \((\bar{x},\bar{y})\in \operatorname{graph}(F) \), then F is a Dηsubinvex setvalued 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
Let \(X=\mathbb{R}\), \(Y=\mathbb{R}^{2}\), \(D=\mathbb{R}^{2}_{+} \), and \(F: \mathbb{R}\rightarrow2^{\mathbb{R}{^{2}}}\) be defined by
and the epigraph of F
Let \(0=\bar{x}\in\mathbb{R}\) and \((0,0)=\bar{y}=(\bar{y_{1}},\bar {y_{2}})\in F(\bar{x})\), we get
and
Thus, \(DF(0,(0,0))\) exists for any \(x\in\mathbb{R}\). However, for any map \(\eta: \mathbb{R}\times\mathbb{R}\rightarrow\mathbb{R}\), we have
Hence,
which indicates that F is not Dηinvex at \((\bar{x},\bar {y})= (0,(0,0) )\) with respect to any mapping η.
Now, choosing any η, \(e=(1,1)\in\operatorname{int}(\mathbb{R}^{2}_{+})\), and \(\varepsilon\geq4\) (or, \(e=(4,4)\in\operatorname{int}(\mathbb{R}^{2}_{+})\) and \(\varepsilon\geq1\)), we have
and
So, we get
Hence, F is a Dηsubinvex setvalued mapping at \((0,(0,0) )\) with respect to \(\varepsilon\cdot e\).
3 Sufficient optimality conditions
Let \(F: X\rightarrow2^{Y}\) and \(G: X\rightarrow2^{Z}\) be two setvalued maps with \(\operatorname{dom}(F)=\operatorname{dom}(G)=X\). In Section 3 and Section 4, we consider the following setvalued optimization problem:
A point \((\bar{x},\bar{y})\in X\times Y\) is said to be a feasible point of the problem (SOPI) if \(\bar{x}\in X\), \(\bar{y}\in F(\bar{x})\), and \(G(\bar{x})\cap(E)\neq\emptyset\). Let \(\Omega=\{x\in X: (x,y) \mbox{ is a feasible point of the problem (SOPI)}\}\). Then the weakly approximate minimizer for the setvalued optimization problem (SOPI) is defined in the following way.
Definition 3.1
(see [21])
(i) A point \(\bar{x}\in\Omega \) is said to be a weak efficient solution of the problem (SOPI), if there exists \(\bar{y}\in F(\bar{x})\) such that
and the pair \((\bar{x},\bar{y})\in\operatorname{graph}(F)\) is said to be a weak minimizer of (SOPI).
(ii) Let \(\varepsilon\geq0\) and \(e\in\operatorname{int} (D)\). A point \(\bar {x}\in\Omega\) is said to be a weak \(\varepsilon\cdot e\)efficient solution of the problem (SOPI), if there exists \(\bar{y}\in F(\bar {x})\) such that
and the pair \((\bar{x},\bar{y})\in\operatorname{graph}(F)\) is said to be a weak \(\varepsilon\cdot e\)minimizer of (SOPI).
Theorem 3.2
(Sufficient optimality condition)
Let \(e\in \operatorname{int}(D)\), \(\varepsilon\geq0\), and \((\bar{x},\bar{y})\in\operatorname {graph}(F)\) be a feasible point of the problem (SOPI) and \(\bar{z}\in G(\bar{x})\cap(E)\). Assume that the contingent epiderivatives \(DF(\bar {x},\bar{y})\) and \(DG(\bar{x},\bar{z})\) exist with \(\operatorname{dom} (DF(\bar{x},\bar{y}) )=\operatorname{dom} (DG(\bar{x},\bar{z}) )=X\). Let F be Dηsubinvex at \((\bar{x},\bar{y})\) with respect to \(\varepsilon\cdot e\) and G be Eηinvex at \((\bar{x},\bar{z})\) with respect to the same η. If there exists \((y^{*},z^{*})\in D^{*}\times E^{*}\) with \(y^{*}\neq0_{Y^{*}}\) such that
and
then \((\bar{x},\bar{y})\) is a weak \(\varepsilon\cdot e\)minimizer of the problem (SOPI).
Proof
Assuming that \((\bar{x},\bar{y})\) is not a weak \(\varepsilon\cdot e\)minimizer of the problem (SOPI),
Hence, there exist \(\hat{x}\in\Omega\) and \(\hat{y}\in F(\hat{x})\) such that
Noticing that \(y^{*}\in D^{*}\backslash\{0_{Y^{*}}\}\), we get
On the other hand, since F is Dηsubinvex at \((\bar{x},\bar {y})\) with respect to \(\varepsilon\cdot e\), we obtain
Therefore,
So, we get
and then
Again, since G is Eηinvex at \((\bar{x},\bar{z})\), we have
Because \(\hat{x}\in\Omega\), there exists an element \(\hat{z}\in G(\hat {x})\cap(E)\) such that
It follows from \(z^{*}\in E^{*}\) that
Since \(z^{*T}\hat{z}\leq0\) and \(z^{*T}\bar{z}=0\), we get \(z^{*T}(\hat {z}\bar{z})\leq0\) and
So, we have from (3.3) and (3.4)
which contradicts (3.1). Hence, \((\bar{x},\bar{y})\) is a weak \(\varepsilon\cdot e\)minimizer of the problem (SOPI). □
4 Duality theorems
4.1 MondWeir type duality
For the primal problem (SOPI), this subsection considers the MondWeir dual problem (MWD):
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
Let \(\varepsilon\geq0\) and \(e\in \operatorname{int}(D)\). A feasible point \((x',y',z',y^{*},z^{*})\) of the problem (MWD) is called a weak \(\varepsilon\cdot e\)maximizer of (MWD) if
Theorem 4.2
(Weak duality)
Let \(e\in\operatorname{int}(D)\), \(\varepsilon\geq0\), \((\bar{x},\bar{y})\), and \((x',y',z',y^{*},z^{*})\) be feasible points for (SOPI) and (MWD), respectively. Suppose that F is Dηsubinvex at \((x',y')\) with respect to \(\varepsilon\cdot e\) and G is Eηinvex at \((x',z')\) with respect to the same η. Then we have
Proof
We proceed by contradiction. Suppose that
Since \(y^{*}\in D^{*}\backslash\{0_{Y^{*}}\}\), we have
And because x̄ is feasible point for (SOPI), we get \(G(\bar {x})\cap(E)\neq\emptyset\). Hence, there exists \(\bar{z}\in G(\bar {x})\cap(E)\) such that \(z^{*T}\bar{z}\leq0\). On the other hand, we have from the dual constraint of (MWD) \(z^{*T}z'\leq0\). Therefore,
Now, since F is Dηsubinvex at \((x',y')\) with respect to \(\varepsilon\cdot e\) and G is Eηinvex at \((x',z')\), we derive
and
Therefore, we obtain
and
Furthermore, we find from (4.4), (4.5), and (4.6) that
which contradicts the dual constraint of (MWD). So, (4.1) is satisfied and this completes the 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 (SOPI). 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 (SOPI) and (MWD) holds, then \((\bar{x},\bar{y},\bar{z},y^{*},z^{*})\) is a weak \(\varepsilon\cdot e\)maximizer of (MWD).
Proof
Since (3.1) and (3.2) hold, it is obvious that \((\bar{x},\bar{y},\bar{z},y^{*},z^{*})\) is a feasible point for (MWD). Afterwards, we will prove that
In fact, assume that there exists \(y'\in K_{1}\) such that
This contradicts the weak duality theorem, Theorem 4.2, between (SOPI) and (MWD). □
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 Dηsubinvex at \((x',y')\) with respect to \(\varepsilon\cdot e\) and G is Eηinvex at \((x',z')\) with respect to the same η. Then \((x',y')\) is a weak \(\varepsilon\cdot e\)minimizer of the problem (SOPI).
Proof
Firstly, it is clearly that \((x',y')\) is a feasible point of the problem (SOPI). Next, assuming that \((x',y')\) is not a weak \(\varepsilon\cdot e\)minimizer of the problem (SOPI),
Hence, there are \(\hat{x}\in\Omega\) and \(\hat{y}\in F(\hat{x})\) such that
Noticing that \(y^{*}\in D^{*}\backslash\{0_{Y^{*}}\}\), we have
Again, because \(\hat{x}\in\Omega\), we get \(G(\hat{x})\cap(E)\neq \emptyset\). Taking \(\hat{z}\in G(\hat{x})\cap(E)\), we derive from \(z^{*}\in E^{*}\) that \(z^{*T}\hat{z}\leq0\). By the constraint of (MWD), we have \(z^{*T}z'\geq0\). Therefore, we get
Together with (4.7), we obtain
On the other hand, since F is Dηsubinvex at \((x',y')\) with respect to \(\varepsilon\cdot e\) and G is Eηinvex at \((x',z')\), it follows that
and
Thus, combining with (4.8), we get
So,
which contradicts the dual constraint of (MWD). Hence, \((x',y')\) is a weak \(\varepsilon\cdot e\)minimizer of the problem (SOPI). □
4.2 Wolfe type duality
Let us fix a point \(d_{0}\in D\backslash\{0_{Y}\}\) and consider the following problem (WD), called the Wolfe type dual problem of (SOPI):
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
Let \(\varepsilon\geq0\) and \(e\in \operatorname{int}(D)\). A feasible point \((x',y',z',y^{*},z^{*})\) of the problem (WD) is called to be a weak \(\varepsilon\cdot e\)maximizer of (WD) if
Theorem 4.6
(Weak duality)
Let \(e\in\operatorname{int}(D)\), \(\varepsilon\geq0\), \((\bar{x},\bar{y})\) and \((x',y',z',y^{*},z^{*})\) be feasible points for (SOPI) and (WD), respectively. Suppose that F is Dηsubinvex at \((x',y')\) with respect to \(\varepsilon\cdot e\) and G is Eηinvex at \((x',z')\) with respect to the same η. Then we have
Proof
Assume that
Since \(G(\bar{x})\cap(E)\neq\emptyset\), let \(\bar{z}\in G(\bar{x})\cap (E)\), \(z^{*T}\bar{z}\leq0\). Hence, we get \(z^{*T}\bar{z}\cdot d_{0}\in D\) and
Noticing that \(y^{*}\in D^{*}\backslash\{0_{Y^{*}}\}\) and \(y^{*T}d_{0}=1\), we have
This proves that inequality (4.4) holds. So, the rest of the proof follows from the same arguments as that of the weak duality theorem, Theorem 4.2, for the problem (MWD), we can still get
which also contradicts the dual constraint of (WD). Thus, \(\bar {y}y'z^{*T}z'\cdot d_{0}+\varepsilon\cdot e\notin\operatorname{int}(D)\), as desired. □
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 (SOPI). 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 (SOPI) and (WD) holds, then \((\bar{x},\bar {y},\bar{z},y^{*},z^{*})\) is a weak \(\varepsilon\cdot e\)maximizer of (WD).
Proof
Since (3.1) and (3.2) are fulfilled, it is obviously that \((\bar{x},\bar{y},\bar{z},y^{*},z^{*})\) is a feasible point for (WD). Next, we show that
Let \((x^{1},y^{1},z^{1},y_{1}^{*},z_{1}^{*})\) be a feasible point for (WD) such that \(y^{1}+z_{1}^{*T}z_{1}\cdot d_{0} \in K_{2}\) and
We find from \(z^{*T}\bar{z}=0\) that
This contradicts the weak duality theorem, Theorem 4.6, between (SOPI) and (WD). □
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 Dηsubinvex at \((x',y')\) with respect to \(\varepsilon\cdot e\) and G is Eηinvex at \((x',z')\) with respect to the same η. Then \((x',y')\) is a weak \(\varepsilon\cdot e\)minimizer of the problem (SOPI).
Proof
Obviously, \((x',y')\) is a feasible point of the problem (SOPI). Let \((x',y')\) be not a weak \(\varepsilon\cdot e\)minimizer of the problem (SOPI), then
Hence, there exist \(\hat{x}\in\Omega\) and \(\hat{y}\in F(\hat{x})\) such that
Noticing that \(y^{*}\in D^{*}\backslash\{0_{Y^{*}}\}\), we have
Again, since \(\hat{x}\in\Omega\), we get \(G(\hat{x})\cap(E)\neq \emptyset\). Let \(\hat{z}\in G(\hat{x})\cap(E)\), it follows from \(z^{*}\in E^{*}\) that \(z^{*T}\hat{z}\leq0\). Noticing that \(z^{*T}z'= 0\), we get
Hence, we obtain
which illustrates that (4.8) is satisfied. By the same arguments as that of the converse duality theorem, Theorem 4.4, for the problem (MWD), we also have
This also contradicts the dual constraint of (WD). Hence, \((x',y')\) is a weak \(\varepsilon\cdot e\)minimizer of the problem (SOPI). □
5 Vector optimization and variational inequality
This section is devoted to a discussion of the relationship between approximate solutions of setvalued 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 setvalued mapping. Considering the following setvalued optimization problem (SOPII):
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].
Definition 5.1
(i) The pair \((\bar{x},\bar{y})\in\operatorname{graph}(F)\) is called a weak efficient solution of the problem \((\mathrm{VVIP})_{\eta}\), if we have
(ii) Let \(e\in\operatorname{int}(D)\) and \(\varepsilon\geq0\). The pair \((\bar{x},\bar{y})\) is called a weak \(\varepsilon\cdot e\)efficient solution of the problem \((\mathrm{VVIP})_{\eta}\), if we have
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 (SOPII), then \((\bar{x},\bar{y})\) is a weak \(\varepsilon\cdot e\)efficient solution of \((\mathrm{VVIP})_{\eta}\).
Proof
Since the pair \((\bar{x},\bar{y})\in \operatorname{graph}(F)\) is a weak \(\varepsilon\cdot e\)minimizer of (SOPII), one has
We proceed by contradictions. Assume that there is an \(x'\in S\) with \(y'= D F(\bar{x},\bar{y}) (\eta(x',\bar{x}) ) \) such that
Denote \(\hat{F}(\cdot):=F(\cdot)+\varepsilon\cdot e\). By the definition of a contingent epiderivative, we get
Then there are a sequence \((x_{n},y_{n})_{n\in\mathbb{N}}\) in \(\operatorname{graph}(F)\) with \((x_{n},y_{n}+\varepsilon\cdot e)_{n\in\mathbb{N}}\) in \(\operatorname{epi}(\hat{F})\) and a sequence \((\lambda_{n})_{n\in\mathbb{N}}\) of positive real numbers with \((\bar{x},\bar{y})=\lim_{n\rightarrow\infty} (x_{n},y_{n}+\varepsilon\cdot e)\) and
Hence, we get
Because of the condition (5.2) and (5.3), there is an \(n_{0}\in\mathbb{N}\) with
This leads to
At the same time, since \((x_{n},y_{n}+\varepsilon\cdot e)\in \operatorname{epi}(\hat{F})\), there exists \(y_{n}'\in F(x_{n})\) and \(d_{n}\in D\) such that \(y_{n}+\varepsilon\cdot e=y_{n}'+\varepsilon\cdot e+d_{n}\). So, we get from (5.4)
Noticing that \(y_{n}'\in F(x_{n})\subset F(S)\), we get
This contradicts (5.1). The proof is completed. □
For the problem (SOPII), 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 (SOPII), 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 setvalued mapping \(F: S\rightarrow2^{Y}\) be Dηsubinvex 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 (SOPII).
Proof
By the assumptions, we get
Assuming that \((\bar{x},\bar{y})\) is not a weak \(\varepsilon\cdot e\)minimizer of (SOPII), then \(\hat{x}\in S\), \(\hat{y}\in F(\hat{x})\) such that
On the other hand, since F is Dηsubinvex on S with respect to \(\varepsilon\cdot e\), we see from Definition 2.7 that there is \(\hat{d}\in D\) such
Thus,
which contradicts (5.5). □
Remark 5.5
Theorem 5.4 generalizes and improves the result of Liu and Gong (see [35], Theorem 7) in the following aspects:

(1)
The constraint set which is a convex subset is extended to the invex set.

(2)
The objective function, that is, a coneconvex setvalued mapping, is extended to conesubinvex.
Remark 5.6
This note only presents the relationships between a kind of generalized variationallike inequalities and setvalued optimization problem. However, we do not discuss the relationships of other kinds of variationallike inequalities and setvalued optimization, and the existence problems of variationallike 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 setvalued optimization. We present the notion of conesubinvex setvalued maps and investigate its properties. A sufficient optimality condition and two types dual theorems are established for weakly approximate minimizers under the assumption of conesubinvexity. We also discuss the relationships between a kind of vector variational inequality and setvalued optimization. Under the assumption conesubinvexity, it shows that the weakly approximate minimizers of setvalued 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 setvalued map, which includes the contingent epiderivative as its special case. They provided necessary and sufficient conditions for a solution of a setvalued problems and some existence results for solutions of setvalued optimization problems and a generalized vector \(\mathbb{T}\)inequality problem under the assumption of coneconvexity for setvalued maps. It is possible to extend the notion of conesubinvexity in the setting of \(\mathbb {T}\)epiderivative and to deal with similar problems for approximate solutions in setvalued optimization, such as optimality conditions and duality. This must be an interesting and meaningful work.
References
Aubin, JP, Frankowsa, H: SetValued Analysis. Birkhäuser, Boston (1990)
Luc, DT: Theory of Vector Optimization. Springer, Berlin (1989)
Jahn, J: Vector Optimization: Theory, Applications and Extensions. Springer, Berlin (2003)
Chen, GY, Huang, XX, Yang, XQ: Vector Optimization: SetValued and Variational Analysis. Springer, Berlin (2005)
Jahn, J, Rauh, R: Contingent epiderivatives and setvalued optimization. Math. Methods Oper. Res. 46, 193211 (1997)
Jahn, J, Rauh, R: The existence of contingent epiderivatives for setvalued maps. Appl. Math. Lett. 16(8), 11791185 (2003)
Cheng, GY, Jahn, J: Optimality conditions for setvalued optimization problem. Math. Methods Oper. Res. 48, 187200 (1998)
RodríguezMarín, L, Sama, M: About contingent epiderivatives. J. Math. Anal. Appl. 327(2), 754762 (2007)
RodríguezMarín, L, Sama, M: Variational characterization of the contingent epiderivative. J. Math. Anal. Appl. 335(2), 13741382 (2007)
Gong, XH: Optimality conditions for Henig and globally proper efficient solutions with ordering cone has empty interior. J. Math. Anal. Appl. 307, 1231 (2005)
Jiménez, B, Novo, V, Sama, M: Scalarization and optimality conditions for strict minimizers in multiobjective optimization via contingent epiderivatives. J. Math. Anal. Appl. 352(2), 788798 (2009)
Yu, GL: Optimality of global proper efficiency for conearcwise connected setvalued optimization using contingent epiderivative. AsiaPac. J. Oper. Res. 30(3), 1340004 (2013)
Hanson, MA: On sufficiency of KuhnTucker conditions. J. Math. Anal. Appl. 80, 545550 (1981)
Luc, DT, Malivert, C: Invex optimization problems. Bull. Aust. Math. Soc. 49, 4766 (1992)
Sach, PH, Craven, BD: Invex multifunction and duality. Numer. Funct. Anal. Optim. 12, 575591 (1991)
Lalitha, CS, Dutta, J, Govll, MG: Optimality criteria in setvalued optimization. J. Aust. Math. Soc. 75, 221231 (2003)
Alonso, M, RodríguezMarín, L: Optimality conditions for a nonconvex setvalued optimization problem. Comput. Math. Appl. 56, 8289 (2008)
Oveisiha, M, Zafarani, J: On characterization of solution sets of setvalued pseudoinvex optimization problems. J. Optim. Theory Appl. 163(2), 387398 (2014)
Das, K, Nahak, C: Sufficient optimality conditions and duality theorems for setvalued optimization problem under generalized cone convexity. Rend. Circ. Mat. Palermo 63, 329345 (2014)
Davinder, B: Lagrangian duality for preinvex setvalued functions. J. Math. Anal. Appl. 214, 599612 (1997)
Long, XJ, Peng, JW: Lagrangian duality for vector optimization problems with setvalued mappings. Taiwan. J. Math. 17(1), 287297 (2013)
Azimov, AY: Duality for setvalued multiobjective optimization problems, part 1: mathematical programming. J. Optim. Theory Appl. 137, 6174 (2008)
Heydea, F, Schrage, C: Continuity concepts for setvalued functions and a fundamental duality formula for setvalued optimization. J. Math. Anal. Appl. 397, 772784 (2013)
Sach, PH, Lee, GM, Kim, DS: Strong duality for proper efficiency in vector optimization. J. Optim. Theory Appl. 130, 139151 (2006)
Sach, PH, Tuan, LA: Strong duality with proper efficiency in multiobjective optimization involving nonconvex setvalued maps. Numer. Funct. Anal. Optim. 30, 371392 (2009)
Giannessi, F: Variational Inequalities and Complementarity Problems. Wiley, New York (1980)
AlHomidan, S, Ansari, QH: Relations between generalized vector variationallike inequalities and vector optimization problems. Taiwan. J. Math. 16(3), 987998 (2012)
Ansari, QH, Rezaei, M, Zafarani, J: Generalized vector variationallike inequalities and vector optimization. J. Glob. Optim. 53(2), 271284 (2012)
Jayswal, A, Choudhury, S, Verma, RU: Exponential type vector variationallike inequalities and vector optimization problems with exponential type invexities. J. Appl. Math. Comput. 45, 8797 (2014)
Jayswal, A, Choudhury, S: Exponential type vector variationallike inequalities and nonsmooth vector optimization problems. J. Appl. Math. Comput. (2014). doi:10.1007/s1219001408288
AlonsoDurán, M, RodríguezMarín, L: On approximate solutions in setvalued optimization problems. J. Comput. Appl. Math. 236, 44214427 (2012)
Gao, Y, Hou, SH, Yang, XM: Existence and optimality conditions for approximate solutions to vector optimization problems. J. Optim. Theory Appl. 152, 97120 (2012)
Long, XJ, Li, XB, Zeng, J: Lagrangian conditions for approximate solutions on nonconvex setvalued optimization problems. Optim. Lett. 7, 18471856 (2013)
Zhou, ZA, Yang, XM, Peng, JW: εOptimality conditions of vector optimization problems with setvalued maps based on the algebraic interior in real linear spaces. Optim. Lett. 8, 10471061 (2014)
Liu, W, Gong, XH: Proper efficiency for setvalued vector optimization problems and vector variational inequalities. Math. Methods Oper. Res. 51, 443457 (2000)
Ansari, QH, Jahn, J: \(\mathbb{T}\)Epiderivative of setvalued maps and its application to set optimization and generalized variational inequalities. Taiwan. J. Math. 14(6), 24472468 (2010)
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.
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Yu, G., Kong, X. Approximate solutions for nonconvex setvalued optimization and vector variational inequality. J Inequal Appl 2015, 324 (2015). https://doi.org/10.1186/s1366001508395
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DOI: https://doi.org/10.1186/s1366001508395