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Globally proper efficiency of set-valued optimization and vector variational inequality involving the generalized contingent epiderivative
- Wang Chen^{1} and
- Zhiang Zhou^{1}Email authorView ORCID ID profile
https://doi.org/10.1186/s13660-018-1891-8
© The Author(s) 2018
- Received: 9 August 2018
- Accepted: 25 October 2018
- Published: 3 November 2018
Abstract
In this paper, firstly, a new property of the cone subpreinvex set-valued map involving the generalized contingent epiderivative is obtained. As an application of this property, a sufficient optimality condition for constrained set-valued optimization problem in the sense of globally proper efficiency is derived. Finally, we establish the relations between the globally proper efficiency of the set-valued optimization problem and the globally proper efficiency of the vector variational inequality.
Keywords
- Set-valued optimization
- Generalized contingent epiderivative
- Cone subpreinvex set-valued map
- Globally proper efficiency
- Vector variational inequality
MSC
- 90C26
- 90C29
- 90C30
- 90C46
1 Introduction
It is well known that convexity plays a crucial role in set-valued optimization. To generalize convexity of set-valued maps, some scholars introduced different kinds of generalized convex set-valued maps. Bhatia and Mehra [1] introduced the cone preinvex set-valued function which is a generalization of the cone convex set-valued map and derived a Lagrangian type duality for a fractional programming problem involving the cone preinvex set-valued map. Jia [2] defined the cone subpreinvex set-valued map and discussed the optimality condition and duality of set-valued optimization problems. Qiu [3] proposed the generalized cone preinvex set-valued map and used the cone-directed contingent derivative given by [4] to obtain necessary and sufficient optimality conditions of set-valued optimization problems in the sense of weak efficiency and strong efficiency, respectively. Zhou et al. [5], in real normed spaces, introduced a new concept of generalized cone convex set-valued map and established optimality conditions of set-valued optimization problem in the sense of Henig proper efficiency by applying the contingent epiderivative proposed by [6] and the generalized cone convexity of set-valued maps. Other generalized convexity of set-valued maps and their applications can be found in [7–10] and references therein.
On the other hand, Giannessi [11] initially introduced the notion of vector variational inequality in the finite dimensional Euclidean space. Since then, different types of extensions and generalizations of variational inequalities have been proposed in different settings, see [6, 12–27] and references therein. In [6], optimality conditions are given by a certain kind of vector variational inequality in real normed spaces. We notice that such vector variational inequality is characterized by the contingent epiderivative of a set-valued map. Later, by virtue of the contingent epiderivative of set-valued maps, Liu and Gong [13] investigated the relations between the proper efficiency of set-valued optimization problem (for short, (SVOP)) and the proper efficiency of vector variational inequality based on the convexity assumption. By considering the generalized cone preinvexity and the contingent epiderivative of set-valued maps, Yu [20] disclosed the relations between Henig global efficiency of (SVOP) and Henig global efficiency of a kind of vector variational inequality. Yu and Kong [21] discussed the relations 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 characterized by the contingent epiderivative. We remark that the contingent epiderivative and the cone convexity (or the generalized cone convexity) of set-valued maps are used to deal with the relations between proper efficiency of (SVOP) and proper efficiency of vector variational inequality. However, it is worth noting that Chen and Jahn [12] pointed out that the existence of the contingent epiderivative of a set-valued map in a general setting is still an open question. To overcome this difficulty, Chen and Jahn [12] introduced a generalized contingent epiderivative of set-valued maps and derived the existence of the generalized contingent epiderivative under some standard assumptions. Therefore, it is interesting to study the optimality conditions of (SVOP) in the sense of globally proper efficiency via the generalized contingent epiderivative.
Motivated by the works [2, 5, 6, 13, 20, 21, 28], in this paper, we first propose a property of the cone subpreinvex set-valued map involving the generalized contingent epiderivative of set-valued maps which improves and generalizes the corresponding results in [5, 12]. A example is presented to illustrate this property. Second, we introduce a new type of generalized vector variational inequality problem (shortly, (GVVIP)) by virtue of the generalized contingent epiderivative of set-valued maps, propose the notion of globally proper efficiency of (GVVIP) and obtain the optimality conditions of the globally proper efficiency of (GVVIP).
This paper is organized as follows. In Sect. 2, we recall some definitions including the generalized contingent epiderivative of set-valued maps, the globally proper efficient point of a set and some generalized cone convexity of set-valued maps. In Sect. 3, a new property of the cone subpreinvex set-valued map is obtained and a sufficient optimality condition of constrained set-valued optimization problem in the sense of globally proper efficiency is derived. In Sect. 4, the relations between the globally proper efficiency of (SVOP) and the globally proper efficiency of (GVVIP) are disclosed.
2 Preliminaries
Throughout the paper, \(\mathbb{R}_{+}^{m}\) represents the nonnegative orthant of \(\mathbb{R}^{m}\) and \(\mathbb{R}_{+}:=\mathbb{R}_{+}^{1}\), where \(\mathbb{R}^{m}\) represents the m-dimensional Euclidean space. Let X be a linear space, Y and Z be two real normed spaces, respectively. For a set \(K\subset Y\), intK denotes the interior of K. Let 0 denote the zero element for every space.
Now, we recall some basic definitions which will be used in the sequel.
Definition 2.1
([29])
Let Ω be a nonempty subset in X, \(\bar{z}\in \operatorname{cl}\varOmega\) and \(h\in X\). The contingent cone of Ω at z̄ is \(T(\varOmega,\bar{z}):=\{h\in X|\exists t_{n}\downarrow 0,\exists h_{n}\rightarrow h,\mbox{ such that }\bar{z}+t_{n}h_{n}\in\varOmega ,\forall n\in\mathbb{N}\}\), or equivalently, \(T(\varOmega,\bar{z}):=\{h\in X|\exists\lambda_{n}\rightarrow+\infty,\exists z_{n}\in\varOmega,\mbox{ such that }\lim_{n\rightarrow\infty}z_{n}=\bar{z}\mbox{ and }\lim_{n\rightarrow \infty}\lambda_{n}(z_{n}-\bar{z})=h\}\).
Definition 2.2
- (i)
A point \(\bar{y}\in S\) is called a minimal point of S iff \((\bar{y}-C)\cap S=\{\bar{y}\}\). The set of all minimal points of S with respect to C is denoted by MinS.
- (ii)
A point \(\bar{y}\in S\) is called a globally proper efficient point of S iff there exists a pointed convex cone \(C'\subset Y\) with \(C\backslash\{0\}\subset\operatorname{int}C'\) such that \((S-\bar{y})\cap(-C'\backslash\{0\})=\emptyset\). The set of all globally proper efficient points of S with respect to C is denoted by \(\operatorname{GPE}(S,C)\).
Remark 2.1
Notice that Yu and Liu [32] presented an equivalent characterization of the globally proper efficient point: For a nonempty subset S in Y, a point \(\bar{y}\in S\) is called a globally proper efficient point of S iff there exists a pointed convex cone \(C'\subset Y\) with \(C\backslash\{0\}\subset\operatorname{int}C'\) such that \((S-\bar{y})\cap(-\operatorname{int}C')=\emptyset\).
Definition 2.3
([30])
- (a)
The cone C is called Daniell iff any decreasing sequence in Y having a lower bound converges to its infimum.
- (b)
A subset A of Y is said to be minorized iff there exists \(y\in Y\) such that \(A\subset\{y\}+C\).
- (c)
The domination property holds for a subset A of Y iff \(A\subset\operatorname{Min}A+C\).
Definition 2.4
([12])
Definition 2.5
([33])
A subset \(\varGamma\subset X\) is called an invex set with respect to \(\eta:\varGamma\times\varGamma\rightarrow X\) iff for each \(x,y\in \varGamma\) and \(\lambda\in[0,1]\), \(y+\lambda\eta(x,y)\in\varGamma\).
Remark 2.2
Obviously, a convex set is an invex set by taking \(\eta(x,y)=x-y\) in Definition 2.5. In general, the converse is not true (see Example 2.1 in [34]).
Definition 2.6
([1])
Remark 2.3
It is clear that the C-convexity of the set-valued map F is the C-preinvexity of the set-valued map F. However, the converse is not necessarily true (see Example 2.1 in [1]).
Now, we will give the concept of cone subpreinvex set-valued maps, which will be needed in the sequel.
Definition 2.7
([2])
Remark 2.4
Clearly, the C-preinvex set-valued map on Γ is a C-subpreinvex set-valued map on Γ. However, Example 2.1 shows that the converse is not necessarily true. Therefore, the C-subpreinvex set-valued map is a proper generalization of the C-preinvex set-valued map.
Example 2.1
Remark 2.5
Zhou et al. [5] introduced the concept of generalized C-convex set-valued functions (see Defintion 2.4 in [5]) and pointed out that the generalized C-convex set-valued map is a proper generalization of the C-convex set-valued map (see Remark 2.1 in [5]). Clearly, if \(\eta(x,y)=x-y\), that is, the invex set Γ is a convex set, then Definition 2.7 reduces to Definition 2.4 in [5]. However, if the invex set Γ is not a convex set in X, then the C-subpreinvexity of set-valued maps cannot imply the generalized C-convexity of set-valued maps. Therefore, the set-valued map F in Example 2.1 is not the generalized C-convex on Γ.
3 Optimality condition
In this section, firstly, we will use the generalized contingent epiderivative of set-valued maps to present a property of the C-subpreinvex set-valued map. Secondly, as an application of this property, we will give a sufficient optimality condition of constrained set-valued optimization problem in the sense of globally proper efficiency.
Theorem 3.1
Proof
Remark 3.1
In Theorem 3.1, the conditions that C is Daniell and \(\varPhi(\eta(x,\bar{x}))\) is minorized guarantee the existence of \(D_{g}F(\bar{x},\bar{y})(\eta(x,\bar{x}))\).
Remark 3.2
Notice that Theorem 3.1 generalizes and improves Lemma 1 in [12] and Theorem 3.1 in [5] in the following aspects: (i) The convex set and the C-convexity of Lemma 1 in [12] are extended to the invex set and the C-subpreinvexity of Theorem 3.1, respectively. (ii) The convex set, the C-convexity and the contingent epiderivative of Theorem 3.1 in [5] are replaced by the invex set, the C-subpreinvexity and the generalized contingent epiderivative of Theorem 3.1, respectively.
The following example is used to illustrate Theorem 3.1.
Example 3.1
Definition 3.1
([31])
\(\bar{x}\in E\) is called a globally proper efficient solution of (SVOP) iff there exists \(\bar{y}\in F(\bar{x})\) such that \(\bar{y}\in\operatorname{GPE}(F(E),C)\). The pair \((\bar{x},\bar{y})\) is called a globally proper efficient element of (SVOP).
Definition 3.2
([35])
Let \(\bar{x}\in E\) and \(\bar{y}\in F(\bar{x})\). \((\bar {x},\bar{y})\) is called a positive properly efficient element of (SVOP) iff there exists \(\mu\in C^{+i}\) such that \(\mu(F(x)-\bar{y})\geq0\) for all \(x\in E\).
Lemma 3.1
([35])
A positive properly efficient element of (SVOP) must be a globally proper efficient element of (SVOP).
By applying Lemma 3.1, Gong et al. [28] obtained a sufficient condition involving multiplier functionals for a globally proper efficient solution of (SVOP) (see Theorem 3.4 in [28]). Next, we will use Lemma 3.1 to establish a sufficient optimality condition characterized by the generalized contingent epiderivative of set-valued maps in the sense of globally proper efficiency.
Theorem 3.2
- (i)
The set-valued map \((F,G)\) is \(C\times D\)-subpreinvex on Γ with respect to η;
- (ii)There exists \((\mu,\nu)\in C^{+i}\times D^{+}\) such thatand$$\begin{aligned} (\mu,\nu) \bigl(D_{g}(F,G) (\bar{x},\bar{y},\bar{z}) \bigl(\eta(x,\bar{x})\bigr)\bigr)\geq 0,\quad \forall x\in\varGamma \end{aligned}$$(3.3)Then \((\bar{x},\bar{y})\) is a globally proper efficient element of (SVOP).$$\begin{aligned} \nu(\bar{z})=0. \end{aligned}$$(3.4)
Proof
4 Generalized vector variational inequality
In this section, we will introduce a generalized vector variational inequality problem (GVVIP) and disclose the close relations between the globally proper efficiency of (SVOP) and the globally proper efficiency of (GVVIP).
Definition 4.1
Remark 4.1
When \(\eta(x,\bar{x})=x-\bar{x}\) and the generalized contingent epiderivative of the set-valued map becomes the contingent epiderivative of the set-valued map, Definition 4.1 reduces to Definition 18 in [13] and Definition 2.10 in [20].
We will use the standard assumptions: Let C be a closed pointed convex cone being Daniell and \(\operatorname{int}C\neq\emptyset\), and suppose that \(\varPhi(\eta(x,\bar{x}))\) given by Theorem 3.1 is minorized and fulfills the domination property for any \(x\in\varGamma\).
Theorem 4.1
Let the standard assumptions hold. If \((\bar{x},\bar{y})\) is a globally proper efficient element of (SVOP), then \((\bar{x},\bar{y})\) is a globally proper efficient element of (GVVIP).
Proof
Remark 4.2
It is worth noting that the contingent epiderivative of Theorem 8 in [13] is replaced with the generalized contingent epiderivative. Therefore, Theorem 4.1 improves Theorem 8 in [13].
Theorem 4.2
Let the standard assumptions hold. Let Γ be a nonempty invex subset in X with respect to η and the set-valued map \(F:\varGamma \rightrightarrows2^{Y}\) be C-subpreinvex on Γ with respect to η. If \((\bar{x},\bar{y})\) is a globally proper efficient element of (GVVIP), then \((\bar{x},\bar{y})\) is a globally proper efficient element of (SVOP).
Proof
5 Conclusions
In this paper, based on the generalized contingent epiderivative of set-valued maps, we obtained a new property of the cone subpreinvex set-valued map. By applying this property, we derived a sufficient optimality condition in the sense of globally proper efficiency in the constrained set-valued optimization problem. We also introduced a new kind of generalized variational inequality problem. Moreover, the relations between the globally proper efficiency of the set-valued optimization problem and the globally proper efficiency of the generalized variational inequality problem are disclosed. These results are new and are extensions of the corresponding ones in set-valued optimization.
Declarations
Authors’ information
Zhiang Zhou (1972-), Professor, Doctor, the major field of interest is in the area of set-valued optimization.
Funding
This research was supported by the National Nature Science Foundation of China (11431004, 11861002, 11471291), the Key Project of Chongqing Frontier and Applied Foundation Research (cstc2017jcyjBX0055, cstc2015jcyjBX0113) and the Graduate Innovation Foundation of Chongqing University of Technology (ycx2018256).
Authors’ contributions
All authors contributed to each part of this work equally, and they all read and approved the final manuscript.
Competing interests
The authors declare that they have no competing interests.
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
References
- Bhatia, D., Mehra, A.: Lagrangian duality for preinvex set-valued functions. J. Math. Anal. Appl. 214(2), 599–612 (1997) MathSciNetView ArticleGoogle Scholar
- Jia, J.H.: Vector optimization of subpreinvex of set-valued maps. J. Chang’an Univ. (Natural Science Edition) 22(4), 100–102 (2002) Google Scholar
- Qiu, J.H.: Cone-directed contingent derivatives and generalized preinvex set-valued optimization. Acta Math. Sci. 27B(1), 211–218 (2007) MathSciNetView ArticleGoogle Scholar
- Corley, H.W.: Optimality conditions for maximizations of set-valued functions. J. Optim. Theory Appl. 58(1), 1–10 (1988) MathSciNetView ArticleGoogle Scholar
- Zhou, Z.A., Yang, X.M., Qiu, Q.S.: Optimality conditions of set-valued optimization problem with generalized cone convex set-valued maps characterized by contingent epiderivative. Acta Math. Appl. Sin. Engl. Ser. 34(1), 11–18 (2018) MathSciNetView ArticleGoogle Scholar
- Jahn, J., Rauh, R.: Contingent epiderivatives and set-valued optimization. Math. Methods Oper. Res. 46(2), 193–211 (1997) MathSciNetView ArticleGoogle Scholar
- Yang, X.M., Li, D., Wang, S.Y.: Near-subconvexlikeness in vector optimization with set-valued functions. J. Optim. Theory Appl. 110(2), 413–427 (2001) MathSciNetView ArticleGoogle Scholar
- Sach, P.H.: New generalized convexity notion for set-valued maps and application to vector optimization. J. Optim. Theory Appl. 125(1), 157–179 (2005) MathSciNetView ArticleGoogle Scholar
- Zhao, K.Q., Yang, X.M., Peng, J.W.: Weak E-optimal solution in vector optimization. Taiwan. J. Math. 17(4), 1287–1302 (2013) MathSciNetView ArticleGoogle Scholar
- Zhou, Z.A., Yang, X.M., Wan, X.: The semi-E cone convex set-valued map and its applications. Optim. Lett. 12(6), 1329–1337 (2018) MathSciNetView ArticleGoogle Scholar
- Giannessi, F.: Variational Inequalities and Complementarity Problems. Wiley, New York (1980) MATHGoogle Scholar
- Chen, G.Y., Jahn, J.: Optimality conditions for set-valued optimization problems. Math. Methods Oper. Res. 48(2), 187–200 (1998) MathSciNetView ArticleGoogle Scholar
- Liu, W., Gong, X.H.: Proper efficiency for set-valued vector optimization problems and vector variational inequalities. Math. Methods Oper. Res. 51(3), 443–457 (2000) MathSciNetView ArticleGoogle Scholar
- Yang, X.M., Yang, X.Q.: Vector variational-like inequality with pseudoinvexity. Optimization 55(1–2), 157–170 (2006) MathSciNetView ArticleGoogle Scholar
- Xiao, Y.B., Huang, N.J., Cho, Y.J.: A class of generalized evolution variational inequalities in Banach space. Appl. Math. Lett. 25(6), 914–920 (2012) MathSciNetView ArticleGoogle Scholar
- Xiao, Y.B., Sofonea, M.: On the optimal control of variational-hemivariational inequalities. J. Math. Anal. Appl. (to appear) Google Scholar
- Long, X.J., Quan, J., Wen, D.J.: Proper efficiency for set-valued optimization problems and vector variational-like inequalities. Bull. Korean Math. Soc. 50(3), 777–786 (2013) MathSciNetView ArticleGoogle Scholar
- Jayswal, A., Choudhury, S., Verma, R.U.: Exponential type vector variational-like inequalities and vector optimization problems with exponential type invexities. J. Appl. Math. Comput. 45(1–2), 87–97 (2014) MathSciNetView ArticleGoogle Scholar
- Xiao, Y.B., Yang, X.M., Huang, N.J.: Some equivalence results for well-posedness of hemivariational inequalities. J. Glob. Optim. 61(4), 789–802 (2015) MathSciNetView ArticleGoogle Scholar
- Yu, G.L.: Henig global efficiency for set-valued optimization and variational inequality. J. Syst. Sci. Complex. 27(2), 338–349 (2014) MathSciNetView ArticleGoogle Scholar
- Yu, G.L., Kong, X.Y.: Approximate solutions for nonconvex set-valued optimization and vector variational inequality. J. Inequal. Appl. 2015, 324 (2015) MathSciNetView ArticleGoogle Scholar
- Li, W., Xiao, Y.B., Huang, N.J., Cho, Y.J.: A class of differential inverse quasi-variational inequalities in finite dimensional spaces. J. Nonlinear Sci. Appl. 10(8), 4532–4543 (2017) MathSciNetView ArticleGoogle Scholar
- Ansari, Q.H., Rezaei, M., Zafarani, J.: Generalized vector variational-like inequalities and vector optimization. J. Glob. Optim. 53(2), 271–284 (2012) MathSciNetView ArticleGoogle Scholar
- Lu, J., Xiao, Y.B., Huang, N.J.: A Stackelberg quasi-equilibrium problem via quasi-variational inequalities. Carpath. J. Math. 34(3), 355–362 (2018) Google Scholar
- Ansari, Q.H., Jahn, J.: \(\mathbb{T}\)-epiderivative of set-valued maps and its application to set optimization and generalized variational inequalities. Taiwan. J. Math. 14(6), 2447–2468 (2010) MathSciNetView ArticleGoogle Scholar
- Shu, Q.Y., Hu, R., Xiao, Y.B.: Metric characterizations for well-posedness of split hemivariational inequalities. J. Inequal. Appl. (2018). https://doi.org/10.1186/s1/s13660-018-1761-4 MathSciNetView ArticleGoogle Scholar
- Hu, R., Xiao, Y.B., Huang, N.J., Wang, X.: Equivalence results of well-posedness for split variational-hemivariational inequalities. J. Nonlinear Convex Anal. (to appear) Google Scholar
- Gong, X.H., Dong, H.B., Wang, S.Y.: Optimality conditions for proper efficient solutions of vector set-valued optimization. J. Math. Anal. Appl. 274(1), 332–350 (2003) MathSciNetView ArticleGoogle Scholar
- Aubin, J.P., Frankowska, H.: Set-Valued Analysis. Birkhäuser, Boston (1990) MATHGoogle Scholar
- Luc, D.T.: Theory of Vector Optimization. Springer, Berlin (1989) View ArticleGoogle Scholar
- Henig, M.I.: Proper efficiency with respect to cones. J. Optim. Theory Appl. 36(3), 387–407 (1982) MathSciNetView ArticleGoogle Scholar
- Yu, G.L., Liu, S.Y.: Optimality conditions of globally proper efficient solutions for set-valued optimization problem. Acta Math. Appl. Sin. 33(1), 150–160 (2010) MathSciNetMATHGoogle Scholar
- Weir, T., Mond, B.: Preinvex functions in multiple-objective optimization. J. Math. Anal. Appl. 136(1), 29–38 (1988) MathSciNetView ArticleGoogle Scholar
- Yang, X.M., Li, D.: On properties of preinvex functions. J. Math. Anal. Appl. 256(1), 229–241 (2001) MathSciNetView ArticleGoogle Scholar
- Dauer, J.P., Gallagher, R.J.: Positive proper efficient points and related cone result in vector optimization theory. SIAM J. Control Optim. 28(1), 158–172 (1990) MathSciNetView ArticleGoogle Scholar
- Huang, Y.W.: Generalized constraint qualifications and optimality conditions for set-valued optimization problems. J. Math. Anal. Appl. 265(2), 309–321 (2002) MathSciNetView ArticleGoogle Scholar