Set-valued Vector Optimization Problems Research Articles

On Sub Convexlike Optimization Problems

In this paper, we show that the sub convexlikeness and subconvexlikeness defined by V. Jeyakumar are equivalent in locally convex topological spaces. We also deal with set-valued vector optimization problems and obtained vector saddle-point theorems and vector Lagrangian theorems.

Nonlinear Scalarization Characterizations of E-Super Efficient Solution of Set-valued Vector Optimization Problem

Abstract: In the study of vector optimization problems, the concept of (weakly) effective solution based on the definition of order cone and its properties play a very important role. The concept of superefficient solution is the unification of several existing concept of properly efficient solutions, while the E- super efficient solution of the vector optimization problem proposed based on the improvement set is an important extension of the classical concept of superefficient solution, which unifies the concepts of superefficient solution and ε-super efficient solution. Therefore, it is of great significance to study the E- super efficient solution and its related properties for the study of vector optimization problems. This paper mainly focuses on the relevant conditions of nonlinear scalarization characterization of E- super efficient solution: firstly, a sufficient and necessary condition for the nonlinear scalarization characterization about the E- super efficient solution of the set-valued vector optimization problem is obtained through the corresponding property of the seminorm p; secondly, the corresponding property of the distance function d is used to obtain another sufficient and necessary condition for the nonlinear scalarization characterization of the E- super efficient solution; then, according to the relationship between the E-optimal solution and the E-super efficient solution, a sufficient condition for the results of nonlinear scalarization characterization of the E- super efficient solution is derived by means of the distance-oriented function Δ-E which based on the improved set E under the condition of E=E+K0; finally, under the condition of 0∈clE, the distance-oriented function Δ-E is used to obtain a necessary condition for the nonlinear scalarization result of E- super efficient solution, and the corresponding sufficient and necessary result is deduced.

On the Existence of Super Efficient Solutions and Optimality Conditions for Set-Valued Vector Optimization Problems

In this paper, by using the normal subdifferential and equilibrium-like function we first obtain some properties for K-preinvex set-valued maps. Secondly, in terms of this equilibrium-like function, we establish some sufficient conditions for the existence of super minimal points of a K-preinvex set-valued map, that is, super efficient solutions of a set-valued vector optimization problem, and also attain necessity optimality terms for a general type of super efficiency.

Semicontinuity of the Minimal Solution Set Mappings for Parametric Set-Valued Vector Optimization Problems

With the help of a level mapping, this paper mainly investigates the semicontinuity of minimal solution set mappings for set-valued vector optimization problems. First, we introduce a kind of level mapping which generalizes one given in Han and Gong (Optimization 65:1337–1347, 2016). Then, we give a sufficient condition for the upper semicontinuity and the lower semicontinuity of the level mapping. Finally, in terms of the semicontinuity of the level mapping, we establish the upper semicontinuity and the lower semicontinuity of the minimal solution set mapping to parametric set-valued vector optimization problems under the C-Hausdorff continuity instead of the continuity in the sense of Berge.

Subdifferential Calculus for Set-Valued Mappings and Optimality Conditions for Multiobjective Optimization Problems

In this work, we provide a generalized formula for the weak subdifferential (resp., for the Benson proper subdifferential) of the sum of two cone-closed and cone-convex set-valued mappings, under the Attouch–Brezis qualification condition. This formula is applied to establish necessary and sufficient optimality conditions in terms of Lagrange/Karush/Kuhn/Tucker multipliers for the existence of the weak (resp., of the Benson proper) efficient solutions of a set-valued vector optimization problem.

Continuity of solutions mappings of parametric set optimization problems

<p style='text-indent:20px;'>Set optimization is an indispensable part of theory and method of optimization, and has been received wide attentions due to its extensive applications in group decision and group game problems. This paper focus on the continuity of the strict (weak) minimal solution set mapping of parametric set-valued vector optimization problems with the lower set less order relation. We firstly introduce a concept of strict lower level mapping of parametric set-valued vector optimization problems. Moreover, the upper and lower semicontinuity of the strict lower level mapping are obtained under some suitable conditions. Lastly, the sufficient condition for the continuity of the strict minimal solution set mappings of parametric set optimization problems are established by a new proof method, which is different from that in [<xref ref-type="bibr" rid="Xu2014">27</xref>,<xref ref-type="bibr" rid="Xu2016">28</xref>].

On the solution continuity of parametric set optimization problems

The aim of this paper is to investigate the continuity of the solution set maps of set-valued vector optimization problems with set optimization criterion. First, we introduce a new concept, which is called a u-lower level map. Then, we give some sufficient conditions for the upper and lower semicontinuities of the generalized lower level map. Finally, by virtue of the semicontinuity of the u-lower level map, we obtain the continuity of the minimal solution set map to parametric set-valued vector optimization problems with set optimization criterion.

A new nonlinear scalarization function and applications

In this paper, a new nonlinear scalarization function, which is a generalization of the oriented distance function, is introduced. Some properties of the function are discussed. Then the function is applied to obtain some new optimality conditions and scalar representations for set-valued vector optimization problems with set optimization criteria. In terms of the function and the image space analysis, some new alternative results for generalized parametric systems are derived.

Continuity of the solution set mappings to a parametric set optimization problem

In this paper, we establish the upper, lower semicontinuity and closedness of the minimal solution and weak minimal solution set mappings to a parametric set-valued vector optimization problem with set optimization criterion under some suitable assumptions.

Optimality conditions for vector optimization problems with variable ordering structures

Our main concern in this article are concepts of nondominatedness w.r.t. a variable ordering structure introduced by Yu [P.L. Yu, Cone convexity, cone extreme points, and nondominated solutions in decision problems with multiobjectives, J. Optim. Theory Appl. 14 (1974), pp. 319–377]. Our studies are motivated by some recent applications e.g. in medical image registration. Restricting ourselves to the case when the values of a cone-valued map defining the ordering structure are Bishop–Phelps cones, we obtain for the first time scalarizing functionals for nondominated elements, Fermat rule, Lagrange multiplier rule and duality results for a single- or set-valued vector optimization problem with a variable ordering structure.

&amp;lt;i&amp;gt;ε&amp;lt;/i&amp;gt;-Optimality in Multivalued Optimization

In this paper we apply the directional derivative technique to characterize D-multifunction, quasi D-multifunction and use them to obtain ε-optimality for set valued vector optimization problem with multivalued maps. We introduce the notions of local and partial-ε-minimum (weak) point and study ε-optimality, ε-Lagrangian multiplier theorem and ε-duality results.

LAGRANGIAN DUALITY FOR VECTOR OPTIMIZATION PROBLEM WITH SET-VALUED MAPPINGS

In this paper, by using a alternative theorem, we establish Lagrangian conditions and duality results for set-valued vector optimization problems when the objective and constant are nearly cone-subconvexlike multifunctions in the sense of $E$-weak minimizer.

Some further applications of KKM theorem in topological semilattices

In this paper, we obtain some further applications of KKM theorem in setting of topological semilattices such as Ky Fan-Kakutani type fixed point theorem, Sion-Neumann type set-valued minimax theorem, set-valued vector optimization problems.

Four types of nonlinear scalarizations and some applications in set optimization

There are two types of criteria of solutions for the set-valued optimization problem, the vectorial criterion and set optimization criterion. The first criterion consists of looking for efficient points of set valued map and is called set-valued vector optimization problem. On the other hand, Kuroiwa–Tanaka–Ha started developing a new approach to set-valued optimization which is based on comparison among values of the set-valued map. In this paper, we treat the second type criterion and call set optimization problem. The aim of this paper is to investigate four types of nonlinear scalarizing functions for set valued maps and their relationships. These scalarizing functions are generalization of Tammer–Weidner’s scalarizing functions for vectors. As applications of the scalarizing functions for sets, we present nonconvex separation type theorems, Gordan’s type alternative theorems for set-valued map, optimality conditions for set optimization problem and Takahashi’s minimization theorems for set-valued map.

Higher-order optimality conditions in set-valued optimization using radial sets and radial derivatives

We propose higher-order radial sets and corresponding derivatives of a set-valued map and prove calculus rules for sums and compositions, which are followed by direct applications in discussing optimality conditions for several particular optimization problems. Our main results are both necessary and sufficient higher-order conditions for weak efficiency in a general set-valued vector optimization problem without any convexity assumptions. Many examples are provided to explain advantages of our results over a number of existing ones in the literature.

Subdifferentials for Set-valued Maps and Optimality Conditions for Set-valued Vector Optimization

Based on the concept of the weak subdifferential for set-valued maps introduced by Sawaragi and Tanino we give the definition of Henig global subdifferentials for set-valued maps.We investigate the existence condition for this kind of subdifferential,and discuss its operational property. Using this concepts,we present the necessary condition and sufficient condition for Henig globally proper efficient solution pair for constrained set-valued vector optimization problem,respectively.

Higher-order radial derivatives and optimality conditions in nonsmooth vector optimization

We propose notions of higher-order outer and inner radial derivatives of set-valued maps and obtain main calculus rules. Some direct applications of these rules in proving optimality conditions for particular optimization problems are provided. Then we establish higher-order optimality necessary conditions and sufficient ones for a general set-valued vector optimization problem with inequality constraints. A number of examples illustrate both the calculus rules and the optimality conditions. In particular, they explain some advantages of our results over earlier existing ones and why we need higher-order radial derivatives.

Scalarization of set-valued optimization problems and variational inequalities in topological vector spaces

This paper deals with set-valued vector optimization problems and set-valued vector variational inequalities in topological vector spaces, and provides some scalarization approaches for these problems by means of the polar cone and Gerstewitz’s scalarization functions.

Conjugate Duality in Constrained Set-Valued Vector Optimization Problems

In this article, under a concept of supremum/infimum of a set, defined in terms of a closure of the set, three kinds of conjugate dual problems are proposed for a constrained set-valued vector optimization problem. Weak duality, strong duality, and stability criteria are investigated. The inclusion relations between the image sets of the dual problems are also discussed.

SCALARIZATION METHODS FOR VECTOR VARIATIONAL INEQUALITIES

1. Introduction and preliminariesIn studying vector problems including vector optimization prob-lems, vectorvariationalinequalityproblemsandvectorequilibriumprob-lems, there have been usually traditional concentrations on scalarizationapproaches which enable us to replace the vector problems under con-sideration with equivalent scalar problems [3, 4, 6, 8-10].In particular, in [10], Slavov discussed some scalarization techniquesand one application of the multi-objective optimization problem into amathematical economics.In [6], Jimenez et al. developed a scalarization method in order toobtain scalar versions of their results on the necessary and sufficientconditions for strict minimizers of a general vector optimization problem,through a variational approach.In [8], Konnov considered a scalarization approach to connect vectorvariational inequalities into an equivalent scalar variational inequalitieswith a set-valued cost mapping. He gave an equivalence between weakand strong solutions of set-valued vector variational inequalities andsuggested a new gap function for vector variational inequalities. Heapplied his results to vector optimization, vector network equilibriumand vector migration equilibrium problems.In [4], Guu et al. extended the scalarization approaches of Giannessiet. al [3] to set-valued vector optimization problems and set-valued weakvector optimization problems. The scalar variational inequalities givenby them is different from those given by Konnov [8].