Optimality conditions for various efficient solutions involving coderivatives: from set-valued optimization problems to set-valued equilibrium problems
نویسنده
چکیده
In this paper, we present a new approach to the study of various efficient solutions of a set-valued equilibrium problem (for short, SEP) through the study of corresponding solutions of a set-valued optimization problem with a geometric constraint (for short, SOP). The solutions under consideration are: efficient solutions, weakly efficient solutions, strongly efficient solutions, and properly efficient solutions such as positive properly efficient solutions, Henig global properly efficient solutions, Henig properly efficient solutions, super efficient solutions, Benson properly efficient solutions. Firstly, we obtain in a unified scheme scalar characterization with the help of Hiriart-Urruty’s signed distance function, necessary/sufficient conditions for weakly, strongly, properly efficient solutions of SOP involving coderivatives and normal cones in the senses of Clarke, Ioffe and Mordukhovich, and recall Bao-Mordukhovich’s necessary conditions for efficient solutions of SOP involving the coderivative and normal cone in the sense of Mordukhovich. Secondly, we show that there is an equivalence between solutions of SOP and solutions of SEP and based on this equivalence we derive similar results for solutions of SEP as well as for solutions of a vector equilibrium problem and a vector variational inequality.
منابع مشابه
Lagrange multipliers for set - valued optimization problems associated with coderivatives ✩
In this paper we investigate a vector optimization problem (P) where objective and constraints are given by set-valued maps. We show that by mean of marginal functions and suitable scalarizing functions one can characterize certain solutions of (P) as solutions of a scalar optimization problem (SP) with single-valued objective and constraint functions. Then applying some classical or recent res...
متن کاملOptimality conditions for Pareto efficiency and proper ideal point in set-valued nonsmooth vector optimization using contingent cone
In this paper, we first present a new important property for Bouligand tangent cone (contingent cone) of a star-shaped set. We then establish optimality conditions for Pareto minima and proper ideal efficiencies in nonsmooth vector optimization problems by means of Bouligand tangent cone of image set, where the objective is generalized cone convex set-valued map, in general real normed spaces.
متن کاملLower semicontinuity for parametric set-valued vector equilibrium-like problems
A concept of weak $f$-property for a set-valued mapping is introduced, and then under some suitable assumptions, which do not involve any information about the solution set, the lower semicontinuity of the solution mapping to the parametric set-valued vector equilibrium-like problems are derived by using a density result and scalarization method, where the constraint set $K$...
متن کاملOn Fixed Point Results for Hemicontractive-type Multi-valued Mapping, Finite Families of Split Equilibrium and Variational Inequality Problems
In this article, we introduced an iterative scheme for finding a common element of the set of fixed points of a multi-valued hemicontractive-type mapping, the set of common solutions of a finite family of split equilibrium problems and the set of common solutions of a finite family of variational inequality problems in real Hilbert spaces. Moreover, the sequence generated by the proposed algori...
متن کاملExistence of Minimizers and Necessary Conditions in Set-Valued Optimization with Equilibrium Constraints
In this paper we study set-valued optimization problems with equilibrium constraints (SOPEOs) described by parametric generalized equations in the form 0 E G(x) + Q(x), where both G and Q are set-valued mappings between infinite-dimensional spaces. Such models particularly arise from certain optimization-related problems governed by set-valued variational inequalities and first-order optimality...
متن کامل