CONE-DIRECTED CONTINGENT DERIVATIVES AND GENERALIZED PREINVEX SET-VALUED OPTIMIZATION

By using cone-directed contingent derivatives, the unified necessary and suffi-cient optimality conditions are given for weakly and strongly minimal elements respectively in generalized preinvex set valued optimization.     

Second-order optimality conditions in set-valued optimization via asymptotic derivatives

In this article we give new second-order optimality conditions in set-valued optimization. We use the second-order asymptotic tangent cones to define second-order asymptotic derivatives and employ them to give the optimality conditions. We extend the well-known Dubovitskii–Milutin approach to set-valued optimization to express the optimality conditions given as an empty intersection of certain cones in the objective space. We also use some duality arguments to give new multiplier rules. By following the more commonly adopted direct approach, we also give optimality conditions in terms of a disjunction of certain cones in the image space. Several particular cases are discussed.     

Optimality conditions for set-valued maps with set optimization

Problems in set-valued optimization can be solved via set optimization. In this paper optimality conditions are studied for set-valued maps with set optimization. Optimality requirements are established for continuous selections using directional derivatives. Necessary and sufficient conditions for the existence of solutions are shown for set-valued maps under generalized convexity assumptions and with the notion of the contingent derivative.     

Nonconvex vector optimization of set-valued mappings

In this paper, we discuss properties, such as monotonicity and continuity, of the Gerstewitz's nonconvex separation functional. With the aid of this functional, necessary and sufficient optimality conditions for nonconvex optimization problems of set-valued mappings are obtained in topological vector spaces.     

Optimality conditions for weak and firm efficiency in set-valued optimization

This paper concerns the study of weak and firm local efficiency in constrained mathematical problems governed by set-valued mappings. We derive optimality conditions by means of the Bouligand derivative and by means of the Mordukhovich coderivative as well.     

On approximate solutions in set-valued optimization problems

In this paper we introduce several concepts of approximate solutions of set-valued optimization problems with vector and set optimization. We prove existence results and necessary and sufficient conditions by using limit sets.     

Weak subdifferential for set-valued mappings and its applications

In this paper, the existence theorems of two kinds of weak subgradients for set-valued mappings, which are the generalizations of Theorem 7 in [G.Y. Chen, J. Jahn, Optimality conditions for set-valued optimization problems, Math. Methods Oper. Res. 48 (2) (1998) 187–200] and Theorem 4.1 in [J.W. Peng, H.W.J. Lee, W.D. Rong, X.M. Yang, Hahn–Banach theorems and subgradients of set-valued maps, Math. Methods Oper. Res. 61 (2005) 281–297], respectively, are proved by virtue of a Hahn–Banach extension theorem. Moreover, some properties of the weak subdifferential for set-valued mappings are obtained by using a so-called Sandwich theorem. Finally, necessary and sufficient optimality conditions are discussed for set-valued optimization problems, whose constraint sets are determined by a fixed set and a set-valued mapping, respectively.     

Weak Clarke epiderivative in set-valued optimization

A new notion of weak Clarke epiderivative for a set-valued map is introduced using the concept of Clarke tangent cone. The existence, characterization and properties of weak Clarke epiderivative are then studied. Finally optimality criteria are established for a constrained set-valued optimization problem in terms of weak Clarke epiderivative.     

Sensitivity of set-valued discrete systems

Consider the surjective, continuous map f:X→X and the continuous map of K(X) into itself induced by f, where X is a compact metric space and K(X) is the space of all non-empty compact subsets of X endowed with a Hausdorff metric. In this paper we give examples showing that sensitivity of f does not imply sensitivity of . Furthermore, we prove that if f is a surjective, continuous interval map, then is sensitive if and only if f is sensitive.     

Higher order optimality conditions for Henig efficient solutions in set-valued optimization

In this paper, higher order generalized contingent epiderivative and higher order generalized adjacent epiderivative of set-valued maps are introduced. Necessary and sufficient conditions for Henig efficient solutions to a constrained set-valued optimization problem are given by employing the higher order generalized epiderivatives.     

Sensitivity analysis in multiobjective optimization

Sensitivity analysis in multiobjective optimization is dealt with in this paper. Given a family of parametrized multiobjective optimization problems, the perturbation map is defined as the set-valued map which associates to each parameter value the set of minimal points of the perturbed feasible set in the objective space with respect to a fixed ordering convex cone. The behavior of the perturbation map is analyzed quantitatively by using the concept of contingent derivatives for set-valued maps. Particularly, it is shown that the sensitivity is closely related to the Lagrange multipliers in multiobjective programming.This research was made while the author stayed at the International Institute for Applied Systems Analysis, Laxenburg, Austria.The author would like to thank an anonymous referee for his helpful suggestions; particularly, he pointed out that Proposition 2.2 and Theorem 2.1 are valid also in infinite-dimensional spaces.     

集值优化Benson真有效元的二阶刻画

引进了一种新的二阶组合切锥, 利用它引进了一种新的二阶组合切导数, 称为二阶组合径向切导数, 并讨论了它的性质及它与二阶组合切导数的关系, 借助二阶径向组合切导数, 分别建立了集值优化取得Benson真有效元的最优性充分和必要条件.     

集值优化强有效解的广义二阶锥方向导数刻画

在实赋范线性空间中考虑集值优化问题的强有效性．借助Henig扩张锥和基泛函的性质,利用广义二阶锥方向相依导数,得到受约束于集值映射的优化问题,取得强有效元的二阶最优性必要条件．当目标函数为近似锥一次类凸映射时,利用强有效点的标量化定理,得到集值优化问题,取得强有效元的二阶充分条件．     

Sensitivity analysis in vector optimization

For a vector optimization problem that depends on a parameter vector, the sensitivity analysis of perturbation, proper perturbation, and weak perturbation maps is dealth with. Each of the perturbation maps is defined as a set-valued map which associates to each parameter value the set of all minimal, properly minimal, and weakly minimal points of the perturbed feasible set in the objective space with respect to a fixed ordering cone. Using contingent cones in a finite-dimensional Euclidean space, we investigate the relationship between the contingent derivatives of the three types of perturbation maps and three types of minimal point sets for the contingent derivative of the feasible-set map in the objective space. These results provide quantitative informations on the behavior of the perturbation maps.The authors would like to thank the referees for their valuable comments and suggestions.     

Contingent derivative of the perturbation map in multiobjective optimization

Consider a parametrized multiobjective optimization problem with parameteru. LetG(u) be the objective space image of the feasible region, and letW(u)=MinG(u) (the perturbation map) be the efficient set in the objective space. The purpose of this paper is to investigate the relationship between the contingent derivativeDW ofW with respect tou and the contingent derivativeDG ofG with respect tou. Tanino (Ref. 1) proves that MinDGDW under certain conditions. In this paper, we prove that MinDG=MinDW under weaker conditions than Tanino's and that MinDG=DW under certain conditions. The paper does this by introducing a weaker notion of set-valued derivative. Along the way, the paper improves another of Tanino's results by using weaker conditions.The author would like to thank two anonymous referees for helpful comments. The author would also like to thank Professor P. L. Yu for encouragement and useful suggestions.     

Nonconvex scalarization in set optimization with set-valued maps

The aim of this work is to obtain scalar representations of set-valued optimization problems without any convexity assumption. Using a criterion of solution introduced by Kuroiwa [D. Kuroiwa, Some duality theorems of set-valued optimization with natural criteria, in: Proceedings of the International Conference on Nonlinear Analysis and Convex Analysis, World Scientific, River Edge, NJ, 1999, pp. 221-228], which is based on ordered relations between sets, we characterize this type of solutions by means of nonlinear scalarization. The scalarizing function is a generalization of the Gerstewitz's nonconvex separation function. As applications of our results we give two existence theorems for set-valued optimization problems.     

A Hausdorff-type distance,a directional derivative of a set-valued map and applications in set optimization

Abstract

In this paper, we follow Kuroiwa’s set approach in set optimization, which proposes to compare values of a set-valued objective map F with respect to various set order relations. We introduce a Hausdorff-type distance relative to an ordering cone between two sets in a Banach space and use it to define a directional derivative for F. We show that the distance has nice properties regarding set order relations and the directional derivative enjoys most properties of the one of a scalar single-valued function. These properties allow us to derive necessary and/or sufficient conditions for various types of maximizers and minimizers of F.     

On optimization problems with set-valued objective maps

In this paper, we consider constrained optimization problems with set-valued objective maps. First, we define three types of quasi orderings on the set of all non-empty subsets in n-dimensional Euclidean space and investigate their properties. Next, by using these orderings, we define the concepts of the convexities to set-valued maps and investigate their properties. Finally, based on these results, we define the concepts of optimal solutions to constrained optimization problems with set-valued objective maps and characterize their properties.     

Gap functions for a system of generalized vector quasi-equilibrium problems with set-valued mappings

In this paper, some gap functions for three classes of a system of generalized vector quasi-equilibrium problems with set-valued mappings (for short, SGVQEP) are investigated by virtue of the nonlinear scalarization function of Chen, Yang and Yu. Three examples are then provided to demonstrate these gap functions. Also, some gap functions for three classes of generalized finite dimensional vector equilibrium problems (GFVEP) are derived without using the nonlinear scalarization function method. Furthermore, a set-valued function is obtained as a gap function for one of (GFVEP) under certain assumptions.