Epiderivatives Of Set-valued Maps Research Articles

New higher-order weak lower inner epiderivatives and application to Karush–Kuhn–Tucker necessary optimality conditions in set-valued optimization

The purpose of the paper is to establish higher-order Karush–Kuhn–Tucker higher-order necessary optimality conditions for set-valued optimization where the derivatives of objective and constraint functions are separated. We first introduce concepts of higher-order weak lower inner epiderivatives for set-valued maps and discuss some useful properties about new epiderivatives, for instance, convexity, subadditivity and chain rule. With the help of the new concept and its properties, we establish higher-order Karush–Kuhn–Tucker necessary optimality conditions which is the classical type Karush–Kuhn–Tucker optimality conditions and improve and enhance some recent existing results in the literatures. Several examples are provided to illustrate our results. Finally, we provide weak and strong duality theorems in set-valued optimization.

New second-order radial epiderivatives and applications to optimality conditions

In this paper, we introduce the second-order weakly composed radial epiderivative of set-valued maps, discuss its relationship to the second-order weakly composed contingent epiderivative, and obtain some of its properties. Then we establish the necessary optimality conditions and sufficient optimality conditions of Benson proper efficient solutions of constrained set-valued optimization problems by means of the second-order epiderivative. Some of our results improve and imply the corresponding ones in recent literature.

New second-order radial epiderivatives and nonconvex set-valued optimization problems

In this paper, we introduce the generalized second-order composed radial epiderivative of set-valued maps and establish a few relations between the epiderivative and the generalized second-order composed contingent epiderivative. We also investigate some of its properties. Then, by virtue of the generalized second-order composed radial epiderivative, we obtain the necessary optimality conditions and sufficient optimality conditions for weakly efficient solutions of unconstrained set-valued optimization problems without convexity conditions. Furthermore, the necessary optimality conditions and sufficient optimality conditions are obtained for Henig efficient solutions of constrained set-valued optimization problems. Some of obtained results improve or imply the corresponding ones in recent literature.

Higher-Order Generalized Radial Epiderivative and Its Applications to Set-Valued Optimization Problems

In the paper, we introduce the higher-order generalized radial epiderivative of set-valued maps. Then, its basic properties are studied. Finally, we apply this epiderivative to optimality conditions for set-valued optimization problems.

Second-Order Optimality Conditions for Set-Valued Vector Equilibrium Problems

In this article, by using the generalized second-order contingent (adjacent) epiderivatives of set-valued maps, we obtain necessary optimality conditions and sufficient optimality conditions for weakly efficient solutions, Henig efficient solutions to the set-valued vector equilibrium problems with constraints. Some results of this article improve the corresponding results in literatures by lessening the assumption of convexity.

$\mathbb{T}$-Epiderivatives of Set-valued Maps and Its Application to Set Optimization and Generalized Variational Inequalities

In this paper, we first define a $\mathbb{T}$-cone which is a unified version of several cones, namely, contingent cone, radial cone, $C$-tangent cone, Clarke tangent cone, $S$-cone, adjacent cone, etc. Then, we define the $\mathbb{T}$-epiderivative of a set-valued map which includes the contingent epiderivative, radial epiderivative, $S$-epiderivative, adjacent epiderivative etc. as special cases. We present several properties of such an epiderivative. The generalized vector $\mathbb{T}$-variational inequality problem is also considered. We provide necessary and sufficient conditions for a solution of a set optimization problem. Several existence results for solutions of set optimization problems and a generalized vector $\mathbb{T}$-variational inequality problem are given.

Generalized Higher-Order Optimality Conditions for Set-Valued Optimization under Henig Efficiency

In this paper, generalized mth-order contingent epiderivative and generalized mth-order epiderivative of set-valued maps are introduced, respectively. By virtue of the generalized mth-order epiderivatives, generalized necessary and sufficient optimality conditions are obtained for Henig efficient solutions to a set-valued optimization problem whose constraint set is determined by a fixed set. Generalized Kuhn–Tucker type necessary and sufficient optimality conditions are also obtained for Henig efficient solutions to a set-valued optimization problem whose constraint set is determined by a set-valued map.

Some Calculus Rules for Contingent Epiderivatives

This article presents some calculus rules for contingent epiderivatives of set-valued maps. Among other results the main emphasis is focused on a formula for scalar multiplication, sum formulae and chain rules. The calculus of contingent cones and some inversion theorems are used as a tool. Some applications are also given.