Coderivatives and Aubin property of efficient point and efficient solution set-valued maps in parametric vector optimization

Xiaowei Xue

doi:10.1186/s13660-020-02499-2

Abstract

The aim of this paper is computing the coderivatives of efficient point and efficient solution set-valued maps in a parametric vector optimization problem. By using a method different from the existing literature we establish an upper estimate and explicit expression for the coderivatives of an efficient point set-valued map where the independent variable can take values in the whole space. As an application, we give some characterizations on the Aubin property of an efficient point map and an explicit expression of the coderivative for an efficient solution map. We provide several examples illustrating the main results.

Highlights

Consider the following parametric vector-valued optimization problem: MinK f (p, x) | x ∈ C(p), (1)where f : Rm × Rn → Rs is a vector-valued map, C : Rm ⇒ Rn is a set-valued map, K is a pointed closed convex cone of Rs that induces a partial ordering K, x ∈ Rn is a decision variable, and p ∈ Rm is a parameter
By using the tangent derivatives for set-valued maps, which are generated by tangent cones to their graphs, sensitivity results are obtained for vector optimization problems with kinds of structure; see, for example, [2, 7, 8, 11, 14, 16, 22, 23] and references therein
Sensitivity results of scalar optimization problems are obtained by the coderivatives generated by normal cones to the graphs of set-valued maps; we refer the readers to [10, 15, 19, 20, 25] for just a few of them

Summary

Introduction

Consider the following parametric vector-valued optimization problem: MinK f (p, x) | x ∈ C(p) , (1)where f : Rm × Rn → Rs is a vector-valued map, C : Rm ⇒ Rn is a set-valued map, K is a pointed closed convex cone of Rs that induces a partial ordering K , x ∈ Rn is a decision variable, and p ∈ Rm is a parameter. By using the tangent derivatives for set-valued maps, which are generated by tangent cones to their graphs, sensitivity results are obtained for vector optimization problems with kinds of structure; see, for example, [2, 7, 8, 11, 14, 16, 22, 23] and references therein.

Objectives

Results

Conclusion