Second-order lower radial tangent derivatives and applications to set-valued optimization

Bihang Xu,Zhenhua Peng,Yihong Xu

doi:10.1186/s13660-016-1275-x

Bihang Xu, Zhenhua Peng + Show 1 more

Open Access

https://doi.org/10.1186/s13660-016-1275-x

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Abstract

We introduce the concepts of second-order radial composed tangent derivative, second-order radial tangent derivative, second-order lower radial composed tangent derivative, and second-order lower radial tangent derivative for set-valued maps by means of a radial tangent cone, second-order radial tangent set, lower radial tangent cone, and second-order lower radial tangent set, respectively. Some properties of second-order tangent derivatives are discussed, using which second-order necessary optimality conditions are established for a point pair to be a Henig efficient element of a set-valued optimization problem, and in the expressions the second-order tangent derivatives of the objective function and the constraint function are separated.

Highlights

1 Introduction In recent years, first-order optimality conditions in the set-valued optimization have attracted a great deal of attention, and various derivative-like notions have been utilized to express these optimality conditions
Gong et al [ ] introduced the concept of radial tangent cone and presented several kinds of necessary and sufficient conditions for some proper efficiencies, Kasimbeyli [ ] gave necessary and sufficient optimality conditions based on the concept of the radial epiderivatives
Second-order optimality conditions and higher-order optimality conditions for vector optimization problems have been extensively studied in the literature

Summary

Introduction

First-order optimality conditions in the set-valued optimization have attracted a great deal of attention, and various derivative-like notions have been utilized to express these optimality conditions. The second-order radial composed tangent derivative of F at (x, y) in the direction (u , v) is the set-valued map R F(x, y, u , v) : X → Y defined by graph R F(x, y, u , v) = R R epi F, (x, y) , (u , v). The second-order lower radial composed tangent derivative of F at (x, y) in the direction (u , v) is the set-valued map Rl F(x, y, u , v) : X → Y defined by graph Rl F(x, y, u , v) = Rl Rl epi F, (x, y) , (u , v). The second-order lower radial tangent derivative of F at (x, y) in the direction (u , v) is the set-valued map R l F(x, y, u , v) : X → Y defined by graph R l F(x, y, u , v) = R l epi F, (x, y), (u , v).

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