Prederivatives of gamma paraconvex set-valued maps and Pareto optimality conditions for set optimization problems

Hui Huang,Jixian Ning

doi:10.1186/s13660-017-1519-4

Abstract

Prederivatives play an important role in the research of set optimization problems. First, we establish several existence theorems of prederivatives for γ-paraconvex set-valued mappings in Banach spaces with gamma>0. Then, in terms of prederivatives, we establish both necessary and sufficient conditions for the existence of Pareto minimal solution of set optimization problems.

Highlights

We say that G : X ⇒ Y is a set-valued mapping if G(x) is a subset of Y for all x ∈ X
It is well known that the prederivative is an effective tool in dealing with nondifferentiable mapping of nonsmooth analysis
In set optimization problems, Pareto minimizers are more suitable than weak minimizers and strong minimizers in practice [, ]

Summary

Introduction

Let Y be a finite dimensional Banach space, C ⊆ Y be a nonempty closed convex cone, G : X ⇒ Y be a C-convex set-valued mapping, x ∈ int(dom(G)). Let C ⊆ Y be a nonempty closed convex cone, η > , γ ≥ , r > , α > , G : X ⇒ Y be a C-γ -paraconvex set-valued mapping with modulus r and x + αBX ⊆ Dom(G).

Pareto minimizer and prederivative

Conclusion