Notes on Lipschitz Properties of Nonlinear Scalarization Functions with Applications

Fang Lu,Chun-Rong Chen

doi:10.1155/2014/792364

Abstract

Various kinds of nonlinear scalarization functions play important roles in vector optimization. Among them, the one commonly known as the Gerstewitz function is good at scalarizing. In linear normed spaces, the globally Lipschitz property of such function is deduced via primal and dual spaces approaches, respectively. The equivalence of both expressions for globally Lipschitz constants obtained by primal and dual spaces approaches is established. In particular, when the ordering cone is polyhedral, the expression for calculating Lipschitz constant is given. As direct applications of the Lipschitz property, several sufficient conditions for Hölder continuity of both single-valued and set-valued solution mappings to parametric vector equilibrium problems are obtained using the nonlinear scalarization approach.

Highlights

In the development of vector optimization, the theory and the methods of scalarization have always played important roles [1,2,3,4,5]
Motivated by the work reported in [25, 31, 38], this paper aims to give some applications of the properties of ξq to the Holder continuity of solutions for parametric VEPs
Whether the two Lipschitz constants L = L󸀠 hold, we show it as follows

Summary

Introduction

In the development of vector optimization, the theory and the methods of scalarization have always played important roles [1,2,3,4,5]. It is necessary to further exploit applications of the globally Lipschitz property of ξq together with other useful properties for studying Holder continuity of parametric VEPs. Motivated by the work reported in [25, 31, 38], this paper aims to give some applications of the properties of ξq to the Holder continuity of solutions for parametric VEPs. To our aim, the nonlinear scalarization function ξq as a fundamental tool will play key roles such that, its globally Lipschitz property, monotonicity, and sublinearity will be fully exploited.

Lipschitz Properties of Nonlinear

Applications to the Hölder Continuity

Conclusions