Stability analysis for generalized semi-infinite optimization problems under functional perturbations

Xiaodong Fan; Tian Qin

doi:10.3934/jimo.2018201

Article Contents

2020, Volume 16, Issue 3: 1221-1233. Doi: 10.3934/jimo.2018201

This issue Previous Article An integrated bi-objective optimization model and improved genetic algorithm for vehicle routing problems with temporal and spatial constraints Next Article An integrated approach based on Fuzzy Inference System for scheduling and process planning through multiple objectives

Stability analysis for generalized semi-infinite optimization problems under functional perturbations

Xiaodong Fan^, and
Tian Qin

Department of Mathematics, Bohai University, Jinzhou, Liaoning 121013, China

* Corresponding author: Xiaodong Fan (E-mail address: bhdxfxd@163.com)
* Corresponding author: Xiaodong Fan (E-mail address: bhdxfxd@163.com)

Received: June 2017

Revised: October 2017

Early access: December 2018

Published: March 2020

The first author is supported by National Natural Science Foundation of China (No. 61572082), Natural Science Foundation of Liaoning Province of China (No. 20170540004, 20170540012) and Educational Commission of Liaoning Province of China (No. LZ2016003)

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Abstract

The concepts of essential solutions and essential solution sets for generalized semi-infinite optimization problems (GSIO for brevity) are introduced under functional perturbations, and the relations among the concepts of essential solutions, essential solution sets and lower semicontinuity of solution mappings are discussed. We show that a solution is essential if and only if the solution is unique; and a solution subset is essential if and only if it is the solution set itself. Some sufficient conditions for the upper semicontinuity of solution mappings are obtained. Finally, we show that every GSIO problem can be arbitrarily approximated by stable GSIO problems (the solution mapping is continuous), i.e., the set of all stable GSIO problems is dense in the set of all GSIO problems with the given topology.

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Mathematics Subject Classification: Primary: 90C29, 90C31; Secondary: 49K40.

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