Unified vector quasiequilibrium problems via improvement sets and nonlinear scalarization with stability analysis

Hong-Zhi Wei; Xin Zuo; Chun-Rong Chen

doi:10.3934/naco.2019036

Article Contents

2020, Volume 10, Issue 1: 107-125. Doi: 10.3934/naco.2019036

This issue Previous Article Characterization of efficient solutions for a class of PDE-constrained vector control problems Next Article

Unified vector quasiequilibrium problems via improvement sets and nonlinear scalarization with stability analysis

1.
College of Mathematics and Statistics, Chongqing University, Chongqing, 401331, China
2.
Department of Basic, Yinchuan Energy College, Yinchuan, 750105, China

^*Corresponding author: Chun-Rong Chen
^*Corresponding author: Chun-Rong Chen

Received: November 2018

Revised: April 2019

Published: March 2020

This research was supported by the National Natural Science Foundation of China (Grant numbers: 11301567 and 11571055) and the Fundamental Research Funds for the Central Universities (Grant number: 106112017CDJZRPY0020).

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Abstract

This paper has two objectives. The first one is to propose a new vector quasiequilibrium problem where the ordering relation is defined via an improvement set $ D $, and its weak version, also their Minty-type dual problems and the corresponding set-valued cases. These models provide unified frameworks to deal with well-known exact and approximate vector quasiequilibrium problems with vector-valued or set-valued mappings. The second one is to study solution stability in the sense of Hölder continuity of the unique solution to parametric unified (resp. weak) vector quasiequilibrium problems, by employing the Gerstewitz scalarization techniques. In particular, we deduce a new stability result for the typical vector optimization problem related with (resp. weak) $ D $-optimality, by considering perturbations of both the objective function and the feasible set.

Keywords:

Improvement sets,
Nonlinear scalarization,
Unified vector quasiequilibrium problems,
Hölder continuity,
Vector optimization.

Mathematics Subject Classification: Primary: 49K40, 90C31, 47J20, 90C29.

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