#### Title

Hybrid Approximate Proximal Method with Auxiliary Variational Inequality for Vector Optimization

#### Document Type

Technical Report

#### Abstract

This paper studies the general vector optimization problem of finding weakly efficient points for mappings in a Banach space Y, with respect to the partial order induced by a closed, convex, and pointed cone C C Y with nonempty interior. In order to find a solution of this problem, we introduce an auxiliary variational inequality problem for monotone, Lipschitz-continuous mapping. The approximate proximal method in vector optimization is extended to develop a hybrid approximate proximal method for the general vector optimization problem by the combination of extragradient method for finding a solution to the variational inequality problem and approximate proximal point method for finding a root of a maximal monotone operator. In this hybrid approximate proximal method, the subproblems consist of finding approximate solutions to the variational inequality problem for monotone, Lipschitz-continuous mapping, and finding weakly efficient points for suitable regularizations of the original mapping. We present both an absolute and a relative version in which the subproblems are solved only approximately. Weak convergence of the generated sequence to a weak efficient point is established under quite mild conditions. In addition, we also discuss an extension to Bregman-function-based hybrid approximate proximal algorithms for finding weakly efficient points for mappings.

#### Number in Series

2009.01

#### Disciplines

Applied Mathematics | Mathematics

#### Recommended Citation

Ceng, L C.; Mordukhovich, Boris S.; and Yao, Jen-Chih, "Hybrid Approximate Proximal Method with Auxiliary Variational Inequality for Vector Optimization" (2009). *Mathematics Research Reports*. 63.

https://digitalcommons.wayne.edu/math_reports/63

## Comments

This research was partially supported by the National Science Foundation of China (10771141), Ph.D. Program Foundation of Ministry of Education of China (20070270004), and Science and Technology Commission of Shanghai Municipality grant (075105118); by the USA National Science Foundation under grant DMS-0603896; and by NSC grant 97-2115M-110-001.