Variable Metric Method for Unconstrained Multiobjective Optimization Problems

TL;DR

This paper introduces a variable metric method for unconstrained multiobjective optimization that guarantees convergence to Pareto critical points and demonstrates superlinear convergence with numerical validation.

Contribution

It develops a new variable metric approach that does not require convexity and introduces a nonmonotone line search for faster convergence.

Findings

Convergence to Pareto critical points is proven.

The method achieves local superlinear convergence.

Numerical results confirm the method's effectiveness.

Abstract

In this paper, we propose a variable metric method for unconstrained multiobjective optimization problems (MOPs). First, a sequence of points is generated using different positive definite matrices in the generic framework. It is proved that accumulation points of the sequence are Pareto critical points. Then, without convexity assumption, strong convergence is established for the proposed method. Moreover, we use a common matrix to approximate the Hessian matrices of all objective functions, along which, a new nonmonotone line search technique is proposed to achieve a local superlinear convergence rate. Finally, several numerical results demonstrate the effectiveness of the proposed method.