A Barzilai-Borwein Descent Method for Multiobjective Optimization Problems

TL;DR

This paper introduces a Barzilai-Borwein descent method for multiobjective optimization that improves convergence by dynamically adjusting gradient steps, addressing issues caused by objective function imbalances.

Contribution

It proposes a novel BBDMO algorithm with theoretical convergence guarantees and demonstrates its efficiency through numerical experiments.

Findings

BBDMO effectively accelerates convergence in multiobjective optimization.

The method achieves Pareto critical points under both monotone and nonmonotone line searches.

Numerical results confirm BBDMO's superior performance over existing methods.

Abstract

The steepest descent method proposed by Fliege et al. motivates the research on descent methods for multiobjective optimization, which has received increasing attention in recent years. However, empirical results show that the Armijo line search often gives a very small stepsize along the steepest direction, which decelerates the convergence seriously. This paper points out that the issue is mainly due to the imbalances among objective functions. To address this issue, we propose a Barzilai-Borwein descent method for multiobjective optimization (BBDMO) that dynamically tunes gradient magnitudes using Barzilai-Borwein's rule in direction-finding subproblem. With monotone and nonmonotone line search techniques, it is proved that accumulation points generated by BBDMO are Pareto critical points, respectively. Furthermore, theoretical results indicate the Armijo line search can achieve a better stepsize in BBDMO. Finally, comparative results of numerical experiments are reported to illustrate the efficiency of BBDMO and verify the theoretical results.