A Barzilai-Borwein descent method for multiobjective optimization problems

Abstract

The steepest descent method proposed by Fliege and Svaiter has motivated 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 results in a very small stepsize along the steepest descent direction, which decelerates the convergence seriously. This paper points out the issue is mainly due to imbalances among objective functions. To address this issue, we propose a Barzilai-Borwein descent method for multiobjective optimization (BBDMO), which dynamically tunes gradient magnitudes using Barzilai-Borwein’s rule in direction-finding subproblem. We emphasize that the BBDMO produces a sequence of new descent directions compared to Barzilai-Borwein’s method proposed by Morovati et al. With monotone and nonmonotone line search techniques, we prove that accumulation points generated by BBDMO are Pareto critical points, respectively. Furthermore, theoretical results indicate that 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.