A Subspace Minimization Barzilai-Borwein Method for Multiobjective Optimization Problems

TL;DR

This paper introduces the SMBBMO, a novel multiobjective optimization method that combines subspace minimization with Barzilai-Borwein steps, ensuring convergence and effectiveness on large-scale, ill-conditioned problems.

Contribution

The paper proposes a new subspace minimization Barzilai-Borwein method for multiobjective optimization, with convergence guarantees and improved performance on challenging problems.

Findings

Ensures global and Q-linear convergence under mild conditions.

Effective on large-scale and ill-conditioned problems.

Outperforms existing methods in numerical experiments.

Abstract

Nonlinear conjugate gradient methods have recently garnered significant attention within the multiobjective optimization community. These methods aim to maintain consistency in conjugate parameters with their single-objective optimization counterparts. However, the preservation of the attractive conjugate property of search directions remains uncertain, even for quadratic cases, in multiobjective conjugate gradient methods. This loss of interpretability of the last search direction significantly limits the applicability of these methods. To shed light on the role of the last search direction, we introduce a novel approach called the subspace minimization Barzilai-Borwein method for multiobjective optimization problems (SMBBMO). In SMBBMO, each search direction is derived by optimizing a preconditioned Barzilai-Borwein subproblem within a two-dimensional subspace generated by the last search direction and the current Barzilai-Borwein descent direction. Furthermore, to ensure the global convergence of SMBBMO, we employ a modified Cholesky factorization on a transformed scale matrix, capturing the local curvature information of the problem within the two-dimensional subspace. Under mild assumptions, we establish both global and $Q$-linear convergence of the proposed method. Finally, comparative numerical experiments confirm the efficacy of SMBBMO, even when tackling large-scale and ill-conditioned problems.