A Subspace Minimization Barzilai-Borwein Method for Multiobjective Optimization Problems

Nonlinear conjugate gradient methods have recently garnered significant attention within the multiobjective optimization community. These methods maintain consistency in conjugate parameters with their single-objective optimization counterparts. However, the desirable conjugate property of search directions remains uncertain, even for quadratic cases, in multiobjective conjugate gradient methods. This loss of the conjugate property significantly limits the applicability of these methods. To elucidate the role of the last search direction, we develop a 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 efficiency of SMBBMO, even when tackling large-scale and ill-conditioned problems.