Preconditioned Barzilai-Borwein Methods for Multiobjective Optimization Problems

Preconditioning is a powerful approach for solving ill-conditioned problems in optimization, where a preconditioning matrix is used to reduce the condition number and speed up the convergence of first-order method. Unfortunately, it is impossible to capture the curvature of all objective functions with a single preconditioning matrix in multiobjective optimization. Instead, second-order methods for multiobjective optimization problems (MOPs) use different matrices for objectives in direction-finding subproblems, leading to a prohibitive per-iteration cost. To balance per-iteration cost and better curvature exploration, we propose a preconditioned Barzilai-Borwein descent method for MOPs (PBBMO). In the direction-finding subproblems, we employ a scale matrix to explore the curvature of an implicit scalarization function. The Barzilai-Borwein method is then applied to the matrix metric to tune the gradients of the objective functions, which can also be considered as an extra diagonal preconditioner based on the scale matrix for each objective, and mitigates the effect of imbalances among objectives. From a preconditioning perspective, we use BFGS update formula to approximate a trade-off of Hessian matrices. Under mild assumption, we give a simple convergence analysis for the Barzilai-Borwein quasi-Newton method. Finally, comparative numerical results confirm the efficiency of the proposed method, even when applied to large-scale and ill-conditioned problems.



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