Preconditioned Barzilai-Borwein Methods for Multiobjective Optimization Problems

Preconditioning is a powerful strategy for addressing ill-conditioned problems in optimization. It involves utilizing a preconditioning matrix to reduce the condition number and speed up the convergence of first-order methods. However, in multiobjective optimization, capturing the curvature of all objective functions using a single preconditioning matrix is challenging. Consequently, second-order methods tailored for multiobjective optimization problems (MOPs) employ distinct matrices for each of the objectives in direction-finding subproblems, resulting in expensive per-step costs. To strike a balance between per-step costs and better curvature exploration, we develop a “preconditioning” $$+$$ + “preconditioning” strategy to devise a preconditioned Barzilai-Borwein descent method for MOPs (PBBMO). Specifically, this method integrates a single scaling matrix to capture the local geometry of an implicit scalarization problem, leading to reduced per-step costs. We then incorporate the Barzilai-Borwein rule relative to the matrix metric to tune the gradients within the direction-finding subproblem. This can be interpreted as an additional diagonal preconditioner tailored to each objective for better curvature exploration. From a preconditioning perspective, we employ the BFGS update formula to approximate a trade-off of Hessian matrices. Subsequently, we develop a Barzilai-Borwein quasi-Newton method with Wolfe line search for MOPs. Under mild assumptions, we provide a convergence analysis for the Barzilai-Borwein quasi-Newton method. Finally, comparative numerical results validate the efficiency of the proposed method, even when applied to higher-dimensional and ill-conditioned problems.