Preconditioned Barzilai-Borwein Methods for Multiobjective Optimization Problems

Jian Chen (), Wang Chen (), Liping Tang () and Xinmin Yang ()
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Jian Chen: Chongqing Normal University
Wang Chen: Chongqing Normal University
Liping Tang: Chongqing Normal University
Xinmin Yang: Chongqing Normal University

Journal of Optimization Theory and Applications, 2026, vol. 208, issue 1, No 9, 43 pages

Abstract: Abstract 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.

Keywords: Multiobjective optimization; Preconditioning; Barzilai-Borwein’s method; BFGS; 90C29; 90C30 (search for similar items in EconPapers)
Date: 2026
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DOI: 10.1007/s10957-025-02824-z

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