Descent Methods for Vector Optimization Problems: A Majorization-Minimization Perspective

Abstract: In this paper, we develop a unified framework and convergence analysis of first-order methods for vector optimization problems (VOPs) from a majorization-minimization perspective. By selecting different surrogate functions, the general method can be reduced to various existing first-order methods, both with and without line search. Our unified convergence analysis reveals that the slow convergence of the steepest descent method is primarily attributed to the significant gap between the surrogate and objective functions. Consequently, narrowing this surrogate gap can enhance the performance of first-order methods. Interestingly, we elucidate that selecting a tighter surrogate function is equivalent to using an appropriate base of the dual cone in the direction-finding subproblem. Building on this insight, we employ the Barzilai-Borwein method to narrow the surrogate gap and propose a Barzilai-Borwein descent method for VOPs (BBDVO) with polyhedral cones. By reformulating the subproblem, we provide a novel perspective on the Barzilai-Borwein descent method, bridging the gap between this method and the steepest descent method. Furthermore, by selecting an appropriate base and reformulating the subproblem in the Newton method, we also provide a novel perspective on the preconditioned Barzilai-Borwein method. Finally, several numerical experiments are presented to validate the efficiency of the BBDVO.