Nonmonotone Wolfe-type quasi-Newton methods for multiobjective optimization problems

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This paper introduces two nonmonotone quasi-Newton algorithms with Wolfe line searches for unconstrained multiobjective optimization problems. The proposed algorithm is applicable to general nonconvex problems, making it versatile in various optimization scenarios. Our approach utilizes the BFGS quasi-Newton method as the foundation for solving unconstrained multiobjective optimization problems. Notably, we incorporate nonmonotone line searches, allowing for objective function value increases in certain iterations. We explore two well-known types of nonmonotone line searches: one that considers the maximum of recent function values and the other that calculates their average. Under reasonable assumptions, we establish global convergence for the algorithm. The efficiency of the proposed method is evaluated through empirical analysis, which includes the computation of relative efficiency and the generation of performance profiles using the methodology developed by Dolan and Moré. Through extensive numerical experiments, we demonstrate that the nonmonotone quasi-Newton algorithm requires fewer function evaluations and fewer iterations compared to the monotone quasi-Newton algorithm. These findings highlight the efficiency and effectiveness of our proposed approach in addressing unconstrained multiobjective optimization problems.