A nonmonotone Polak-Ribière-Polyak conjugate gradient method for multiobjective optimization problems

Multiobjective optimization problems arise in numerous real-world applications where multiple conflicting objectives must be optimized simultaneously. A key challenge in solving such problems is efficiently computing Pareto critical points without imposing restrictive assumptions, such as convexity, on the objective functions. While conjugate gradient methods have been widely studied for single-objective optimization, their extension to multiobjective settings remains an active research area, particularly with nonmonotone line search techniques that enhance robustness. This study proposes a Polak-Ribière-Polyak conjugate gradient method for unconstrained multiobjective optimization, where the objective functions are continuously differentiable. The method employs an average-type nonmonotone Armijo-like line search to determine the step-size, improving flexibility and convergence behavior. Under mild assumptions (without convexity requirements), we establish the asymptotic convergence of the method, proving that every limit point of the generated iterate sequence is Pareto critical. To validate the method's effectiveness, we apply it to benchmark test problems and compare its performance with existing multiobjective conjugate gradient methods, such as the Hager-Zhang and Liu-Storey conjugate gradient approaches. Furthermore, to isolate and assess the specific contribution of the nonmonotone line search strategy, we provide an additional comparison between the proposed method and its monotone counterpart, which employs a standard Armijo-type line search. The quality of the Pareto front approximations generated by the considered methods is evaluated using the Hypervolume and Inverted Generational Distance performance indicators on a set of representative convex and nonconvex test problems. The numerical results demonstrate that the proposed nonmonotone Polak-Ribière-Polyak conjugate gradient method consistently outperforms the competing methods in terms of both computational efficiency and the quality of the Pareto front approximation.