A proximal gradient method with adaptive backtracking for weakly smooth multiobjective optimization

In this paper, we propose a proximal gradient method with adaptive linesearch for multiobjective optimization problems whose objective functions are weakly smooth, i.e., they have H\"older continuous gradients. The proposed method is parameter-free as we do not require prior knowledge of parameters related to the weak smoothness of the objective function; the proposed linesearch finds an appropriate step-size that adapts to the weak smoothness. The complexity guarantee analyzed in this paper for the non-convex case is compatible with related works and our algorithm accepts coercer stationarity measure compared to existing methods. We also establish a novel complexity result for the convex case which improves the one in non-convex case.