LAUSR

In 2019, Tanabe et al. proposed proximal gradient methods for multiobjective optimization. Under reasonable assumptions, they proved that the accumulation points of the sequences generated by these methods are Pareto stationary. However, the convergence rates were not established in that paper. Here, we show global convergence rates for multiobjective proximal gradient methods, matching what is known in scalar optimization. More specifically, by using merit functions as a way to measure the complexity, we prove the rates for non-convex problems ($O(\sqrt{1 / k})$), convex ones ($O(1 / k)$), and strongly convex ones ($O(r^k)$ for some~$r \in (0, 1)$). We also extend the so-called Polyak-{\L}ojasiewicz (PL) inequality for multiobjective optimization and establish the rate for multiobjective problems that satisfy such inequalities ($O(r^k)$ for some~$r \in (0, 1)$).