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Convex-composite optimization, which minimizes an objective function represented by the sum of a differentiable function and a convex one, is widely used in machine learning and signal/image processing. Fast Iterative Shrinkage Thresholding Algorithm (FISTA) is a typical method for solving this problem and has a global convergence rate of $O(1 / k^2)$. Recently, this has been extended to multi-objective optimization, together with the proof of the $O(1 / k^2)$ global convergence rate. However, its momentum factor is classical, and the convergence of its iterates has not been proven. In this work, introducing some additional hyperparameters $(a, b)$, we propose another accelerated proximal gradient method with a general momentum factor, which is new even for the single-objective cases. We show that our proposed method also has a global convergence rate of $O(1/k^2)$ for any $(a,b)$, and further that the generated sequence of iterates converges to a weak Pareto solution when $a$ is positive, an essential property for the finite-time manifold identification. Moreover, we report numerical results with various $(a,b)$, showing that some of these choices give better results than the classical momentum factors.

Many descent algorithms for multiobjective optimization have been developed in the last two decades. Tanabe et al. (Comput Optim Appl 72(2):339--361, 2019) proposed a proximal gradient method for multiobjective optimization, which can solve multiobjective problems, whose objective function is the sum of a continuously differentiable function and a closed, proper, and convex one. Under reasonable assumptions, it is known that the accumulation points of the sequences generated by this method are Pareto stationary. However, the convergence rates were not established in that paper. Here, we show global convergence rates for the multiobjective proximal gradient method, matching what is known in scalar optimization. More specifically, by using merit functions to measure the complexity, we present the convergence rates for non-convex ($O(\sqrt{1 / k})$), convex ($O(1 / k)$), and strongly convex ($O(r^k)$ for some $r \in (0, 1)$) problems. We also extend the so-called Polyak-{\L}ojasiewicz (PL) inequality for multiobjective optimization and establish the linear convergence rate for multiobjective problems that satisfy such inequalities ($O(r^k)$ for some $r \in (0, 1)$)., Comment: will appear in Optim. Lett