On the Metric Resolvent: Nonexpansiveness, Convergence Rates and Applications: On the Metric Resolvent: Nonexpansiveness, Convergence Rates and Applications - Xue, Feng (Journal of the Operations Research Society of China (Internet), 2025)
In this paper, we study the nonexpansive properties of metric resolvent and present the convergence analysis for the associated fixed-point iterations of both Banach–Picard and Krasnosel’skiĭ–Mann types. A by-product of our expositions also extends the proximity operator and Moreau’s decomposition identity to arbitrary metric. It is further shown that many classes of the first-order operator splitting algorithms, including the alternating direction methods of multipliers, primal–dual hybrid gradient and Bregman iterations, can be expressed by the fixed-point iterations of a simple metric resolvent, and thus, the convergence can be easily obtained within this unified framework.