Various Notions of Nonexpansiveness Coincide for Proximal Mappings of Functions
Honglin Luo, Xianfu Wang, Xinmin Yang · SIAM Journal on Optimization · 2024
Abstract. Proximal mappings are essential in splitting algorithms for both convex and nonconvex optimization. In this paper, we show that proximal mappings of every prox-bounded function are nonexpansive if and only if they are firmly nonexpansive if and only if they are averaged if and only if the function is convex. Lipschitz proximal mappings of prox-bounded functions are also characterized via hypoconvex or strongly convex functions. Our results generalize a recent result due to Rockafellar.