Various Notions of Nonexpansiveness Coincide for Proximal Mappings of Functions
本文证明了每个近端有界函数的近端映射是非扩张的当且仅当它是强非扩张的当且仅当它是平均映射当且仅当该函数是凸的，并利用次凸或强凸函数刻画了Lipschitz近端映射。
.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.Keywordsaveraged mappingfirmly nonexpansive mappinghypoconvex functionlimiting subdifferentialnonexpansive mappingproximal mappingprox-bounded functionMSC codes49J5347H0947H0526B25