Generalized monotone operators and their averaged resolvents
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Publication:6314640
DOI10.1007/S10107-020-01500-6arXiv1902.09827MaRDI QIDQ6314640
Xianfu Wang, Heinz Bauschke, Walaa M. Moursi
Publication date: 26 February 2019
Abstract: The correspondence between the monotonicity of a (possibly) set-valued operator and the firm nonexpansiveness of its resolvent is a key ingredient in the convergence analysis of many optimization algorithms. Firmly nonexpansive operators form a proper subclass of the more general - but still pleasant from an algorithmic perspective - class of averaged operators. In this paper, we introduce the new notion of conically nonexpansive operators which generalize nonexpansive mappings. We characterize averaged operators as being resolvents of comonotone operators under appropriate scaling. As a consequence, we characterize the proximal point mappings associated with hypoconvex functions as cocoercive operators, or equivalently; as displacement mappings of conically nonexpansive operators. Several examples illustrate our analysis and demonstrate tightness of our results.
Convex programming (90C25) Monotone operators and generalizations (47H05) Contraction-type mappings, nonexpansive mappings, (A)-proper mappings, etc. (47H09) Duality theory (optimization) (49N15)
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