Generalized monotone operators and their averaged resolvents

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...

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Bibliographic Details
Published in:Mathematical programming 2021-09, Vol.189 (1-2), p.55-74
Main Authors: Bauschke, Heinz H., Moursi, Walaa M., Wang, Xianfu
Format: Article
Language:English
Subjects:
Algorithms
Calculus of Variations and Optimal Control
Optimization
Combinatorics
Full Length Paper
Mathematical and Computational Physics
Mathematical Methods in Physics
Mathematics
Mathematics and Statistics
Mathematics of Computing
Numerical Analysis
Operators
Optimization
Theoretical
Tightness
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Summary: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.
ISSN:0025-5610
1436-4646
DOI:10.1007/s10107-020-01500-6