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Generalized monotone operators and their averaged resolvents

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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.

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Notes

  1. This is also known as weakly convex function.

  2. Let \(\beta >0\) and let \(T:X\rightarrow X\). Recall that T is \(\beta \)-cocoercive if \(\beta T\) is firmly nonexpansive, i.e., \((\forall (x,y)\in X\times X)\) \(\langle x-y,Tx-Ty\rangle \ge \beta \Vert Tx-Ty\Vert ^2\).

  3. This is also known as \(\alpha \)-negatively averaged (see [15, Definition 3.7]).

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Acknowledgements

The authors thank the editors and three anonymous referees for careful reading and constructive comments. HHB, WMM, and XW were partially supported by the Natural Sciences and Engineering Research Council of Canada.

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Authors and Affiliations

  1. Mathematics, University of British Columbia, Kelowna, BC, V1V 1V7, Canada

    Heinz H. Bauschke & Xianfu Wang

  2. Department of Combinatorics and Optimization, University of Waterloo, Waterloo, ON, N2L 3G1, Canada

    Walaa M. Moursi

Authors
  1. Heinz H. Bauschke
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  2. Walaa M. Moursi
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  3. Xianfu Wang
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Correspondence to Heinz H. Bauschke.

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Dedicated to Marco López on the occasion of his 70th birthday.

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Bauschke, H.H., Moursi, W.M. & Wang, X. Generalized monotone operators and their averaged resolvents. Math. Program. 189, 55–74 (2021). https://doi.org/10.1007/s10107-020-01500-6

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