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Monotone operator theory in convex optimization

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Abstract

Several aspects of the interplay between monotone operator theory and convex optimization are presented. The crucial role played by monotone operators in the analysis and the numerical solution of convex minimization problems is emphasized. We review the properties of subdifferentials as maximally monotone operators and, in tandem, investigate those of proximity operators as resolvents. In particular, we study new transformations which map proximity operators to proximity operators, and establish connections with self-dual classes of firmly nonexpansive operators. In addition, new insights and developments are proposed on the algorithmic front.

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  1. Department of Mathematics, North Carolina State University, Raleigh, NC, 27695-8205, USA

    Patrick L. Combettes

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This work was supported by the National Science Foundation under Grant CCF-1715671.

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Combettes, P.L. Monotone operator theory in convex optimization. Math. Program. 170, 177–206 (2018). https://doi.org/10.1007/s10107-018-1303-3

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