Monotone operator theory in convex optimization

Monotone operator theory in convex optimization Math. Program., Ser. B https://doi.org/10.1007/s10107-018-1303-3 FULL LENGTH PAPER Patrick L. Combettes Received: 6 February 2018 / Accepted: 17 May 2018 © Springer-Verlag GmbH Germany, part of Springer Nature and Mathematical Optimization Society 2018 Abstract Several aspects of the interplay between monotone operator theory and con- vex 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. Keywords Firmly nonexpansive operator · Monotone operator · Operator splitting · Proximal algorithm · Proximity operator · Proximity-preserving transformation · Self-dual class · Subdifferential Mathematics Subject Classification 47H25 · 49M27 · 65K05 · 90C25 1 Introduction and historical overview In this paper, we examine various facets of the role of monotone operator theory in convex optimization and of the interplay between the two fields. Throughout, H is a real Hilbert space with scalar product · http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Mathematical Programming Springer Journals

Monotone operator theory in convex optimization

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Publisher
Springer Berlin Heidelberg
Copyright
Copyright © 2018 by Springer-Verlag GmbH Germany, part of Springer Nature and Mathematical Optimization Society
Subject
Mathematics; Calculus of Variations and Optimal Control; Optimization; Mathematics of Computing; Numerical Analysis; Combinatorics; Theoretical, Mathematical and Computational Physics; Mathematical Methods in Physics
ISSN
0025-5610
eISSN
1436-4646
D.O.I.
10.1007/s10107-018-1303-3
Publisher site
See Article on Publisher Site

Abstract

Math. Program., Ser. B https://doi.org/10.1007/s10107-018-1303-3 FULL LENGTH PAPER Patrick L. Combettes Received: 6 February 2018 / Accepted: 17 May 2018 © Springer-Verlag GmbH Germany, part of Springer Nature and Mathematical Optimization Society 2018 Abstract Several aspects of the interplay between monotone operator theory and con- vex 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. Keywords Firmly nonexpansive operator · Monotone operator · Operator splitting · Proximal algorithm · Proximity operator · Proximity-preserving transformation · Self-dual class · Subdifferential Mathematics Subject Classification 47H25 · 49M27 · 65K05 · 90C25 1 Introduction and historical overview In this paper, we examine various facets of the role of monotone operator theory in convex optimization and of the interplay between the two fields. Throughout, H is a real Hilbert space with scalar product ·

Journal

Mathematical ProgrammingSpringer Journals

Published: Jun 5, 2018

References

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