# Monotone operator theory in convex optimization

@article{Combettes2018MonotoneOT, title={Monotone operator theory in convex optimization}, author={Patrick L. Combettes}, journal={Mathematical Programming}, year={2018}, volume={170}, pages={177-206} }

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…

## 48 Citations

### Multivariate Monotone Inclusions in Saddle Form

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A novel approach to monotone operator splitting based on the notion of a saddle operator, which achieves full splitting, exploits the specific attributes of each operator, is asynchronous, and requires to activate only blocks of operators at each iteration, as opposed to activating all of them.

### Proximal Splitting Algorithms for Convex Optimization: A Tour of Recent Advances, with New Twists

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- 2019

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### Bregman Forward-Backward Operator Splitting

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- 2019

We propose an iterative method for finding a zero of the sum of two maximally monotone operators in reflexive Banach spaces. One of the operators is single-valued, and the method alternates an…

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- Computer Science, Mathematics
- 2020

This overview of recent proximal splitting algorithms within a unified framework, which consists in applying splitting methods for monotone inclusions in primal-dual product spaces, with well-chosen metric, is presented.

### Resolvent and Proximal Compositions

- Mathematics
- 2022

. We introduce the resolvent composition, a monotonicity-preserving operation between a linear operator and a set-valued operator, as well as the proximal composition, a convexity-preserving…

### Douglas-Rachford Splitting for Pathological Convex Optimization

- Computer Science
- 2018

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- 2021

In this paper, we develop rapidly convergent forward-backward algorithms for computing zeroes of the sum of finitely many maximally monotone operators. A modification of the classical…

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This work presents several existing proximal splitting algorithms and derives new ones, within a unified framework, which consists in applying splitting methods for monotone inclusions, like the forward-backward algorithm, in primal-dual product spaces with well-chosen metric, to derive new convergence theorems with larger parameter ranges.

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A variant of the ADMM algorithm is formulated that is able to handle convex optimization problems involving an additional smooth function in its objective, and which is evaluated through its gradient.

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