# Resolvent splitting for sums of monotone operators with minimal lifting

@article{Malitsky2021ResolventSF, title={Resolvent splitting for sums of monotone operators with minimal lifting}, author={Yura Malitsky and Matthew K. Tam}, journal={Mathematical Programming}, year={2021} }

In this work, we study ﬁxed point algorithms for ﬁnding a zero in the sum of n ≥ 2 maximally monotone operators by using their resolvents. More precisely, we consider the class of such algorithms where each resolvent is evaluated only once per iteration. For any algorithm from this class, we show that the underlying ﬁxed point operator is necessarily deﬁned on a d -fold Cartesian product space with d ≥ n − 1. Further, we show that this bound is unimprovable by providing a family of examples for…

## 12 Citations

### Resolvent splitting with minimal lifting for composite monotone inclusions

- Mathematics
- 2021

In this paper we propose a resolvent splitting with minimal lifting for finding a zero of the sum of n > 2 maximally monotone operators involving the composition with a linear bounded operator. The…

### A primal-dual splitting algorithm for composite monotone inclusions with minimal lifting

- Mathematics, Computer ScienceNumerical Algorithms
- 2022

This work establishes the first primal-dual splitting algorithm for composite monotone inclusions with minimal lifting, which reduces the dimension of the product space where the underlying fixed point operator is defined, in comparison to other algorithms, without requiring additional evaluations of the resolvent operators.

### Frugal Splitting Operators: Representation, Minimal Lifting and Convergence

- MathematicsArXiv
- 2022

We consider frugal splitting operators for ﬁnite sum monotone inclusion problems, i.e., splitting operators that use exactly one direct or resolvent evaluation of each operator of the sum. A novel…

### Frugal and Decentralised Resolvent Splittings Defined by Nonexpansive Operators

- Mathematics, Computer ScienceArXiv
- 2022

A general framework for frugal resolvent splitting is developed which simultaneously covers and extends several important schemes in the literature and yields a new resolent splitting algorithm which is suitable for decentralised implementation on regular networks.

### The splitting algorithms by Ryu and by Malitsky-Tam applied to normal cones of linear subspaces converge strongly to the projection onto the intersection

- Mathematics
- 2021

Finding a zero of a sum of maximally monotone operators is a fundamental problem in modern optimization and nonsmooth analysis. Assuming that resolvents of the operators are available, this problem…

### A product space reformulation with reduced dimension for splitting algorithms

- MathematicsComputational Optimization and Applications
- 2022

In this paper we propose a product space reformulation to transform monotone inclusions described by finitely many operators on a Hilbert space into equivalent two-operator problems. Our approach…

### The splitting algorithms by Ryu, by Malitsky-Tam, and by Campoy applied to normal cones of linear subspaces converge strongly to the projection onto the intersection

- Mathematics
- 2022

Finding a zero of a sum of maximally monotone operators is a fundamental problem in modern optimization and nonsmooth analysis. Assuming that the resolvents of the operators are available, this…

### A Direct Proof of Convergence of Davis–Yin Splitting Algorithm Allowing Larger Stepsizes

- Computer ScienceSet-Valued and Variational Analysis
- 2022

A direct proof is provided that guarantees its convergence when the stepsizes are smaller than four times the cocoercivity constant, thus doubling the size of the interval established by Davis and Yin's splitting algorithm.

### Graph and distributed extensions of the Douglas-Rachford method

- Computer Science, MathematicsArXiv
- 2022

This paper describes how the graph-based extensions of the DRS method can be leveraged to design new fully distributed protocols and shows interesting connections with the underlying graph topology and highly competitive performances with state-of-the-art distributed optimization approaches.

### Distributed forward-backward methods for ring networks

- Mathematics, Computer ScienceComputational Optimization and Applications
- 2022

In this work, we propose and analyse forward-backward-type algorithms for finding a zero of the sum of finitely many monotone operators, which are not based on reduction to a two operator inclusion…

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