Resolvent splitting for sums of monotone operators with minimal lifting

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...

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Bibliographic Details
Published in:Mathematical programming Vol. 201; no. 1-2; pp. 231 - 262
Main Authors: Malitsky, Yura, Tam, Matthew K.
Format: Journal Article
Language:English
Published: Berlin/Heidelberg Springer Berlin Heidelberg 01.09.2023
Springer
Subjects:
Algorithms
Calculus of Variations and Optimal Control; Optimization
Combinatorics
Full Length Paper
Mathematical and Computational Physics
Mathematical Methods in Physics
Mathematics
Mathematics and Statistics
Mathematics of Computing
Numerical Analysis
Theoretical
90C30
Decentralised optimisation
Splitting algorithm
65K10
Monotone operator
47H05
ADMM
ISSN:0025-5610, 1436-4646, 1436-4646
Online Access:Get full text
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Summary: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 which d = n - 1 is attained. This family includes the Douglas–Rachford algorithm as the special case when n = 2 . Applications of the new family of algorithms in distributed decentralised optimisation and multi-block extensions of the alternation direction method of multipliers (ADMM) are discussed.
ISSN:0025-5610
1436-4646
1436-4646
DOI:10.1007/s10107-022-01906-4