Resolvent Splitting for Sums of Monotone Operators with Minimal Lifting

Abstract

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.

Distributed Forward-Backward Methods without Central Coordination

“…. , [16] proposed a splitting algorithm with (n − 1)-fold lifting for finding a zero of the sum of n 2 maximally monotone operators; see also [7] for recent extensions. In this section, we adapt the methodology developed in [1] to obtain a splitting method of forward-backward-type for the inclusion (7) by modifying the splitting method in [16] without increasing the dimension of the ambient space.…”

“…If 9) reduces to the resolvent splitting algorithms proposed by the authors in [16]. This has been further studied in [5] for the particular case in which the operators A i are normal cones of closed linear subspaces.…”

“…• They do not rely on existing product space reformulation: Instead, we extend the framework for backward operators, proposed in [16], which in turn is a generalisation of [21] for n > 3.…”

Resolvent splitting with minimal lifting for composite monotone inclusions

“…By using different techniques, frugal resolvent splittings for (P n (A, Id)) with (n − 1)−fold liftings are proposed independently in [8] and [4]. These algorithms enjoy a lower dimensionality which reduces the computational resources needed for solving large scale problems.…”

“…The resolvent of each monotone operator, the linear operator, and its adjoint are computed exactly once in the proposed algorithm. In the case when the linear operator is the identity, we recover the resolvent splitting with minimal lifting developed in [8]. We also derive a new resolvent splitting for solving the composite monotone inclusion in the case n = 2 with minimal 1−fold lifting.…”