A Product Space Reformulation with Reduced Dimension for Splitting Algorithms

TL;DR

This paper introduces a novel product space reformulation that reduces the dimensionality of splitting algorithms for monotone inclusions, enabling more efficient parallel computations without additional convergence assumptions.

Contribution

It presents a new reformulation based on Pierra's approach that decreases the dimension of the product space, leading to improved parallel splitting algorithms.

Findings

Reduced computational complexity demonstrated in numerical experiments

New parallel variants of splitting algorithms with fewer variables

Convergence established without extra assumptions

Abstract

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 relies on Pierra's classical reformulation with a different decomposition, which results on a reduction of the dimension of the outcoming product Hilbert space. We discuss the case of not necessarily convex feasibility and best approximation problems. By applying existing splitting methods to the proposed reformulation we obtain new parallel variants of them with a reduction in the number of variables. The convergence of the new algorithms is straightforwardly derived with no further assumptions. The computational advantage is illustrated through some numerical experiments.