On the optimal relaxation parameter of graph-based splitting methods for subspaces

TL;DR

This paper analyzes graph-based splitting algorithms for intersecting linear subspaces, revealing that the optimal relaxation parameter is 1 when the graph and subgraph are identical, extending classical results.

Contribution

It generalizes the optimal relaxation parameter result from Douglas-Rachford to a broader class of graph-based splitting methods.

Findings

Optimal relaxation parameter is exactly 1 when graph and subgraph coincide.

Extends known results for Douglas-Rachford to more general algorithms.

Uses properties of iso-averaged linear operators in the analysis.

Abstract

In this paper, we investigate the behavior of the family of graph-based splitting algorithms specialized to the problem of finding a point in the intersection of linear subspaces. The algorithms in this family, which encompasses several classical methods such as the Douglas-Rachford algorithm, are defined by a connected graph and a subgraph. Our main result establishes that when the graph and subgraph coincide, the optimal relaxation parameter is exactly $1$, thereby extending known results for the Douglas-Rachford algorithm to a much broader class of methods. Our analysis hinges on some properties of iso-averaged linear operators, which are defined as the average of an isometry and the identity, and are characterized by a specific symmetry of the norm of their relaxation.