Graph splitting methods: Fixed points and strong convergence for linear subspaces

TL;DR

This paper analyzes the fixed points and convergence properties of graph splitting methods for linear subspaces, providing explicit formulas and unifying various existing results.

Contribution

It offers a general analysis framework for fixed points in graph splitting methods, especially for normal cones of linear subspaces, with explicit limit point formulas.

Findings

Explicit formulas for limit points of graph splitting schemes.

Unification of existing results and derivation of new ones.

Analysis of fixed points for operators involving linear subspaces.

Abstract

In this paper, we develop a general analysis for the fixed points of the operators defining the graph splitting methods from [SIAM J. Optim., 34 (2024), pp. 1569-1594] by Bredies, Chenchene and Naldi. We particularize it to the case where the maximally monotone operators are normal cones of closed linear subspaces and provide an explicit formula for the limit points of the graph splitting schemes. We exemplify these results on some particular algorithms, unifying in this way some results previously derived as well as obtaining new ones.