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

Francisco J. Aragón-Artacho, Heinz H. Bauschke, Rubén Campoy, César López-Pastor

TL;DR

This work addresses the fixed-point structure and convergence of graph-splitting methods for solving sums of maximally monotone operators. It adopts a graph-based, preconditioned proximal-point framework to derive explicit fixed-point sets and projection formulas, unifying several known schemes (e.g., Douglas--Rachford, Ryu, Malitsky--Tam) and enabling the design of new variants. The main contributions include explicit expressions for the fixed points when the operators are normal cones of closed linear subspaces, characterization of limit points, and strong convergence results in the linear-subspace setting. The results provide a systematic, graph-driven approach that extends prior analyses and offers practical tools for producing and analyzing new splitting algorithms in feasibility problems.

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.

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

TL;DR

This work addresses the fixed-point structure and convergence of graph-splitting methods for solving sums of maximally monotone operators. It adopts a graph-based, preconditioned proximal-point framework to derive explicit fixed-point sets and projection formulas, unifying several known schemes (e.g., Douglas--Rachford, Ryu, Malitsky--Tam) and enabling the design of new variants. The main contributions include explicit expressions for the fixed points when the operators are normal cones of closed linear subspaces, characterization of limit points, and strong convergence results in the linear-subspace setting. The results provide a systematic, graph-driven approach that extends prior analyses and offers practical tools for producing and analyzing new splitting algorithms in feasibility problems.

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.

Paper Structure

This paper contains 33 sections, 15 theorems, 106 equations, 1 figure, 2 tables, 2 algorithms.

Table of Contents

  1. Keywords:
  2. 2020 Mathematics Subject Classification:
  3. Introduction
  4. Preliminaries
  5. Operator Theory
  6. Preconditioned proximal-point problems
  7. Graph Theory
  8. On the graph splitting framework
  9. Description of the operators and the algorithms
  10. The preconditioner $\mathcal{M}$:
  11. The operator $\mathcal{A}$:
  12. The operators $\mathcal{T}$ and $\widetilde{\mathcal{T}}$:
  13. Computing fixed points
  14. The case of linear subspaces
  15. Fixed points specializations
  16. ...and 18 more sections

Key Result

Proposition 2.18

Let $G$ be an algorithmic graph of order $n$ and let $Z\in\mathbb{R}^{n\times(n-1)}$ be an onto decomposition of its Laplacian satisfying eq:Zdecom. Then, the linear and continuous operator $Z_{\scaleto{\otimes}{4pt}}:\mathcal{H}^{n-1}\to\mathcal{H}^n$ given by eq: KronMatrix satisfies Consequently, $\operatorname{ran} Z_{\scaleto{\otimes}{4pt}}$ is closed and it holds

Figures (1)

  • Figure 1: Graph (left) and subgraph (right) defining Ryu's splitting

Theorems & Definitions (50)

  • Definition 2.1
  • Definition 2.2
  • Definition 2.4
  • Definition 2.5
  • Remark 2.7
  • Definition 2.8
  • Definition 2.11
  • proof
  • Definition 2.14
  • Example 2.15
  • ...and 40 more