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A general approach to distributed operator splitting

Minh N. Dao, Matthew K. Tam, Thang D. Truong

TL;DR

This work develops a general approach of forward-backward splitting methods for solving monotone inclusion problems involving both set-valued and single-valued operators, where the latter may lack cocoercivity.

Abstract

Splitting methods have emerged as powerful tools to address complex problems by decomposing them into smaller solvable components. In this work, we develop a general approach of forward-backward splitting methods for solving monotone inclusion problems involving both set-valued and single-valued operators, where the latter may lack cocoercivity. Our proposed approach, based on some coefficient matrices, not only encompasses several important existing algorithms but also extends to new ones, offering greater flexibility for different applications. Moreover, by appropriately selecting the coefficient matrices, the resulting algorithms can be implemented in a distributed and decentralized manner.

A general approach to distributed operator splitting

TL;DR

This work develops a general approach of forward-backward splitting methods for solving monotone inclusion problems involving both set-valued and single-valued operators, where the latter may lack cocoercivity.

Abstract

Splitting methods have emerged as powerful tools to address complex problems by decomposing them into smaller solvable components. In this work, we develop a general approach of forward-backward splitting methods for solving monotone inclusion problems involving both set-valued and single-valued operators, where the latter may lack cocoercivity. Our proposed approach, based on some coefficient matrices, not only encompasses several important existing algorithms but also extends to new ones, offering greater flexibility for different applications. Moreover, by appropriately selecting the coefficient matrices, the resulting algorithms can be implemented in a distributed and decentralized manner.

Paper Structure

This paper contains 11 sections, 8 theorems, 65 equations, 7 figures, 1 algorithm.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. A general approach to distributed operator splitting
  4. Weighted graph-based forward-backward algorithms
  5. Some notations from graph theory.
  6. A graph-based selection for coefficient matrices $M$, $N$, and $D$.
  7. Numerical experiments
  8. Experiments with cocoercivity
  9. Data generated.
  10. Experiments without cocoercivity
  11. Acknowledgements.

Key Result

Proposition 2.2

Let $T$ be a conically $\rho$-quasiaveraged operator with full domain and $\operatorname{Fix} T \neq \varnothing$. Let $z^0 \in \mathcal{H}$. For each $k\in\mathbb{N}$, set where $(\zeta_k)_{k\in\mathbb{N}}$ is a sequence in $[0, 1/\rho]$ such that $\liminf_{k\to+\infty} \zeta_k(1-\rho\zeta_k) > 0$. Then the following hold:

Figures (7)

  • Figure 1: Possible 3-node connected graphs. Edge numbers, directions, and weights refer to $G'$
  • Figure 2: A weighted sequential graph
  • Figure 3: Weighted star graphs with $n$ nodes
  • Figure 4: A complete graph $G'$ of 7 nodes
  • Figure 5: Impact of varying $\gamma$ and $\lambda_k$ on the performance.
  • ...and 2 more figures

Theorems & Definitions (20)

  • Definition 2.1
  • Proposition 2.2: Krasnosel'skiı̆--Mann iterations
  • Lemma 2.3
  • Lemma 2.4: GS09
  • Remark 3.1
  • Remark 3.3: Implications of the assumption
  • Remark 3.4: Simple selections for $P$, $Q$, and $R$
  • Lemma 3.5: Fixed points and zeros
  • Lemma 3.6: Conical (quasi)averagedness
  • Remark 3.7: A variant of the key inequalities
  • ...and 10 more