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

Francisco J. Aragón-Artacho, César López-Pastor

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.

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

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 , 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.

Paper Structure

This paper contains 6 sections, 7 theorems, 47 equations, 3 figures.

Table of Contents

  1. Keywords
  2. MSC2020
  3. Introduction
  4. Relaxation of normal and iso-averaged maps
  5. Graph-based splitting algorithms applied to subspaces
  6. Acknowledgments

Key Result

Lemma 2.1

A linear map $T:\mathbb{E}\to\mathbb{E}$ is normal if and only if $T_\theta$ is normal for all $\theta\in\mathbb{R}$. $\blacktriangleleft$$\blacktriangleleft$

Figures (3)

  • Figure 1: Graph of the nonconvex function $f(\theta)=\left\lVert T^2_\theta x\right\rVert$ (blue) given in \ref{['not normal']}
  • Figure 2: Graphs of $|\lambda_\theta|$ for $|\lambda|\in\{0.1,0.25,0.5,0.75,0.9\}$
  • Figure 3: Geometrical interpretation of \ref{['ex:geometric']}, where $a_1=(-1,6)$, $a_2=(-3,1)$, $a_3=(-3,-4)$, $\bm{v}^0=((1,-10),(-8,1))$, and $\theta=0.2$

Theorems & Definitions (21)

  • Lemma 2.1
  • proof
  • Proposition 2.2
  • proof
  • Example 2.3: Necessity of normality in \ref{['convex']}
  • Definition 2.4
  • Proposition 2.5
  • proof
  • Example 2.6: Iso-averaged maps might not be normal in Hilbert spaces
  • Example 2.7: The relaxed iso-averaged map might not be iso-averaged
  • ...and 11 more