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Cyclic Relaxed Douglas-Rachford Splitting for Inconsistent Nonconvex Feasibility

Thi Lan Dinh, G. S. Matthijs Jansen, D. Russell Luke

TL;DR

The paper advances the theory of projection-type methods for inconsistent and potentially nonconvex feasibility by analyzing a cyclic relaxed Douglas-Rachford scheme. It derives a concrete fixed-point structure showing that fixed points lie in convex combinations of projected data from the involved sets, and it proves that the shadows of these fixed points align with fixed points of the cyclic projections in the affine setting. By establishing almost nonexpansiveness and alpha-firm nonexpansiveness properties under prox-regularity and Shapiro$^+$-type regularity, the authors obtain local convergence guarantees with quantifiable rates through a gauge function, including linear-rate results under linear subregularity. The results provide theoretical justification for using projection-based fixes in inconsistent nonconvex feasibility and offer practical insight into the behavior of shadow sequences and convergence diagnostics in such settings.

Abstract

We study the cyclic relaxed Douglas-Rachford algorithm for possibly nonconvex, and inconsistent feasibility problems. This algorithm can be viewed as a convex relaxation between the cyclic Douglas-Rachford algorithm first introduced by Borwein and Tam [2014] and the classical cyclic projections algorithm. We characterize the fixed points of the cyclic relaxed Douglas-Rachford algorithm and show the relation of the {\em shadows} of these fixed points to the fixed points of the cyclic projections algorithm. Finally, we provide conditions that guarantee local quantitative convergence estimates in the nonconvex, inconsistent setting.

Cyclic Relaxed Douglas-Rachford Splitting for Inconsistent Nonconvex Feasibility

TL;DR

The paper advances the theory of projection-type methods for inconsistent and potentially nonconvex feasibility by analyzing a cyclic relaxed Douglas-Rachford scheme. It derives a concrete fixed-point structure showing that fixed points lie in convex combinations of projected data from the involved sets, and it proves that the shadows of these fixed points align with fixed points of the cyclic projections in the affine setting. By establishing almost nonexpansiveness and alpha-firm nonexpansiveness properties under prox-regularity and Shapiro-type regularity, the authors obtain local convergence guarantees with quantifiable rates through a gauge function, including linear-rate results under linear subregularity. The results provide theoretical justification for using projection-based fixes in inconsistent nonconvex feasibility and offer practical insight into the behavior of shadow sequences and convergence diagnostics in such settings.

Abstract

We study the cyclic relaxed Douglas-Rachford algorithm for possibly nonconvex, and inconsistent feasibility problems. This algorithm can be viewed as a convex relaxation between the cyclic Douglas-Rachford algorithm first introduced by Borwein and Tam [2014] and the classical cyclic projections algorithm. We characterize the fixed points of the cyclic relaxed Douglas-Rachford algorithm and show the relation of the {\em shadows} of these fixed points to the fixed points of the cyclic projections algorithm. Finally, we provide conditions that guarantee local quantitative convergence estimates in the nonconvex, inconsistent setting.

Paper Structure

This paper contains 6 sections, 18 theorems, 86 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Notation and definitions
  3. Analysis of Fixed Points
  4. Characterization of fixed points
  5. Analysis of the shadows - affine approximations
  6. Quantitative convergence analysis

Key Result

Lemma 1

Let $T: \mathbb{E} \rightrightarrows \mathbb{E}$, $U\subset\mathbb{E}$ and let $\alpha\in (0,1)$. The following are equivalent. Consequently, if $T$ is pointwise a$\alpha$-fne at $y$ on $U$ with violation $\epsilon$ then $T$ is pointwise almost nonexpansive at $y$ on $U$ with violation at most $\epsilon$.

Figures (1)

  • Figure 1: The sum of the gaps between the sets $A_i$. Here $\mathbb{E}=\mathbb{R}^2$, $A_i$ are circles.

Theorems & Definitions (41)

  • Definition 1: super-regularity at a distance
  • Remark 1
  • Lemma 1: characterizations of $\alpha$-firmly nonexpensive operators
  • Lemma 2
  • Lemma 3: projectors/reflectors of $\epsilon$-super-regular sets, Proposition 3.4, luke2020convergence
  • Lemma 4
  • proof
  • Theorem 1: fixed points
  • proof
  • Remark 2
  • ...and 31 more