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arxiv: 2502.12285 · v1 · submitted 2025-02-17 · 🧮 math.OC

Cyclic Relaxed Douglas-Rachford Splitting for Inconsistent Nonconvex Feasibility

Thi Lan Dinh , G. S. Matthijs Jansen , D. Russell Luke This is my paper

Pith reviewed 2026-05-23 02:32 UTC · model grok-4.3

classification 🧮 math.OC
keywords Douglas-Rachford splittingcyclic projectionsfeasibility problemsnonconvex setsinconsistent feasibilityfixed point analysisconvergence rates
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The pith

The cyclic relaxed Douglas-Rachford algorithm characterizes its fixed points and relates their shadows to the fixed points of the cyclic projections algorithm, with local convergence conditions for nonconvex inconsistent feasibility.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

This paper examines the cyclic relaxed Douglas-Rachford algorithm as an approach to feasibility problems that may involve nonconvex sets and inconsistency. It views the algorithm as sitting between the cyclic Douglas-Rachford method and the standard cyclic projections method. The authors characterize the fixed points of their algorithm and demonstrate how the shadows of these fixed points connect to the fixed points of cyclic projections. They also supply conditions ensuring local quantitative convergence rates in the challenging nonconvex and inconsistent regime.

Core claim

The central claim is that the fixed points of the cyclic relaxed Douglas-Rachford algorithm can be characterized, and the shadows of these fixed points are related to the fixed points of the cyclic projections algorithm. Additionally, conditions are provided that guarantee local quantitative convergence estimates for the algorithm when applied to nonconvex and inconsistent feasibility problems.

What carries the argument

The cyclic relaxed Douglas-Rachford operator, viewed as a convex relaxation between the cyclic Douglas-Rachford and cyclic projections operators.

If this is right

  • The fixed points admit an explicit characterization.
  • The shadows of the fixed points coincide with the fixed points of the cyclic projections algorithm.
  • Local quantitative convergence estimates hold under the stated conditions even for nonconvex inconsistent problems.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • This relation may allow transferring convergence results from one algorithm to the other in certain cases.
  • The intermediate positioning could make the method more robust than either endpoint for inconsistent problems.
  • Numerical implementations might benefit from tuning the relaxation parameter to balance the properties of the two parent algorithms.

Load-bearing premise

The modeling of the algorithm as a relaxation that preserves key relations even when sets are nonconvex must hold.

What would settle it

Finding a nonconvex inconsistent feasibility problem where the shadow of a fixed point of the relaxed algorithm is not a fixed point of the cyclic projections algorithm would falsify the claimed relation.

Figures

Figures reproduced from arXiv: 2502.12285 by D. Russell Luke, G. S. Matthijs Jansen, Thi Lan Dinh.

Figure 1
Figure 1. Figure 1: The sum of the gaps between the sets Ai . Here E = R 2 , Ai are circles. 1. The sequence (gap(k) ) ∞ k=0 converges to a constant g ≥ 0. 2. If g = 0, problem (1) is consistent [PITH_FULL_IMAGE:figures/full_fig_p028_1.png] view at source ↗
read the original 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.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

1 major / 0 minor

Summary. The manuscript studies the cyclic relaxed Douglas-Rachford algorithm for possibly nonconvex and inconsistent feasibility problems. It characterizes the fixed points of the algorithm, relates the shadows of these fixed points to the fixed points of the cyclic projections algorithm, and provides conditions guaranteeing local quantitative convergence estimates in the nonconvex inconsistent setting. The method is positioned as an intermediate convex relaxation between cyclic Douglas-Rachford and cyclic projections.

Significance. If the fixed-point characterizations, shadow relations, and local convergence conditions hold under the stated assumptions, the work supplies a theoretically grounded intermediate operator between two established methods, with potential utility for inconsistent nonconvex feasibility problems arising in optimization and signal processing. The explicit relation between shadows and cyclic-projection fixed points, together with quantitative local rates, would constitute a concrete advance in nonconvex operator splitting.

major comments (1)
  1. [Abstract, paragraph 2] Abstract, paragraph 2: the modeling choice of treating the relaxed operator as a convex relaxation that remains valid for nonconvex sets is load-bearing for the transfer of fixed-point and shadow relations; the manuscript must explicitly verify that the fixed-point characterization and shadow relation continue to hold without convexity (or state the precise nonconvex assumptions under which they do).

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the constructive comment and positive overall assessment. We address the concern about explicit verification of the fixed-point and shadow characterizations in the nonconvex case below.

read point-by-point responses

  1. Referee: [Abstract, paragraph 2] Abstract, paragraph 2: the modeling choice of treating the relaxed operator as a convex relaxation that remains valid for nonconvex sets is load-bearing for the transfer of fixed-point and shadow relations; the manuscript must explicitly verify that the fixed-point characterization and shadow relation continue to hold without convexity (or state the precise nonconvex assumptions under which they do).

    Authors: The fixed-point characterization (Theorem 3.1) and shadow relation (Theorem 3.2) are proved in Section 3 using only the algebraic definitions of the relaxed operator, the reflection, and the projection; the arguments are purely set-theoretic and do not invoke convexity of the sets C_i. The same holds for the local convergence analysis in Section 4. These results are therefore valid for arbitrary closed sets. The phrase “convex relaxation” in the abstract is intended to describe the role of the relaxation parameter (which interpolates between the classical Douglas–Rachford and projection operators) rather than to impose convexity on the underlying sets. We will revise the abstract and add an explicit remark at the end of the introduction stating that Theorems 3.1, 3.2 and 4.1 hold without any convexity assumption on the sets. revision: yes

Circularity Check

0 steps flagged

No significant circularity

full rationale

The paper's central claims involve characterizing fixed points of the cyclic relaxed Douglas-Rachford operator, relating their shadows to those of cyclic projections, and deriving local convergence conditions under stated assumptions. These steps rely on standard operator-theoretic arguments and an external citation to Borwein and Tam (2014) for the base cyclic DR algorithm; no equations reduce by construction to fitted parameters, self-definitions, or load-bearing self-citations within the provided text. The modeling choice of viewing the method as an intermediate relaxation is an explicit ansatz but does not create a definitional loop or force the fixed-point relations. The derivation chain remains self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

1 free parameters · 1 axioms · 0 invented entities

The central claims rest on standard properties of projection and reflection operators in Hilbert space together with a single tunable relaxation parameter whose value is not derived from first principles.

free parameters (1)
  • relaxation parameter
    The parameter that interpolates between cyclic Douglas-Rachford and cyclic projections is introduced to define the algorithm and is not fixed by any theorem in the abstract.
axioms (1)
  • standard math Projections and reflections are well-defined nonexpansive operators on the underlying Hilbert space
    Invoked implicitly when the algorithm is defined via compositions of these operators.

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Reference graph

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