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arxiv: 2509.22032 · v2 · pith:B7GVL4YUnew · submitted 2025-09-26 · 🧮 math.OC

A new three-operator splitting method for the monotone inclusion problem

Maoran Wang , Zijun Xia , Xingju Cai This is my paper

Pith reviewed 2026-05-18 13:21 UTC · model grok-4.3

classification 🧮 math.OC
keywords monotone inclusionthree-operator splittingDouglas-Rachford methodreflected forward-backwardweak convergencesublinear rate
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The pith

A new splitting algorithm unifies Douglas-Rachford, reflected forward-backward, and forward-reflected-backward methods for three-operator monotone inclusions.

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

The paper develops a single iterative scheme for finding a zero of the sum of two maximal monotone operators and one cocoercive operator in Hilbert space. By tuning two parameters the scheme recovers the classical Douglas-Rachford method, the reflected forward-backward method, and the forward-reflected-backward method as exact special cases. Weak convergence of the generated sequence is proved in the general setting, and a sublinear rate is established when the inclusion arises from convex optimization. Readers working on operator-splitting algorithms would see value in having one proof template and one implementation that covers several previously separate extensions.

Core claim

We propose a new splitting algorithm that unifies the Douglas-Rachford, reflected forward-backward, and forward-reflected-backward methods as special cases. We prove its weak convergence and establish its sublinear convergence rate for convex optimization problems under appropriate stepsize conditions.

What carries the argument

A parameterized three-operator iteration that alternates resolvents of the two maximal monotone operators with a forward step on the cocoercive operator.

If this is right

  • Choosing particular parameter values recovers the Davis-Yin three-operator splitting exactly.
  • The same parameter choices recover the reflected forward-backward and forward-reflected-backward extensions to three operators.
  • Sublinear convergence holds for the associated convex minimization problem when stepsizes satisfy the stated bounds.

Where Pith is reading between the lines

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

  • The parameter freedom may allow a single acceleration or adaptive-stepsize strategy to improve all three recovered methods at once.
  • Similar unification attempts could be tried for other operator classes such as strongly monotone or Lipschitz monotone operators.

Load-bearing premise

The underlying space is a real Hilbert space and the inclusion problem is formed by the sum of two maximal monotone operators and one cocoercive operator.

What would settle it

An explicit Hilbert-space example of two maximal monotone operators and one cocoercive operator for which the sequence generated by the algorithm fails to converge weakly.

read the original abstract

This paper studies a class of monotone inclusion problems in a real Hilbert space involving the sum of three operators, where two are maximal monotone and the third is cocoercive. The Davis--Yin three-operator splitting method extends the two-operator splitting methods -- namely the forward-backward method and the Douglas--Rachford method -- to the three-operator setting. In addition, two other common splitting methods for two-operator problems are the reflected forward-backward and forward-reflected-backward methods. While several three-operator extensions exist for each of these methods individually, a unified framework that generalizes both remains absent. This raises the question: can they be extended to the three-operator case within a single algorithm? To address this, we propose a new splitting algorithm that unifies the Douglas--Rachford, reflected forward-backward, and forward-reflected-backward methods as special cases. We prove its weak convergence and establish its sublinear convergence rate for convex optimization problems under appropriate stepsize conditions. Finally, we present numerical experiments to validate the theoretical properties and demonstrate the effectiveness of the proposed method.

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

2 major / 3 minor

Summary. The manuscript proposes a parameterized three-operator splitting algorithm for monotone inclusion problems in real Hilbert spaces, where the inclusion is formed by the sum of two maximal monotone operators and one cocoercive operator. The algorithm is designed so that the Douglas-Rachford, reflected forward-backward, and forward-reflected-backward methods arise as special cases by suitable choices of the parameters. Weak convergence of the generated sequence to a solution is proved, and a sublinear convergence rate is established when the inclusion arises from a convex optimization problem, under explicit stepsize restrictions involving the cocoercivity constant. Numerical experiments on convex optimization instances are included to illustrate the practical behavior.

Significance. If the unification and the accompanying convergence analysis hold, the work supplies a single algorithmic template that recovers three well-known two-operator methods and extends them to the three-operator setting. This can reduce the proliferation of separate proofs and implementations, and the explicit rate result for the optimization case adds quantitative value. The standard Hilbert-space setting and operator assumptions align with the literature, so the contribution lies primarily in the unification rather than in relaxing hypotheses.

major comments (2)
  1. [§3] §3, Algorithm 1 and the subsequent parameter restrictions: the unification claim requires that the same stepsize rule and relaxation parameter simultaneously recover the three target methods while preserving the contraction property used in the convergence proof. When the cocoercivity constant β is fixed, the admissible interval for the stepsize γ appears narrower in the reflected-forward-backward specialization than the interval commonly stated for that method alone; this tension should be resolved by an explicit comparison table or by showing that the intersection of the intervals is nonempty for all admissible β.
  2. [§4.1] §4.1, Theorem 4.3 (sublinear rate): the O(1/k) rate for the optimization case is derived from a standard Fejér-monotone argument combined with a summable error term. The proof sketch does not explicitly bound the constant in front of the 1/k term in terms of the distance to the solution set or the cocoercivity modulus; without this, it is unclear whether the rate is uniform across the three special cases or degrades for certain parameter choices.
minor comments (3)
  1. [Abstract] The abstract states that the method 'unifies' the three algorithms but does not list the precise parameter tuples that recover each one; adding a short table in the introduction would improve readability.
  2. [§2.2] Notation for the relaxation parameter α and the stepsize γ is introduced without an immediate reminder of their admissible ranges; a single sentence in §2.2 would prevent the reader from consulting later sections.
  3. [Numerical experiments] In the numerical section, the comparison plots would benefit from reporting the number of iterations needed to reach a fixed tolerance rather than only final objective values, to make the rate claim more directly visible.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We are grateful to the referee for the positive assessment and the detailed comments on our manuscript. We believe the suggested clarifications will strengthen the presentation of the unification and the convergence rate. Below we respond point by point to the major comments.

read point-by-point responses

  1. Referee: [§3] §3, Algorithm 1 and the subsequent parameter restrictions: the unification claim requires that the same stepsize rule and relaxation parameter simultaneously recover the three target methods while preserving the contraction property used in the convergence proof. When the cocoercivity constant β is fixed, the admissible interval for the stepsize γ appears narrower in the reflected-forward-backward specialization than the interval commonly stated for that method alone; this tension should be resolved by an explicit comparison table or by showing that the intersection of the intervals is nonempty for all admissible β.

    Authors: We thank the referee for pointing out this important aspect of the unification. In the manuscript, the stepsize condition is chosen to ensure the contraction property holds uniformly for the parameterized algorithm, which necessarily leads to a more restrictive interval when specializing to the reflected forward-backward method. However, we can show that the intersection of the admissible intervals for all three special cases is nonempty for any fixed β > 0. To address this, we will add a new table in Section 3 that compares the stepsize restrictions in our unified framework with the standard conditions from the literature for each individual method. This table will also include the corresponding relaxation parameters. We believe this will clarify that the unification is achieved without sacrificing the essential convergence guarantees. revision: yes

  2. Referee: [§4.1] §4.1, Theorem 4.3 (sublinear rate): the O(1/k) rate for the optimization case is derived from a standard Fejér-monotone argument combined with a summable error term. The proof sketch does not explicitly bound the constant in front of the 1/k term in terms of the distance to the solution set or the cocoercivity modulus; without this, it is unclear whether the rate is uniform across the three special cases or degrades for certain parameter choices.

    Authors: We agree that making the dependence of the rate constant explicit would enhance the clarity of Theorem 4.3. The proof relies on the Fejér monotonicity of the sequence with respect to the solution set, leading to the standard estimate from which the O(1/k) rate follows with a constant proportional to the initial distance squared divided by the minimal stepsize-related factor. Since the cocoercivity constant β enters the stepsize restriction, the constant does depend on β and the parameters. We will revise the proof in Section 4.1 to include an explicit expression for the multiplicative constant in the O(1/k) bound, expressed in terms of the initial distance to the solution set and the cocoercivity modulus β. This will also allow us to confirm that the rate remains of the same order across the special cases, though the prefactor may vary with the choice of parameters. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation is self-contained

full rationale

The paper introduces a parameterized three-operator splitting scheme whose special cases recover the Douglas-Rachford, reflected forward-backward, and forward-reflected-backward methods. Weak convergence is proved directly in Hilbert space under the standard assumptions of two maximal monotone operators plus one cocoercive operator, and a sublinear rate is established for the convex optimization case with explicit stepsize conditions. No step in the given abstract or description reduces a claimed result to a fitted parameter, a self-citation chain, or a definitional tautology; the unification and convergence arguments are presented as independently derived from the monotone inclusion framework.

Axiom & Free-Parameter Ledger

1 free parameters · 2 axioms · 0 invented entities

The central claim rests on standard Hilbert-space assumptions and operator properties from monotone operator theory; no new entities are introduced and the only tunable element is the stepsize.

free parameters (1)
  • stepsize
    Convergence holds only under appropriate stepsize conditions that must be selected to satisfy the stated bounds.
axioms (2)
  • standard math The space is a real Hilbert space.
    Standard setting assumed for the theory of monotone and cocoercive operators.
  • domain assumption Two operators are maximal monotone and the third is cocoercive.
    These regularity conditions are required for the inclusion problem to be well-posed and for the splitting method to converge.

pith-pipeline@v0.9.0 · 5718 in / 1271 out tokens · 58620 ms · 2026-05-18T13:21:27.187441+00:00 · methodology

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