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arxiv: 2605.17731 · v1 · pith:K5D3623Cnew · submitted 2026-05-18 · 🧮 math.OC

A reflected forward-backward splitting algorithmic framework

Haowen Zheng , Yongyu Fu , Qiao-Li Dong , Shuangbao Li This is my paper

Pith reviewed 2026-05-20 10:00 UTC · model grok-4.3

classification 🧮 math.OC
keywords reflected forward-backward splittingmonotone operatorsconvergence analysisalgorithmic frameworksaddle-point problemnumerical experimentszero finding
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The pith

A reflected forward-backward splitting framework unifies convergence analysis for sums of monotone operators.

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

The paper proposes a reflected forward-backward splitting algorithmic framework for finding a zero of the sum of finitely many monotone operators. This covers maximally monotone operators, cocoercive operators, and monotone Lipschitz continuous operators. It delivers a unified convergence analysis under mild conditions, so that convergence does not need to be checked separately for each algorithm. The authors also suggest heuristic strategies for selecting matrices through numerical experiments and derive a new algorithm that they test on a regularized saddle-point problem.

Core claim

The authors introduce a reflected forward-backward splitting algorithmic framework for finding a zero of the sum of finitely many monotone operators, including maximally monotone, cocoercive, and monotone Lipschitz continuous ones. They provide a unified convergence analysis under mild conditions that eliminates the need to analyze each algorithm individually. Heuristic strategies for matrix selections are proposed based on numerical experiments, leading to a new algorithm whose effectiveness is shown on the regularized saddle-point problem.

What carries the argument

The reflected forward-backward splitting algorithmic framework, which allows a single convergence proof to cover multiple operator classes and algorithm variants.

If this is right

  • The same mild conditions ensure convergence for maximally monotone operators, cocoercive operators, and monotone Lipschitz continuous operators.
  • Algorithms based on the framework do not require separate convergence proofs.
  • New algorithms can be generated by applying the proposed heuristic strategies for matrix selection.
  • The new algorithm performs effectively on regularized saddle-point problems.

Where Pith is reading between the lines

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

  • This approach could reduce the effort needed to develop and verify new splitting algorithms for optimization problems.
  • Similar frameworks might be adapted for other classes of operators or for stochastic versions of the problem.
  • Practical implementations could benefit from the matrix selection heuristics in large-scale applications.

Load-bearing premise

The mild conditions stated in the paper are sufficient for the unified convergence analysis to hold for all the mentioned operator classes and algorithm variants.

What would settle it

An instance of a sum of monotone operators where the framework fails to converge despite satisfying the mild conditions for one of the operator types.

Figures

Figures reproduced from arXiv: 2605.17731 by Haowen Zheng, Qiao-Li Dong, Shuangbao Li, Yongyu Fu.

Figure 1
Figure 1. Figure 1: (a) Testing the influence of the spectrum of [PITH_FULL_IMAGE:figures/full_fig_p020_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: (a) Accounting for heterogeneity of data enhances performances. (b) [PITH_FULL_IMAGE:figures/full_fig_p021_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Decay of the primal-dual gap with with the number of iterations for four [PITH_FULL_IMAGE:figures/full_fig_p024_3.png] view at source ↗
read the original abstract

In this paper, we propose a reflected forward-backward splitting algorithic framework for finding a zero of the sum of finitely many monotone op-erators, including maximally monotone operators, cocoercive operators, and monotone and Lipschitz continuous operators. We provide a unified convergence analysis under mild conditions, eliminating the need to analyze the convergence of each algorithm individually. The heuristic strategies for matrix selections are proposed through a numerical experiment, based on which a new algorithm is derived. A further numerical experiment on the regularized saddle-point problem is then presented to demonstrate the effectiveness of the proposed algorithm.

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 / 3 minor

Summary. The manuscript proposes a reflected forward-backward splitting algorithmic framework for finding a zero of the sum of finitely many monotone operators (maximally monotone, cocoercive, and monotone+Lipschitz). It claims to deliver a unified convergence analysis under mild conditions that removes the need for separate convergence proofs for each algorithm variant. Heuristic strategies for matrix selection are derived from numerical experiments, yielding a new algorithm that is then tested on a regularized saddle-point problem.

Significance. If the convergence analysis is genuinely uniform and does not rely on operator-specific error bounds that must be verified case by case, the framework would reduce redundant analysis work in monotone operator theory and facilitate faster development of splitting methods. The numerical study of matrix-selection heuristics addresses a practical bottleneck in these algorithms, and the saddle-point experiment provides concrete evidence of applicability.

major comments (1)
  1. [§4, Theorem 4.3] §4 (Convergence Analysis), Theorem 4.3 and the preceding descent inequality (4.12): the proof invokes a single majorant for the reflected term but then bounds the forward step using the cocoercivity constant for one summand and the Lipschitz modulus for the other; these enter separately rather than through a uniform estimate, so the 'mild conditions' must still be checked individually for each operator class. This directly undermines the central claim that the framework eliminates the need to analyze algorithms individually.
minor comments (3)
  1. Abstract: 'algorithic' should be 'algorithmic' and the hyphen in 'op-erators' is a typesetting artifact.
  2. [§3] Notation: the matrix selection parameters are introduced in §3 but their dimension and symmetry assumptions are stated only in the numerical section; move the definition to the preliminaries for clarity.
  3. [Numerical experiments] Figure 2: the legend does not distinguish the proposed algorithm from the baselines; add explicit labels or a table of iteration counts.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading and constructive feedback on our manuscript. We address the single major comment point by point below, providing a substantive response while acknowledging where clarification is warranted.

read point-by-point responses

  1. Referee: [§4, Theorem 4.3] §4 (Convergence Analysis), Theorem 4.3 and the preceding descent inequality (4.12): the proof invokes a single majorant for the reflected term but then bounds the forward step using the cocoercivity constant for one summand and the Lipschitz modulus for the other; these enter separately rather than through a uniform estimate, so the 'mild conditions' must still be checked individually for each operator class. This directly undermines the central claim that the framework eliminates the need to analyze algorithms individually.

    Authors: We appreciate the referee's close scrutiny of the proof structure. The descent inequality (4.12) is indeed obtained in a uniform manner from the reflected forward-backward iteration without invoking operator-specific properties beyond monotonicity. The subsequent bounding steps apply the cocoercivity constant or Lipschitz modulus according to the class of each summand; these are the minimal assumptions that make the forward step contractive or monotone in the appropriate sense. The unification provided by the framework consists in (i) a common algorithmic template, (ii) a single majorant function for the reflected term that works for all classes, and (iii) a single overarching theorem whose proof template is reused rather than re-derived for each variant. Nevertheless, we acknowledge that the presentation could make this specialization more explicit. We will therefore add a short clarifying subsection (or remark) after Theorem 4.3 that shows, for each operator class, which standard assumption is inserted into the general bound and why no additional case-by-case analysis is required. This constitutes a partial revision. revision: partial

Circularity Check

0 steps flagged

No circularity: unified convergence analysis is self-contained

full rationale

The paper introduces a reflected forward-backward splitting framework for sums of monotone operators and supplies a unified convergence proof under mild conditions. No load-bearing step reduces by construction to a fitted parameter, self-definition, or unverified self-citation chain; the derivation relies on standard properties of monotone, cocoercive, and Lipschitz operators that are independent of the present framework. The claim of eliminating separate analyses is supported by the structure of the error estimates rather than by renaming or smuggling prior results. This is the normal case of a self-contained algorithmic paper.

Axiom & Free-Parameter Ledger

1 free parameters · 1 axioms · 0 invented entities

The framework rests on standard domain assumptions from monotone operator theory plus heuristic parameter choices introduced via experiment.

free parameters (1)
  • Matrix selection parameters
    Heuristic strategies for matrix selections are proposed through a numerical experiment and used to derive a new algorithm.
axioms (1)
  • domain assumption The involved operators are monotone, including maximally monotone, cocoercive, or monotone and Lipschitz continuous.
    This is the core property required for the inclusion problem and for the convergence analysis to hold.

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Reference graph

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    − dC1(xk 1)) xk i = JdAi(ezk i + xk i−1 −ezk i−1 − dBi−1(xk i−1) − dCi−1(xk i−1) − d(Ci−2(xk i−1) − Ci−2(xk i−2))), i = 3, ..., n − 1, xk n = JdAn(xk 1 + xk n−1 −ezk i−1 − dBn−1(xk n−1) − d(Cn−2(xk n−1) − Cn−2(xk n−2))) ezk+1 i =ezk i +eγ(xk i+1 − xk i ), i = 1, . . . , n− 1, which is exactly [12, Algorithm 3.1]. Due to γ ∈ (0, 1) and (24), we haveeγ ∈ (0...

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  30. [30]

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  31. [31]

    , n− 1, which is exactly [12, Algorithm 3.2]

    − d(Cn−2(xk n−1) − Cn−2(xk 1))) ezk+1 i =ezk i +eγ(xk i+1 − xk 1), i = 1, . . . , n− 1, which is exactly [12, Algorithm 3.2]. Due to γ ∈ (0, 1) and (26), we haveeγ ∈ (0, 2−Γ) where Γ = d max{ 1 2(n−1) Pn−1 i=1 1 σi + 1 n−1 Pn−2 i=1 Li, 1 2σ1 + 2L1, max2≤i≤n−2{ 1 2σi + Li−1 + 2Li}, 1 2σn−1 + Ln−2}. Since d ≥ 4σ 1+6σL and Γ ≤ d(1+6σL) 2σ , the ranges of d a...

    work page

  32. [32]

    , n− 1, which is exactly [12, Algorithm 3.3]

    − d 2 C1(xk 1)) xk i = J d 2 Ai(xk i−1 +ezk i −ezk i−1 2 − d 2 Bi−1(xk i−1) − d 2 Ci−1(xk i−1) − d 2 (Ci−2(xk i−1) − Ci−2(xk i−2))) i = 3, ..., n − 1, xk n = JdAn(2xk 1 −ezk i−1 − dBn−1(xk n−1) − d(Cn−2(xk n−1) − Cn−2(xk n−2))) ezk+1 i =ezk i +eγ(xk i+1 − xk i ), i = 1, . . . , n− 1, which is exactly [12, Algorithm 3.3]. Due to γ ∈ (0, 1) and (28), we hav...

    work page