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arxiv: 2512.10366 · v2 · pith:N56PKYT7new · submitted 2025-12-11 · 🧮 math.OC

Primal-dual splitting for structured composite monotone inclusions with or without cocoercivity

Minh N. Dao , Hung M. Phan , Matthew K. Tam , Thang D. Truong This is my paper

Pith reviewed 2026-05-16 23:43 UTC · model grok-4.3

classification 🧮 math.OC
keywords primal-dual splittingmonotone inclusionscocoercivitycomposite operatorsconvergence analysisgraph-based algorithmsdecentralized optimizationfused LASSO
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The pith

A primal-dual splitting algorithm solves structured monotone inclusions without requiring cocoercivity on single-valued operators.

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

The paper introduces a primal-dual splitting algorithm for composite monotone inclusions that combine multiple set-valued operators, their compositions with bounded linear operators, and single-valued operators that may lack cocoercivity. This single method unifies several existing splitting algorithms under one convergence analysis while serving as a template to derive new variants that incorporate graph-based structures. It works directly on the original problem dimensions instead of lifting to a higher-dimensional product space and supports operator-specific stepsizes and resolvent parameters. The result is a wider range of allowable stepsizes than in recent comparable methods. The approach is shown on a decentralized fused LASSO problem for cancer detection.

Core claim

The proposed primal-dual splitting algorithm converges for the broad class of structured composite monotone inclusions under monotonicity of the set-valued operators and boundedness of the linear operators, with parameters chosen to satisfy explicit inequalities. This unifies contemporary algorithms, reduces dimensionality relative to product-space techniques, accommodates different cocoercive or Lipschitz constants together with different resolvent parameters, and permits larger stepsize ranges than recent methods, all within a transparent single analysis.

What carries the argument

The primal-dual splitting iteration that directly processes the structured inclusion with finitely many set-valued operators, linear compositions, and possibly non-cocoercive single-valued terms.

If this is right

  • Several existing primal-dual and forward-backward splitting algorithms arise as direct special cases.
  • New algorithms with arbitrary graph-based operator structures can be generated under the same convergence guarantee.
  • Problem dimensionality stays lower than in standard product-space reformulations of the inclusion.
  • The allowable stepsize interval is strictly larger than the intervals obtained by recent comparable methods.
  • Each operator can use its own cocoercive constant, Lipschitz constant, and resolvent parameter.

Where Pith is reading between the lines

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

  • The graph-based blueprint could directly support distributed optimization over networks without extra reformulation steps.
  • The transparent analysis may simplify the derivation of stochastic or inexact variants for large-scale data problems.
  • Larger stepsizes could reduce the number of iterations needed in regularized regression tasks such as the demonstrated LASSO example.
  • The framework suggests a route to handle time-varying or parameter-dependent operators while preserving the same proof structure.

Load-bearing premise

The set-valued operators are monotone, the linear operators are bounded, and the stepsizes together with resolvent parameters satisfy the stated inequalities that guarantee convergence.

What would settle it

A concrete monotone inclusion instance, with parameters chosen to meet the paper's conditions, on which the iterates fail to converge to a solution would disprove the convergence claim.

Figures

Figures reproduced from arXiv: 2512.10366 by Hung M. Phan, Matthew K. Tam, Minh N. Dao, Thang D. Truong.

Figure 1
Figure 1. Figure 1: Coefficient matrix structure It is worthwhile mentioning that condition (9) is weaker and easier to verify than the condition in [18, Remark 3.1(i)]. Moreover, based on [PITH_FULL_IMAGE:figures/full_fig_p008_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Impact of varying γˆ on the performance of the sequential and complete graph algorithms. changing ηˆ. The corresponding results for the sequential graph algorithm and the complete graph algorithm are shown in Figure 3a and Figure 3b. The experiment indicates that they achieve the best performance when ηˆ is small. Next, we investigate how the algorithms behave as λˆ varies while (a) γˆ = 0.5, λˆ = 0.9 (b) … view at source ↗
Figure 3
Figure 3. Figure 3: Impact of varying ηˆ on the performance of the sequential and complete graph algorithms. γˆ = 0.5, ηˆ = 0.1 are set. Figure 4a and Figure 4b show the result for the sequential graph algorithm 29 [PITH_FULL_IMAGE:figures/full_fig_p029_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Impact of varying λˆ on the performance of the sequential and complete graph algorithms. Now, we compare the performance of three algorithms using γˆ = 0.5, ηˆ = 0.1, and λˆ = 0.9. In Figure 5a, the vector xtrue is represented by red color, followed by xcomplete in blue, xsequential in orange, and xstar in green. All three algorithms converge to the same solution which is resemble to the original vector. T… view at source ↗
Figure 5
Figure 5. Figure 5: The performance of the sequential graph, star graph, and complete graph algorithms. [PITH_FULL_IMAGE:figures/full_fig_p030_5.png] view at source ↗
read the original abstract

In this paper, we propose a primal-dual splitting algorithm for a broad class of structured composite monotone inclusions that involve finitely many set-valued operators, compositions of set-valued operators with bounded linear operators, and single-valued operators possibly without cocoercivity. The proposed algorithm is not only a unification for several contemporary algorithms but also a blueprint to generate new algorithms with graph-based structures using a single transparent convergence analysis. Our approach reduces dimensionality compared with the standard product space technique, which typically reformulates the original problem as the sum of two maximally monotone operators in order to apply splitting methods. It accommodates different cocoercive or Lipschitz constants as well as different resolvent parameters, and yields a larger allowable stepsize range than recent methods. We demonstrate the practicality of the approach by a numerical experiment on cancer detection using the decentralized fused LASSO problem.

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 / 2 minor

Summary. The manuscript proposes a primal-dual splitting algorithm for structured composite monotone inclusions involving finitely many set-valued operators, compositions with bounded linear operators, and single-valued operators possibly without cocoercivity. It claims to unify several existing algorithms, serve as a blueprint for new graph-based algorithms via a single transparent convergence analysis, reduce dimensionality relative to product-space reformulations, accommodate varying cocoercive/Lipschitz constants and resolvent parameters, and yield larger allowable stepsizes than recent methods. The approach is illustrated by a numerical experiment on decentralized fused LASSO for cancer detection.

Significance. If the convergence analysis holds with the claimed generality and stepsize improvements, the work would provide a useful unified framework in monotone operator theory, simplifying analysis for composite inclusions and enabling new algorithm constructions with reduced dimensionality. The explicit handling of non-cocoercive single-valued operators and the graph-based blueprint could facilitate applications in distributed optimization. The numerical demonstration on a practical problem is a positive feature, though stronger quantitative validation would increase impact.

major comments (2)
  1. [§4] §4 (Convergence analysis): The claim of convergence for single-valued operators 'possibly without cocoercivity' requires an explicit statement of any Lipschitz continuity assumption on those operators. The stepsize and resolvent parameter inequalities (e.g., those involving the constants in the Lyapunov function) appear to encode such a bound implicitly; without declaring it in the problem formulation (Section 2), the generality asserted in the abstract and introduction is not fully supported.
  2. [§5] §5 (Numerical experiment): The decentralized fused LASSO experiment for cancer detection is described without quantitative metrics (e.g., objective values, iteration counts, or error measures), baselines, or direct comparison to prior primal-dual methods. This prevents assessment of whether the larger stepsize range translates to measurable practical gains, weakening support for the unification and blueprint claims.
minor comments (2)
  1. [Abstract] Abstract: The phrase 'larger allowable stepsize range than recent methods' should cite the specific compared algorithms and quantify the improvement (e.g., via the ratio of allowable stepsizes).
  2. [§3] Notation: The distinction between the graph-based structure and the standard product-space reformulation could be clarified with a small diagram or explicit dimension count in Section 3.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the constructive comments, which help clarify the presentation of our results. We address each major comment below and will revise the manuscript to strengthen the exposition and supporting evidence.

read point-by-point responses

  1. Referee: [§4] §4 (Convergence analysis): The claim of convergence for single-valued operators 'possibly without cocoercivity' requires an explicit statement of any Lipschitz continuity assumption on those operators. The stepsize and resolvent parameter inequalities (e.g., those involving the constants in the Lyapunov function) appear to encode such a bound implicitly; without declaring it in the problem formulation (Section 2), the generality asserted in the abstract and introduction is not fully supported.

    Authors: We agree that an explicit declaration of the Lipschitz continuity assumption on the single-valued operators is needed in Section 2 to fully align the problem formulation with the generality claimed in the abstract and introduction. While the stepsize conditions in the convergence analysis already incorporate these constants (allowing different values per operator), we will add a clear statement in the problem setup of Section 2 specifying that the single-valued operators are Lipschitz continuous. This revision will not change the analysis or results but will improve transparency. revision: yes

  2. Referee: [§5] §5 (Numerical experiment): The decentralized fused LASSO experiment for cancer detection is described without quantitative metrics (e.g., objective values, iteration counts, or error measures), baselines, or direct comparison to prior primal-dual methods. This prevents assessment of whether the larger stepsize range translates to measurable practical gains, weakening support for the unification and blueprint claims.

    Authors: We acknowledge that the current numerical section provides only a qualitative illustration. In the revision we will augment §5 with quantitative metrics (objective values, iteration counts to tolerance, and error measures), include baseline comparisons against recent primal-dual methods, and report the specific stepsize ranges used. These additions will directly demonstrate the practical advantage of the larger allowable stepsizes and better support the unification claims. revision: yes

Circularity Check

0 steps flagged

Convergence analysis is self-contained using standard monotone-operator tools

full rationale

The paper derives its primal-dual splitting scheme and convergence guarantees directly from standard fixed-point and Lyapunov arguments for monotone inclusions (maximal monotonicity, bounded linear operators, and parameter inequalities). No step reduces by construction to a fitted input, self-definition, or self-citation chain; the dimensionality reduction and stepsize enlargement follow from the explicit product-space reformulation and the choice of resolvent parameters. The unification claim is a consequence of the single analysis rather than an input. External benchmarks (prior splitting methods) are cited for comparison but are not load-bearing for the new proof.

Axiom & Free-Parameter Ledger

1 free parameters · 2 axioms · 0 invented entities

The central claim rests on standard domain assumptions of monotone operator theory together with parameter choices that must satisfy explicit inequalities; no new entities are postulated.

free parameters (1)
  • resolvent parameters and stepsizes
    Different resolvent parameters per operator and stepsizes chosen to satisfy the convergence inequalities; these are free parameters tuned for each instance.
axioms (2)
  • domain assumption Set-valued operators are monotone (typically maximally monotone).
    Invoked throughout the monotone inclusion formulation and convergence proof.
  • domain assumption Linear operators are bounded.
    Required for the compositions with set-valued operators to remain well-defined.

pith-pipeline@v0.9.0 · 5452 in / 1351 out tokens · 56570 ms · 2026-05-16T23:43:45.236999+00:00 · methodology

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Reference graph

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