{"id":"6310f0c9-0f66-4c65-9555-3f1375c3201e","arxiv_id":"2501.07807","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"The ergodic (averaged) sequence of the semi-proximal Peaceman-Rachford splitting method converges to a solution of convex optimization problems with an O(1/k) complexity bound.","lead":"This paper proves that averaging the iterations of the Peaceman-Rachford splitting method, a classical optimization algorithm that can fail to converge point by point, makes the iterates converge for convex optimization problems. The authors also report that the averaged, restarted method outperforms the Douglas-Rachford splitting method on a standard linear programming benchmark.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Ergodic PR convergence rests on Proposition 2, imported from the companion preprint [23], which asserts the pADMM-to-dPPA equivalence and Lipschitz continuity of (M+T)^{-1}; if either assertion fails, Corollary 1 is unsupported.","rationale":"The reader's weakest_assumption identifies exactly the most load-bearing dependency: Proposition 2 from the companion preprint [23]. I find no internal contradiction in Theorem 1 that cannot be repaired by replacing the questionable Opial paragraph with a continuity argument, and the numerical claims, while lacking code and data, are not the central claim. The core issue is that the key transfer from the abstract dPPA result to the pADMM is not proved in this manuscript. If Proposition 2 holds, the central claim is likely correct; if it fails, the proof collapses. Since the reader already marked the verdict CONDITIONAL and my concern does not move it, the appropriate stress-test verdict is UNCHANGED.","tokens_in":17482,"tokens_out":23902,"duration_ms":229698,"concrete_test":"Independently re-derive Proposition 2 without invoking [23]: write the optimality conditions of Steps 1-3 of Algorithm 1 for arbitrary positive semidefinite T1,T2, eliminate the intermediate variables, and verify that the trial point wbar^k equals (M+T)^{-1}Mw^k for M defined in (15). Then derive the Lipschitz continuity of (M+T)^{-1} from the positive definiteness in Assumption 2, or construct a counterexample with T1,T2 satisfying Assumptions 1-2 where the Lipschitz bound or the equality fails. If either derivation requires an assumption not stated in the paper, Corollary 1 is not established.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The paper's central claim is Corollary 1: the ergodic sequence generated by the pADMM with rho in (0,2], including the PR splitting method with semi-proximal terms at rho=2, converges to a KKT point. The proof chain has two load-bearing links: Theorem 1 establishes ergodic convergence for an abstract dPPA, and Proposition 2 (imported verbatim from [23]) asserts that Algorithm 1 is exactly the dPPA with M in (15), and that (M+T)^{-1} is Lipschitz continuous. Both assertions are necessary. Theorem 1 uses the Lipschitz property to bound the trial sequence and to apply Baillon's theorem to the shadow sequence; Corollary 1 uses the operator equality to transfer the dPPA convergence result to pADMM iterates. The manuscript gives no derivation or independent verification of Proposition 2, and [23] is an overlapping-author preprint. If the equivalence or the Lipschitz property fails for some positive semidefinite T1,T2 allowed by Assumptions 1-2, the claimed convergence of the ergodic PR sequence does not follow from the arguments presented. A smaller, repairable defect also appears in Theorem 1: the Opial-type step is not valid for the degenerate M-seminorm, but the conclusion can be reached by continuity of the M-seminorm, so this is not the critical threat.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper proves ergodic convergence of the preconditioned ADMM (pADMM) with relaxation parameter ρ∈(0,2], which includes the Peaceman–Rachford splitting method with semi-proximal terms at ρ=2, for convex optimization problems (1). The proof route is: reformulate the pADMM as a degenerate proximal point algorithm (dPPA) with preconditioner M (Proposition 2, imported from the companion preprint [23]); prove an ergodic convergence theorem for the dPPA (Theorem 1) using an ε-enlargement argument (Proposition 1) and Baillon's theorem; then transfer the result to the pADMM (Corollary 1). The paper also states an O(1/k) ergodic iteration complexity bound (Theorem 2) and reports numerical experiments on Mittelmann's LP benchmark in which a restarted ergodic PR method outperforms DR variants.","tokens_in":17723,"tokens_out":18553,"duration_ms":149589,"significance":"If the main theorem is correct, it settles a problem stated as open in the introduction: the averaged PR splitting method converges for convex optimization without strong monotonicity, and the convergence is for the primal–dual ergodic sequence rather than only a shadow sequence. The O(1/k) complexity bound and the numerical study on 49 LP instances are useful additions, and the analytical example in Appendix A nicely illustrates the obstruction to pointwise convergence. The dPPA ergodic result and its ε-enlargement proof appear to be the original contribution of this manuscript. However, the central bridge from the pADMM to the dPPA is not proved in the manuscript and is taken from an overlapping-author preprint, and two local steps in the proof of Theorem 1 and Proposition 1 need correction. The result is believable but the manuscript is not yet self-contained.","major_comments":[{"comment":"The central claim of the paper rests on results imported without proof from the companion preprint [23] by four of the five authors: the equivalence between the pADMM in Algorithm 1 and the dPPA (6) under the preconditioner M in (15), the Lipschitz continuity of (M+T)^{-1}, and the estimate from [23, Lemma 3.6] used in Theorem 2. Corollary 1 transfers Theorem 1 to the pADMM precisely through this equivalence, so a failure of any of these ingredients would invalidate the main result. The manuscript must either include proofs of these supporting statements or cite a peer-reviewed published version; relying on an unreviewed preprint is not sufficient for a load-bearing step.","section":"Section 3, Proposition 2; also Theorem 2 via [23, Lemma 3.6]"},{"comment":"The displayed chain does not justify the final bound ‖w0−w∗‖_M^2/(2ρ(k+1)). The authors use the estimate ‖w0−w̄_a^k‖_M ≤ 2‖w0−w∗‖_M but then substitute only ‖w0−w∗‖_M into the term 2‖w^{k+1}−w0‖_M‖w0−w̄_a^k‖_M. With a = ‖w^{k+1}−w0‖_M ≤ 2‖w0−w∗‖_M and b = ‖w0−w̄_a^k‖_M ≤ 2‖w0−w∗‖_M, the expression −a^2+2ab is bounded by 4‖w0−w∗‖_M^2, not by ‖w0−w∗‖_M^2. The stated bound (and the constant in Eq. (20) of Theorem 2) should be corrected. A corrected bound with an O(1/k) rate is still sufficient for Theorem 1.","section":"Section 2, proof of Proposition 1(b), Eq. (10)"},{"comment":"The Opial-type argument is not valid for the degenerate seminorm ‖·‖_M. When M has a nontrivial kernel, the strict inequality liminf ‖x_i−w∗‖_M < liminf ‖x_i−w∗_a‖_M can fail even when w∗ ≠ w∗_a, so the subsequent contradiction is not justified. The uniqueness of cluster points can be proved directly: if a subsequence w̄_a^{k_i} converges in norm to w∗, then by continuity of the M-seminorm ‖w∗−w∗_a‖_M = 0; since both w∗ and w∗_a lie in T^{-1}(0), Mw∗ = Mw∗_a and the single-valuedness of (M+T)^{-1} give w∗ = w∗_a. The Opial step should be replaced by this (or an equivalent) argument.","section":"Section 2, proof of Theorem 1, final paragraph"}],"minor_comments":[{"comment":"The norm in the second displayed inequality should be ‖z^k − z∗‖^2_{T2}, not ‖z^k − z∗‖^2_{T1}.","section":"Appendix B, Eq. (29)"},{"comment":"Baillon's theorem yields weak convergence of the Cesàro means; the authors should state explicitly that weak and strong convergence coincide in the finite-dimensional setting before writing ‖u_a^k − C^*w_a^*‖ → 0.","section":"Section 2, proof of Theorem 1"},{"comment":"The running header reads 'Ergodical Peaceman-Rachford Splitting Method'; 'Ergodical' should be 'Ergodic'.","section":"Page 3, running header"},{"comment":"The phrase 'combing with' should be 'combining with'.","section":"Appendix B, proof of Theorem 2"}],"recommendation":"major_revision","confidential_remarks":"The manuscript depends heavily on the companion preprint [23] by the same research group. In addition to the technical corrections above, the editor may wish to require that the authors provide the proofs of the imported results or cite an accepted/published version of [23], and to clarify the exact novelty of this manuscript relative to [23] and [4]. The main result is plausible and the numerical evidence is interesting, but the current level of self-containedness is below what is normally expected for a journal publication."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The main result is almost certainly correct: ergodic convergence of the Peaceman-Rachford splitting method for convex optimization under only KKT nonemptiness and well-defined subproblems, with O(1/k) complexity for semi-proximal terms and KKT residuals. That answers the open question the authors state, and the dPPA ergodic theorem behind it is a real contribution, not a tautology. The paper is also honest about the limits: the Appendix A example shows pointwise divergence and even divergence of the unaveraged ergodic sequence {w^k_a}, so Corollary 1 is the best you can hope for in that direction.\n\nWhat is new here is the actual proof for rho in (0,2] and the clean O(1/k) bounds, which improve on prior results by extending to semi-proximal terms and KKT residuals. The numerical experiments on LP instances suggest the restarted ergodic PR method is practically competitive, though without code or data the empirical claims are suggestive rather than conclusive.\n\nThe soft spot is real and load-bearing. Corollary 1 relies on Proposition 2, which asserts that the pADMM in Algorithm 1 coincides with the dPPA with the preconditioner M in (15) and that (M+T)^{-1} is Lipschitz. That proposition is imported verbatim from the companion preprint [23] by four of the same authors, and the manuscript gives no derivation or independent verification. If that equivalence fails for some positive semidefinite T1 and T2 allowed by Assumptions 1–2, the ergodic convergence result does not follow from the arguments given. A referee should check that link carefully; it is likely correct given the authors' track record, but it has to be pulled into the paper or verified.\n\nSmaller issues: the Opial-type step in Theorem 1 uses a degenerate M-seminorm and is not valid as written; the conclusion can be reached by continuity of the M-seminorm, so this is repairable. The factor-of-2 bound in Proposition 1 is compressed but follows from M-firm nonexpansiveness. These are minor blemishes, not threats to the core argument.\n\nWho should read it: anyone working on operator splitting, ADMM theory, or large-scale convex optimization. It deserves a serious referee. My recommendation: send it to review, and ask the authors to either prove Proposition 2 or give a precise reference with a self-contained statement, fix the Opial step, and release code/data for the numerics. If those conditions are met, this is a solid and citable paper.","headline":"The central claim—ergodic PR convergence for convex programs—is plausible and important; the main risk is the unverified equivalence imported from companion preprint [23], which a referee should check.","tokens_in":18317,"tokens_out":2775,"would_cite":true,"duration_ms":27125,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["90C05","90C06","90C25"],"pacs":[],"model":"deepseek-v4-flash","headline":"The averaged Peaceman-Rachford splitting method provably converges for convex optimization.","keywords":["Peaceman-Rachford splitting","ergodic convergence","convex optimization","degenerate proximal point algorithm","preconditioned ADMM","linear programming","restart strategy"],"falsifier":"Run Algorithm 1 with $\\rho=2$ and nonzero semi-proximal terms $T_1, T_2$ on a small convex problem with known solution, compute the dPPA iterates in (6) with $M$ as in (15) from the same starting point, and compare; any mismatch of the sequences, or any example where $(M+T)^{-1}$ fails to be Lipschitz, would invalidate Proposition 2 and with it Corollary 1.","tokens_in":17244,"feed_emoji":"🧮","tokens_out":10907,"duration_ms":81862,"temperature":0.7,"pith_summary":"The paper proves that the Peaceman-Rachford (PR) splitting method—an alternating algorithm for two-block convex optimization—converges when one looks at the running average of its iterates, a property called ergodic convergence. This resolves an open question: PR splitting is known to be faster than its relative Douglas-Rachford whenever it converges pointwise, but pointwise convergence can fail, and whether even the averaged sequence converged was unknown. The proof works for the relaxation factor $\\rho \\in (0,2]$, which includes the classic PR method ($\\rho=2$) and the Douglas-Rachford method ($\\rho=1$), and it allows semi-proximal terms that make the subproblems easier to solve. The paper also reports numerical experiments on linear programming benchmarks showing that a restarted averaged PR method consistently outperforms both the pointwise and averaged Douglas-Rachford methods. If correct, the result makes the averaged PR method a provably reliable choice for large-scale convex optimization.","feed_headline":"Averaged PR splitting provably converges for convex optimization","feed_subtitle":"A new proof covers relaxation factors up to 2; restarting the averaged method beats Douglas-Rachford on LP benchmarks.","key_machinery":"The carrying mechanism is the equivalence between the pADMM and a degenerate proximal point algorithm (dPPA) acting on the KKT operator $T$ of the convex problem, with a special positive semidefinite preconditioner $M$. The operator $M$ is a block matrix built from the constraint operators $B_1, B_2$, the penalty parameter $\\sigma$, and the semi-proximal terms $T_1, T_2$; the dPPA step is $\\bar w = (M+T)^{-1} M w$, followed by relaxation $w^{k+1} = (1-\\rho) w^k + \\rho \\bar w^k$. Because the resolvent map is firmly nonexpansive in the $M$-seminorm, a distance-decrease inequality bounds the distance to any solution, and the classical nonlinear ergodic theorem supplies convergence of the averaged 'shadow' sequence. The Lipschitz continuity of $(M+T)^{-1}$ is what upgrades convergence of the shadow into convergence of the actual averaged iterates.","core_discovery":"The paper's central claim is that the ergodic sequence $\\{\\bar w_a^k\\}$ generated by the preconditioned alternating direction method of multipliers (pADMM) with relaxation parameter $\\rho \\in (0,2]$ converges to a Karush–Kuhn–Tucker point of the convex problem, provided the KKT system has a solution and the semi-proximal terms keep the subproblems well posed. Since $\\rho=2$ in the pADMM is exactly the Peaceman-Rachford splitting method with semi-proximal terms, this gives the first ergodic convergence proof for PR splitting on convex optimization without strong monotonicity. The authors reach this by reformulating the pADMM as a degenerate proximal point algorithm with a positive semidefinite preconditioner, proving an ergodic convergence theorem for that general inclusion problem, and then transferring the result back to the optimization setting. They also establish an $O(1/k)$ ergodic iteration complexity for the objective error, feasibility violation, and KKT residual, with a constant half that of Douglas-Rachford.","pith_inferences":["Inference: the same dPPA reformulation could yield ergodic convergence for other splitting methods (e.g., primal-dual hybrid gradient variants) whenever they admit a degenerate proximal point representation.","Inference: the restart strategy's success hints that the ergodic sequence has oscillatory errors that averaging cancels; a testable extension is to measure how restart frequency interacts with the $\\rho=2$ factor to see whether the factor-two complexity gap explains the observed speed-up.","Inference: the numerical comparison is on linear programs only; applying the restarted averaged PR method to conic or network optimization would show whether the advantage is generic.","Inference: the paper's complexity bound suggests that the averaged PR method's constant improves by a factor of two over DR, so iteration counts before restart should roughly halve; recording those counts on benchmark instances would directly test the mechanism."],"forward_implications":["The averaged PR splitting method ($\\rho=2$) converges for any convex problem satisfying the two structural assumptions, with no strong monotonicity required.","The same proof covers all $\\rho \\in (0,2]$, so the averaged Douglas-Rachford method ($\\rho=1$) converges as a special case.","The averaged PR iterates have worst-case $O(1/k)$ complexity for objective gap, feasibility violation, and KKT residual, with constants half those of the corresponding Douglas-Rachford bounds.","With a restart strategy, the averaged PR method solves LP benchmark instances faster than both pointwise and averaged DR methods, and solves more instances within a time limit.","The pointwise sequence of the PR method can stay unbounded even when the averaged sequence converges, so averaging is the right object to analyze."],"supporting_citations":[{"why":"Supplies the equivalence between the pADMM and the degenerate proximal point algorithm (Proposition 2), the central bridge of the proof.","marker":"[23]"},{"why":"Introduces the degenerate proximal point algorithm and its pointwise convergence analysis, which the ergodic argument extends.","marker":"[4]"},{"why":"Supplies the nonlinear ergodic theorem used to establish convergence of the averaged shadow sequence in the dPPA proof.","marker":"[3]"},{"why":"Provides the restart strategy for primal-dual hybrid gradient on LP that is adopted here to design the restarted ergodic PR method.","marker":"[2]"},{"why":"Frames the pADMM as a splitting method extending the relaxed Douglas-Rachford/Peaceman-Rachford analysis.","marker":"[11]"},{"why":"Defines the relaxed Douglas-Rachford splitting operator and the PR splitting case rho=2, the algorithm being studied.","marker":"[15]"},{"why":"Introduces ε-enlargements of monotone operators used for the KKT-residual complexity bound.","marker":"[5]"},{"why":"Supplies the convex-analysis results (dual equivalence, KKT characterization) behind Assumption 1.","marker":"[19]"}],"fun_headline_variants":["Ergodic PR splitting converges without strong monotonicity","Restarted PR averaging beats DR on LP benchmarks","First ergodic proof for PR splitting on convex optimization","Averaged PR splitting: O(1/k) ergodic convergence for convex problems"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The entire proof depends on the imported claim that the alternating-direction iterations coincide exactly with a degenerate proximal point iteration whose resolvent $(M+T)^{-1}$ is Lipschitz continuous; if that equivalence fails for some semi-proximal operators $T_1, T_2$, the ergodic convergence theorem for the PR method does not follow.","fun_headline_variants_meta":{"raw":{"variants":["Ergodic PR splitting converges without strong monotonicity","Restarted PR averaging beats DR on LP benchmarks","First ergodic proof for PR splitting on convex optimization","Averaged PR splitting: O(1/k) ergodic convergence for convex problems"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.002023,"raw_usage":{"total_tokens":7845,"prompt_tokens":863,"completion_tokens":6982,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":479,"completion_tokens_details":{"reasoning_tokens":6912}},"tokens_in":479,"tokens_out":6982,"duration_ms":40671,"temperature":1.0,"reasoning_tokens":6912,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-10T20:37:27.358497+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Run Algorithm 1 with $\\rho=2$ and nonzero semi-proximal terms $T_1, T_2$ on a small convex problem with known solution, compute the dPPA iterates in (6) with $M$ as in (15) from the same starting point, and compare; any mismatch of the sequences, or any example where $(M+T)^{-1}$ fails to be Lipschitz, would invalidate Proposition 2 and with it Corollary 1.","supporting_citations":[{"cited_title":"arXiv preprint arXiv:24 03.18618 (2024)","cited_arxiv_id":null,"evidence_quote":"Supplies the equivalence between the pADMM and the degenerate proximal point algorithm (Proposition 2), the central bridge of the proof."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Introduces the degenerate proximal point algorithm and its pointwise convergence analysis, which the ergodic argument extends."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the nonlinear ergodic theorem used to establish convergence of the averaged shadow sequence in the dPPA proof."},{"cited_title":"In: Ad- vances in Neural Information Processing System, vol","cited_arxiv_id":null,"evidence_quote":"Provides the restart strategy for primal-dual hybrid gradient on LP that is adopted here to design the restarted ergodic PR method."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Frames the pADMM as a splitting method extending the relaxed Douglas-Rachford/Peaceman-Rachford analysis."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Defines the relaxed Douglas-Rachford splitting operator and the PR splitting case rho=2, the algorithm being studied."},{"cited_title":"Set-Valued Ana l","cited_arxiv_id":null,"evidence_quote":"Introduces ε-enlargements of monotone operators used for the KKT-residual complexity bound."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the convex-analysis results (dual equivalence, KKT characterization) behind Assumption 1."}],"review_version":1}