{"id":"bf741bf5-7631-4ea3-8787-06d1985fea73","arxiv_id":"2412.02045","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":5.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"Inertial and relaxed versions of the nonlinear forward-backward method converge weakly to solutions of monotone inclusions, and they specialize to accelerated variants of forward-backward, forward-half-reflect-backward, Chambolle-Pock, and Condat-Vu.","lead":"This paper adds momentum, a memory of past steps, and relaxation, an averaging step, to a general class of splitting algorithms for monotone inclusion problems. The authors prove weak convergence of the new iterations and show numerically that the inertial versions speed up image deblurring compared with the standard method.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Invalid Young bound in Proposition 4.2(4) breaks the non-negativity proof of the Lyapunov sequence for λ∈(0,1); Theorem 4.4's weak-convergence claim is unproven in the under-relaxed case.","rationale":"The reader's weakest_assumption was Assumption 4.1, but the more load-bearing issue is internal to the proof of Theorem 4.4: the non-negativity of the Lyapunov sequence C_n, which is essential for the convergence argument, rests on an invalid inequality for λ<1. Assumption 4.1 is imported from [48] and seems standard for the framework; the real soft spot is the Young-type bound in Proposition 4.2(4). The concern is concrete, localized, and falsifiable: if the inequality fails, the proof of Theorem 4.4 does not cover λ∈(0,1). The λ≥1 branch appears unaffected, and the paper's main applications may in practice use λ≥1, so the appropriate action is to require the authors to either correct the bound, add a condition that restores non-negativity, or restrict the theorem to λ∈[1,2). This is why the verdict should remain conditional, albeit for a more specific and substantive reason than the reader's stated assumptions.","tokens_in":24178,"tokens_out":33715,"duration_ms":266402,"concrete_test":"Re-derive the lower bound for C_n in Proposition 4.2(4) using the correct Young coefficient 1/[λ(1+|1−λ|)]. With this correction, check whether the coefficient (1 − ζ/(λ(1+|1−λ|))) can be negative for parameters satisfying (4.22) with λ<1; if yes, construct a one-dimensional numerical instance of Algorithm 2.2 with those parameters and test whether C_n can become negative or the iterates fail to converge. For example, test λ=0.5, ζ=0.8, with α_n and ρ_n chosen to satisfy (4.22); if the Lyapunov sequence is not non-negative or the iterates diverge, the theorem's λ<1 branch is false or at least requires an additional condition.","verdict_should_be":"CONDITIONAL","load_bearing_attack":"In Proposition 4.2, Step (4), the proof of non-negativity of C_n(x) relies on bounding 2λ⟨u_n | x_n−x⟩ by −λ(1+|1−λ|)ζ_{n−1}∥x_n−y_{n−1}∥² − (λζ_{n−1}/(1+|1−λ|))∥x_n−x∥². Since u_n = T_{n−1}p_n − T_{n−1}y_{n−1} with T_{n−1}=γ_{n−1}M_{n−1}−S, Lipschitzness gives ∥u_n∥_{S^{-1}} ≤ ζ_{n−1}∥p_n−y_{n−1}∥_S = (ζ_{n−1}/λ)∥x_n−y_{n−1}∥_S, so 2λ⟨u_n | x_n−x⟩ ≥ −2ζ_{n−1}∥x_n−y_{n−1}∥∥x_n−x∥. Applying Young with c=λ(1+|1−λ|) yields the coefficient ζ_{n−1}/(λ(1+|1−λ|)) on ∥x_n−x∥², not λζ_{n−1}/(1+|1−λ|). For λ∈(0,1), λ/(1+|1−λ|) < 1/(λ(1+|1−λ|)), so the displayed lower bound is false. A concrete counterexample: λ=0.5, ζ=1, T=−Id, x_n=0, x*=1, y_{n−1}=1 gives LHS=−2 but RHS=−1.083. Thus (4.21) and the contradiction argument for C_n≥0 fail in the under-relaxed case, and Theorem 4.4's allowance λ∈(0,1) is not supported by the proof as written.","agreement_with_reader":"disagree"},"referee_report":{"model":"deepseek-v4-flash","summary":"This manuscript studies inertial and relaxed versions of the nonlinear forward-backward with momentum (NFBM) of Morin, Banert, and Giselsson. It proposes two algorithms: Algorithm 2.2, an inertial NFBM with a relaxation step, and Algorithm 2.3, a double-inertial variant with an additional momentum term. For both, it proves weak convergence to a zero of A+C under Assumption 4.1 and the parameter conditions (4.22) and (4.41). It then derives inertial and relaxed versions of forward-backward, forward-half-reflected-backward, Chambolle-Pock, Condat-Vu, and related primal-dual methods, recovering previously known convergence conditions as special cases. Numerical experiments on image restoration compare the variants, and a restart strategy is proposed to preserve acceleration for large step sizes.","tokens_in":24618,"tokens_out":8929,"duration_ms":69712,"significance":"If the convergence theorems are fully correct, the paper gives a valuable unified framework: the same Lyapunov analysis covers inertia, relaxation, and momentum for NFBM and its many special cases. A notable strength is that the special-case reductions are independently checkable and match known results, for example (5.1) recovers the condition in [45] and (5.7) recovers the conditions in [46] and [58]. The algebraic derivations in Section 4 are detailed, and the numerical code is made available. The main caveat is that the proof of the under-relaxed case lambda in (0,1) in Theorem 4.4 contains a false inequality, so the claim as stated is not fully established; the argument for lambda in [1,2) appears sound.","major_comments":[{"comment":"The displayed lower bound in the proof of Proposition 4.2(4) is false for lambda in (0,1). From u_n = T_{n-1}p_n - T_{n-1}y_{n-1} and the zeta_{n-1}-Lipschitz property of T_{n-1}, together with p_n - y_{n-1} = (x_n - y_{n-1})/lambda, one obtains ||u_n||_{S^{-1}} <= (zeta_{n-1}/lambda)||x_n - y_{n-1}||_S, hence 2lambda <u_n | x_n - x> >= -2 zeta_{n-1} ||x_n - y_{n-1}||_S ||x_n - x||_S. Applying Young's inequality with coefficient c = lambda(1+|1-lambda|) gives the term -zeta_{n-1}/(lambda(1+|1-lambda|)) ||x_n - x||_S^2, not the claimed -lambda zeta_{n-1}/(1+|1-lambda|) ||x_n - x||_S^2. For lambda in (0,1) the claimed coefficient is strictly smaller than any coefficient obtainable by this argument, and the concrete example lambda=1/2, zeta=1, x_n=0, x=1, y_{n-1}=1, T=-Id gives LHS=-2 and RHS=-13/12. Therefore the proof that C_n(x) >= 0 via (4.21) fails in the under-relaxed case. Since Theorem 4.4 explicitly allows lambda in (0,1) with (xi_n)_{n>=N0} non-decreasing, the weak-convergence claim in that case is not proved as written. The argument for lambda in [1,2) is unaffected by this particular inequality.","section":"§4.1, Proposition 4.2(4), Eq. (4.21)"}],"minor_comments":[{"comment":"The sentence 'it follows from (2.2)' just before (4.12) should refer to (2.3), since the displayed identity concerns Algorithm 2.2.","section":"§4.1, proof of Proposition 4.2"},{"comment":"The header repeats 'FHRBDI' twice instead of listing FHRBDI and FHRBSDI separately, making the table difficult to read.","section":"Table 3"},{"comment":"The captions and text use 'PNSR' where the standard abbreviation is 'PSNR'.","section":"Figures 3-5"},{"comment":"The phrase 'alpha_n -> alpha in (0,+infinity)' should be 'alpha in [0,1]' in accordance with Algorithm 2.2; the same applies to beta and theta.","section":"§5.2"},{"comment":"The claim that the restart choice alpha_n = alpha for n <= N0 and alpha_n = 0 afterwards satisfies the assumptions of Theorem 4.4 should state explicitly that the theorem's N0 is taken after the restart, since the sequence is non-increasing across the whole index set.","section":"§6, restart strategy"},{"comment":"The labels 'FHRBR' appear to be typos for 'FHRBIR'; the notation should be unified with the text.","section":"Captions of Figures 4 and 5"}],"recommendation":"major_revision","confidential_remarks":"The stress-test concern raised by the reader is valid: the proof of the under-relaxed case lambda in (0,1) in Theorem 4.4 contains a genuine false inequality, and this is a load-bearing part of the stated result. The rest of the paper appears sound, and the special-case reductions provide independent support for the main framework. The authors can resolve this either by repairing the non-negativity proof or by restricting Theorem 4.4 to lambda in [1,2) and adjusting the abstract and Section 5 claims accordingly."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Colleague,\n\nThe paper is a careful and mostly correct extension of the NFBM framework from Morin–Banert–Giselsson. It adds two inertial variants (with relaxation and with double inertia), proves weak convergence under Lyapunov conditions, and shows that the parameter conditions reduce to known ones for FB, FHRB, CP, and CV. The special-case reductions are genuine algebra, not curve-fitting, and the numerical experiments are sensible, if modest.\n\nI looked hard at the stress-test claim that Proposition 4.2(4) contains an invalid Young bound for λ∈(0,1). It does not hold up. The displayed bound in (4.21) is obtained with Young's inequality with parameter t = 1+|1−λ|, which gives exactly the coefficients shown: λ(1+|1−λ|) on ∥x_n−y_{n−1}∥² and λ/(1+|1−λ|) on ∥x_n−x∥². The stress-test used a different parametrization and got a different coefficient; that is a misreading. So the non-negativity proof for C_n in the under-relaxed case stands, and Theorem 4.4's allowance λ∈(0,1) is supported by the proof as written.\n\nThe main limitations are presentation and verification infrastructure: the code URL is missing from the text, the numerical tables report averages without error bars or variance, and the restart strategy, while it satisfies the theorem's hypotheses, is only heuristically motivated. The proofs are long and would benefit from a supplementary derivation of the key Young steps. None of this threatens the central convergence results.\n\nThe paper is for specialists in monotone operator splitting. It will be useful to anyone working on inertial and relaxed variants of FB, FHRB, and primal-dual methods. It deserves a serious referee: the mathematics is original, the reductions to known conditions are reassuring, and I could not find a load-bearing error. My own verdict would be conditional accept with minor revisions.\n\nRecommendation: send it to peer review.","headline":"Solid inertial/relaxed extension of NFBM; the stress-test's Young-bound objection is a false alarm.","tokens_in":25116,"tokens_out":3655,"would_cite":true,"duration_ms":29768,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["47H05","65K05","65K15","90C25"],"pacs":[],"model":"deepseek-v4-flash","headline":"Two inertial variants of the nonlinear forward-backward algorithm with momentum are proved to converge weakly to solutions of monotone inclusions under explicit step-size and momentum conditions.","keywords":["monotone inclusion","nonlinear forward-backward","inertial algorithm","momentum correction","relaxation parameter","weak convergence","forward-half-reflect-backward","image restoration"],"falsifier":"Run Algorithm 2.2 in $\\mathbb{R}^2$ with $A$ the subdifferential of the indicator of the unit square, $C(x)=0.5x$, $S=\\mathrm{Id}$, constant $\\gamma$, $\\alpha_n\\equiv 0.2$, $\\lambda=1.5$, choosing $\\gamma$ so that (4.22) holds, and compute $C_{n+1}(x)-C_n(x)+(\\eta_n-\\xi_{n+1})\\|x_{n+1}-x_n\\|^2$ at each iteration. If this quantity is ever positive, the Lyapunov descent inequality (4.8), which is the core of the proof, is false and the theorem collapses.","tokens_in":23953,"feed_emoji":"⚡","tokens_out":7771,"duration_ms":70562,"temperature":0.7,"pith_summary":"This paper proposes two momentum-accelerated versions of the nonlinear forward-backward with momentum (NFBM) algorithm for solving monotone inclusions of the form $0 \\in (A+C)x$, where $A$ is maximally monotone and $C$ is cocoercive in a real Hilbert space. The first version adds a relaxation parameter $\\lambda$ to an inertial step, and the second adds a double-inertial step with an extra momentum term. The authors prove that, under a strict Lipschitz condition on the warped operators and a bounded step-size regime, the iterates converge weakly to a solution point. Because NFBM specializes to forward-backward, forward-half-reflect-backward, Chambolle\\u2013Pock, and Condat\\u2013V\\u0169, the results yield inertial and relaxed versions of those algorithms and recover known step-size conditions from the literature. Numerical experiments on image restoration show that the inertial variants reduce iteration counts for fixed step sizes, and a restart heuristic gives further gains in CPU time.","feed_headline":"Inertia and relaxation keep monotone-inclusion solvers convergent","feed_subtitle":"New proofs extend weak-convergence guarantees to accelerated FB, FHRB, Chambolle–Pock, and Condat–Vũ.","key_machinery":"The engine of the proof is the Lyapunov function $C_n(x)=\\|x_n-x\\|_S^2 - \\alpha_n\\|x_{n-1}-x\\|_S^2 + 2\\lambda\\langle u_n \\mid x_n-x\\rangle + \\lambda(1+|1-\\lambda|)\\zeta_n\\|p_n-y_{n-1}\\|_S^2 + \\xi_n\\|x_n-x_{n-1}\\|^2$ for Algorithm 2.2, with an analogous function for Algorithm 2.3. The warped resolvent step $(M_n+A)^{-1}$ with momentum correction $u_n$ is the object that generalizes forward-backward, and the coefficients $\\rho_n,\\eta_n,\\xi_n$ encode the step-size, cocoercivity constant, and Lipschitz constants. The central descent inequality $C_{n+1}(x)\\le C_n(x)-(\\eta_n-\\xi_{n+1})\\|x_{n+1}-x_n\\|^2$ is what forces summability of successive differences and eventual weak convergence.","core_discovery":"The central claim is that inertia and relaxation can be added to the nonlinear forward-backward with momentum algorithm without sacrificing convergence. Theorems 4.4 and 4.7 assert that, under Assumption 4.1 and the parameter conditions (4.22) or (4.41), the iterates of Algorithms 2.2 and 2.3 converge weakly to a point in the solution set $Z=\\mathrm{zer}(A+C)$ of the monotone inclusion $0\\in (A+C)x$. The proof constructs a non-increasing Lyapunov function $C_n(x)$ in the $S$-norm that includes momentum-correction terms, derives a descent inequality whose coefficient $\\eta_n-\\xi_{n+1}$ is uniformly positive, and then uses Opial's lemma after showing every weak cluster point lies in $Z$. By choosing the warping operators and metric in specific ways, the same theorems recover inertial and relaxed versions of forward-backward, forward-half-reflect-backward, Chambolle\\u2013Pock, and Condat\\u2013V\\u0169, together with their known step-size restrictions.","pith_inferences":["Because the Lyapunov analysis uses only $S$-norm geometry and the Lipschitz bound on the warped operator, the same descent-inequality template should transfer to fixed-point iterations of averaged operators in other metric spaces, though the authors do not explore that direction.","The restart strategy tested numerically has no convergence proof; a natural next step would be to prove weak convergence for the restarted sequence, which currently holds only because the inertial parameter is eventually zero, making the acceleration heuristic rather than guaranteed.","The numerical observation that larger step-sizes shrink admissible inertia suggests a trade-off: acceleration from momentum and acceleration from step-size compete, and an adaptive scheme that modulates inertia as the step-size approaches its limit might achieve better worst-case rates.","One can test the sharpness of (4.22) on a two-dimensional example where $A$ is a normal cone and $C$ is a linear cocoercive map: if the inequality is necessary as well as sufficient, iterates should fail to converge as soon as $\\eta_n-\\xi_{n+1}$ turns negative."],"forward_implications":["If the theorems hold, every algorithm that is a special case of Algorithm 2.2 or 2.3 inherits a weak-convergence guarantee under explicit parameter inequalities, covering inertial and relaxed FB, FHRB, Chambolle\\u2013Pock, and Condat\\u2013V\\u0169.","The parameter conditions (4.22) and (4.41) give a user-ready recipe: the inertial coefficients must be non-decreasing, and the admissible step-size interval shrinks as the inertial parameters grow for fixed cocoercivity and Lipschitz data.","A direct corollary is that when relaxation is removed ($\\lambda=1$), the inertial FB condition reduces to $1-3\\alpha-\\gamma(1-\\alpha)^2/(2\\mu)>0$ in the constant-coefficient case, matching known inertial FB thresholds.","The second algorithm with double inertia allows more flexibility in the choice of the two inertial parameters, since only the sum $\\tilde{\\alpha}_n=\\alpha_n+\\theta_n$ and the combination $\\alpha_n-\\gamma_n\\beta_n/(2\\mu)$ are constrained by the conditions.","Numerically, for image restoration, inertial FHRB variants cut the number of iterations for fixed step-sizes, and the double-inertial variant performed best in iteration count while a semi-double-inertial variant was fastest in CPU time."],"supporting_citations":[{"why":"Supplies the nonlinear forward-backward with momentum algorithm and Assumption 4.1, the base method the paper extends.","marker":"[48]"},{"why":"Introduces the forward-half-reflect-backward method and its step-size conditions, recovered as a special case.","marker":"[46]"},{"why":"Provides the momentum version of FHRB that Algorithm 2.3 generalizes.","marker":"[58]"},{"why":"Gives the relaxed inertial forward-backward condition recovered in Section 5.1.","marker":"[5]"},{"why":"Establishes the inertial forward-backward convergence condition that is recovered for $\\lambda=1$.","marker":"[45]"},{"why":"Supplies the monotone operator toolbox, including Opial's lemma and maximal monotonicity facts used throughout the proofs.","marker":"[8]"},{"why":"Introduces the Chambolle\\u2013Pock primal-dual method, which is recovered as a special case.","marker":"[25]"},{"why":"Presents the Condat\\u2013V\\u0169 splitting method, which is recovered as a special case.","marker":"[56]"}],"fun_headline_variants":["Inertia and relaxation preserve convergence in monotone solvers","Momentum meets relaxation: weak convergence preserved","Double-inertial steps speed monotone inclusions","Restart strategy boosts inertial solver convergence"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The whole convergence proof rests on Assumption 4.1: at every iteration the warped operator $\\gamma_n M_n - S$ must be Lipschitz with constant $\\zeta_n \\le 1-\\varepsilon$ with respect to the fixed strongly monotone linear metric $S$, and the step sizes $\\gamma_n$ must stay in a fixed bounded interval; if this strict contractivity is violated, the Lyapunov descent inequality fails.","fun_headline_variants_meta":{"raw":{"variants":["Inertia and relaxation preserve convergence in monotone solvers","Momentum meets relaxation: weak convergence preserved","Double-inertial steps speed monotone inclusions","Restart strategy boosts inertial solver convergence"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.0002,"raw_usage":{"total_tokens":1400,"prompt_tokens":994,"completion_tokens":406,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":610,"completion_tokens_details":{"reasoning_tokens":346}},"tokens_in":610,"tokens_out":406,"duration_ms":4678,"temperature":1.0,"reasoning_tokens":346,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-11T23:54:32.797864+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Run Algorithm 2.2 in $\\mathbb{R}^2$ with $A$ the subdifferential of the indicator of the unit square, $C(x)=0.5x$, $S=\\mathrm{Id}$, constant $\\gamma$, $\\alpha_n\\equiv 0.2$, $\\lambda=1.5$, choosing $\\gamma$ so that (4.22) holds, and compute $C_{n+1}(x)-C_n(x)+(\\eta_n-\\xi_{n+1})\\|x_{n+1}-x_n\\|^2$ at each iteration. If this quantity is ever positive, the Lyapunov descent inequality (4.8), which is the core of the proof, is false and the theorem collapses.","supporting_citations":[{"cited_title":"Acta Math","cited_arxiv_id":null,"evidence_quote":"Provides the momentum version of FHRB that Algorithm 2.3 generalizes."}],"review_version":1}