{"id":"bb121776-b303-4f66-a156-997ac874c88a","arxiv_id":"1908.07077","paper_version":5,"verdict":"ACCEPT","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"Warped resolvents generalize classical resolvents, and new convergence theorems unify and extend operator-splitting algorithms for monotone inclusions.","lead":"This paper introduces 'warped' resolvents, a broad generalization of the classical resolvent of a monotone operator, built using an auxiliary kernel operator. It proves weak and strong convergence principles that unify many existing splitting algorithms and yields a new algorithm for complex coupled inclusion problems.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The nonstandard regularity condition (ii)[b] is the true load-bearing point of Theorems 4.2/4.8; it can fail for natural monotone kernels, and the paper verifies it only via strong monotonicity plus Lipschitzianity.","rationale":"The reader's weakest assumption matches the load-bearing point I find: condition (ii)[b] in Theorems 4.2 and 4.8 is the bridge that turns a vanishing inner product into the graph convergence needed for maximal monotonicity. The 2D rotation example shows the assumption is not redundant, but it also does not contradict the theorem, which explicitly assumes (ii)[b]. The paper states the condition clearly and supplies a valid sufficient condition (Remark 4.3), and all applications in Section 5 verify it through strong monotone Lipschitz kernels. Thus the mathematical argument is internally consistent and the accept verdict stands. The only qualification is that the abstract principle's generality beyond strongly monotone Lipschitz kernels is unillustrated; this is a scope caveat, not a correctness defect. No additional red flags were found in the proofs or in the derivation of the applications.","tokens_in":21078,"tokens_out":20107,"duration_ms":201348,"concrete_test":"Run the 2D check: X=R^2, K(x,y)=(-y,x), M=Id, gamma=1, ~x_n=x_n, lambda_n=1, x_0=(1,0). Then K is monotone, K+M is bijective, zerM={0}, and all hypotheses of Theorem 4.2 except (ii)[b] are satisfied. Compute y_0=(1/2,-1/2), y*_0=(-1/2,1/2), and <y_0-x_0 | y*_0> = 0, so the 'else' branch keeps x_1=x_0 and the sequence never approaches 0. This confirms that (ii)[b] is necessary and that the theorem's applicability outside strong monotone Lipschitz kernels is unestablished; a positive control is to rerun with K=Id, where the standard proximal point algorithm converges to 0.","verdict_should_be":"UNCHANGED","load_bearing_attack":"Theorem 4.2's proof is sound only if condition (ii)[b] holds. Fejer monotonicity gives at most epsilon^{-1}||x_{n+1}-x_n|| >= <x_n-y_n | (K_n~x_n-K_n y_n)^#>, and then (4.10)-(4.11) reduce this to the antecedent of (ii)[b]. The theorem then needs the full consequent: ~x_n-y_n -> 0 weakly identifies the weak cluster point of y_n, while K_n~x_n-K_n y_n -> 0, together with gamma_n >= epsilon, makes y*_n -> 0 so that maximal monotonicity of M applies. This bridge is not a formality. For instance, in X=R^2 let K=J be the 90-degree rotation (monotone, Lipschitz, injective, but not strongly monotone), M=Id, gamma=1, ~x_n=x_n, lambda_n=1. Then all hypotheses of Theorem 4.2 except (ii)[b] hold; from any x_0 != 0 the computed y_n satisfies x_n-y_n perpendicular to J(x_n-y_n), so <y_n-x_n | y*_n> = 0 and the iterate stays at x_0, which is not a zero of M. Thus (ii)[b] is genuinely load-bearing and can fail for natural monotone kernels. The paper's only sufficient condition (Remark 4.3) is uniform strong monotonicity plus Lipschitzianity, and every Section 5 application uses exactly that; no example is given of a non-strongly-monotone family satisfying (ii)[b]. The theorem remains correct as a conditional statement, but its advertised breadth as an abstract principle is not supported outside that regime.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper introduces a generalization of resolvents of monotone operators, called warped resolvents, defined by J^K_M = (K+M)^{-1}∘K for a kernel K and a monotone operator M. After establishing well-definedness criteria and basic properties (Propositions 3.8–3.11), the authors prove two abstract convergence principles: weak convergence (Theorem 4.2) and strong convergence to the projection onto the zero set (Theorem 4.8) for iterations that use warped resolvents with possibly time-varying kernels K_n applied at auxiliary points ~x_n, followed by a Fejér-type update. The main hypotheses are ~x_n−x_n→0 and condition (ii)[b], which states that a vanishing normalized inner product forces both ~x_n−y_n⇀0 and K_n~x_n−K_n y_n→0. The principles are then instantiated to recover known algorithms (e.g., Tseng's forward-backward-forward method, Corollary 5.3) and to design new ones, including a perturbed forward-backward-forward algorithm with memory (Corollary 5.2) and a primal-dual method for a coupled system of monotone inclusions (Problem 5.4 and Corollary 5.5).","tokens_in":21507,"tokens_out":12778,"duration_ms":120970,"significance":"If the results are correct, the warped resolvent framework provides a unified perspective on a broad class of splitting algorithms and a modular design tool for new solution methods. The paper is self-contained, with detailed proofs from standard monotone operator theory, and the applications verify the required hypotheses rather than relying on numerical evidence. The main strength is the breadth of the proposed formalism and the concrete new algorithms derived from it. The main weakness is that the crucial condition (ii)[b] in Theorems 4.2 and 4.8 is only shown to hold for uniformly strongly monotone and Lipschitzian kernels (Remark 4.3), and every Section 5 application uses exactly that regime; no example is given of a non-strongly-monotone kernel satisfying (ii)[b], and a simple rotation kernel shows that the condition is not automatic.","major_comments":[{"comment":"Condition (ii)[b] is essential to the proof: at (4.10)–(4.13) it is used to convert the vanishing inner product produced by the Fejér-monotone update into weak convergence of ~x_n−y_n and strong convergence of K_n~x_n−K_n y_n, which together with γ_n≥ε yields y*_n→0 and enables the maximal monotonicity argument. The only sufficient condition offered in Remark 4.3 is uniform α-strong monotonicity plus β-Lipschitzianity, and every application in Section 5 relies on exactly that sufficient condition. The paper does not provide an example of a non-strongly-monotone family satisfying (ii)[b], and in fact the condition is not implied by the other hypotheses. Concretely, in X=R^2 take M=Id, let K be the 90-degree rotation, set γ_n=1, λ_n=1, and ~x_n=x_n. Then all hypotheses of Theorem 4.2 except (ii)[b] hold; the antecedent of (ii)[b] holds (the inner product is identically zero) but the consequent fails because ~x_n−y_n is a nonzero constant, and the iterates remain fixed at the starting point, which is not a zero of M. I recommend that the authors add a remark explicitly noting this non-redundancy, and either prove (ii)[b] under more general checkable hypotheses or qualify the abstract principle's scope to the strongly monotone kernel regime.","section":"Section 4, Theorem 4.2 (and Theorem 4.8), condition (ii)[b]"}],"minor_comments":[{"comment":"The implication notation in (ii)[b] uses an arrow followed by a set of two statements; this is nonstandard and should be restated in words, for example: 'if ⟨~x_n−y_n | (K_n~x_n−K_n y_n)^♯⟩→0, then both ~x_n−y_n⇀0 and K_n~x_n−K_n y_n→0 hold.'","section":"Section 4, Theorem 4.2, condition (ii)[b]"},{"comment":"The sentence 'this is done as in the proof of Theorem 4.2(ii)' should be expanded, because Theorem 4.2 uses the bound ε^{-1}‖x_{n+1}−x_n‖ with λ_n∈[ε,2−ε], whereas Theorem 4.8 has no relaxation parameter and the corresponding bound is simply ‖x_{n+1/2}−x_n‖; the adaptation is straightforward but should be stated explicitly for clarity.","section":"Section 4, proof of Theorem 4.8"},{"comment":"The coercivity condition leading to (3.6) and the inequality involving inf ⟨x,∂ϕ(x+z)⟩ are stated without proof or reference; one sentence of justification, or a citation to the relevant convex-analytic fact, would improve readability.","section":"Section 3, Proposition 3.9(i)[f]"},{"comment":"Reference [14] cites the preprint version of this paper; if a published version exists, it should be cited in place of the preprint to avoid a self-citation of an unpublished report.","section":"References, [14]"}],"recommendation":"major_revision","confidential_remarks":"The paper is mathematically sound as a conditional statement, and the warped resolvent framework is a genuine contribution. The main concern is the unadvertised narrowness of the key condition (ii)[b], which is load-bearing and currently verified only under strong monotonicity of the kernels; I trust the authors can address this in revision with a counterexample or a broader sufficient condition. The heavy self-citation, including the preprint reference [14], is a minor editorial point."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"First: this is a solid paper and the reader's verdict is about right. The warped resolvent definition is a genuine extension of the classical resolvent and it does organize known objects (D-resolvents, K-resolvents, normalized duality resolvents) under one roof, with new properties in Section 3. The convergence principles in Theorems 4.2 and 4.8 and the algorithm in Corollary 5.5 are new. The proofs are careful and use standard monotone-operator machinery; I did not see circular reasoning or fitted assumptions.\n\nThe real soft spot is exactly the one the stress-test note flags. In Theorem 4.2(ii)[b], the implication from a vanishing inner product to `~x_n-y_n` weak convergence plus `K~x_n-Ky_n -> 0` is load-bearing. The proof uses it to identify weak cluster points with zeros of M; without it the iteration can stall. The rotation example is convincing: in R^2, take K=J the 90-degree rotation, M=Id, gamma=1, `~x_n=x_n`, lambda=1. The warped resolvent is well-defined and the update criterion is never active, because `<y_n-x_n | y*_n>=0`, so the iterates stay at a non-zero point. All hypotheses of Theorem 4.2 except (ii)[b] hold. So (ii)[b] is not a technical footnote; it is the theorem.\n\nTo be clear, this is not a proof error. The theorem is a correct conditional statement with an explicit hypothesis. The issue is scope. The paper gives only one sufficient condition for (ii)[b] — uniform strong monotonicity plus Lipschitzianity (Remark 4.3) — and every application in Section 5 uses it. The abstract principle therefore has a narrower demonstrated range than the presentation suggests. A reader wanting to find a genuinely new non-strongly-monotone family satisfying (ii)[b] will have to do extra work, and the paper does not point to any.\n\nThe heavy self-citation is not a problem here; the cited results are prior independent theorems, and the claimed novelty relative to [23] is handled fairly. The paper deserves a serious referee. I would recommend acceptance after a minor revision that clearly states that all known applications fall under the strong-monotone/Lipschitz regime and that either supplies a non-strongly-monotone example satisfying (ii)[b] or adds a remark that no such example is currently known.","headline":"Warped resolvents are a real generalization and the convergence theorems are correct, but the advertised breadth rests on a single nonstandard condition that only strong monotonicity is known to trigger.","tokens_in":21981,"tokens_out":2748,"would_cite":true,"duration_ms":26929,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["47J25","47N10","47H05","90C25"],"pacs":[],"model":"deepseek-v4-flash","headline":"The paper proposes warped resolvents $(K+\\gamma M)^{-1}\\circ K$ as a building block for monotone inclusion algorithms, and proves weak and strong convergence for the resulting proximal iterations.","keywords":["monotone inclusions","warped resolvent","operator splitting","proximal point algorithm","primal-dual algorithms","strong convergence","Fejér monotonicity","maximally monotone operators"],"falsifier":"Take a maximally monotone operator with a known zero set, for instance the normal cone $N_C$ of a closed convex set $C$ in $\\ell^2$, and implement iteration (4.5) with $\\widetilde{x}_n=x_n$ and a family of monotone kernels $K_n$ for which the warped resolvents are easy to compute but which are not strongly monotone and Lipschitz. If a family can be found where the inner products in condition (ii)[b] tend to $0$ while the iterates have a weak cluster point outside $C$, or while $K_n\\widetilde{x}_n-K_n y_n$ fails to converge strongly to $0$, then Theorem 4.2 as stated fails. A sharper falsifier is a single example satisfying every hypothesis of Theorem 4.2 except (ii)[b], with the inner products tending to $0$ and the iterates not converging to a zero of $M$.","tokens_in":20894,"feed_emoji":"🔁","tokens_out":11726,"duration_ms":108172,"temperature":0.7,"pith_summary":"The paper's project is to replace the resolvent $(\\mathrm{Id}+\\gamma M)^{-1}$ inside the proximal point algorithm by a warped resolvent $(K+\\gamma M)^{-1}\\circ K$, where $K$ is an auxiliary monotone operator chosen to fit the structure of the problem. The central results, Theorems 4.2 and 4.8, state that when each warped resolvent is well defined, the auxiliary evaluation points track the iterates, and a kernel regularity condition holds, the iterates converge weakly to a zero of $M$, or strongly to the projection of the starting point onto the zero set. The payoff is a single abstract principle that specializes to classical proximal point and to forward-backward-forward splitting, and that yields new algorithms, including a forward-backward-forward method with memory and a splitting method for a system of coupled primal-dual inclusions. A sympathetic reader would care because the kernel $K$ can be engineered so that otherwise intractable resolvents become computable while convergence is preserved.","feed_headline":"Warped resolvents turn hard inclusions into simple projections","feed_subtitle":"One template covers proximal point, forward-backward-forward splitting, and new primal-dual solvers.","key_machinery":"The central object is the warped resolvent $J^K_{\\gamma M}=(K+\\gamma M)^{-1}\\circ K$ for a monotone kernel $K$; in a Hilbert space it maps $x$ to the unique $p$ satisfying $Kx\\in Kp+\\gamma Mp$, so it has the fixed-point characterization $\\mathrm{Fix}\\, J^K_{\\gamma M}=\\mathrm{zer}\\, M$ and produces graph points $(p,\\gamma^{-1}(Kx-Kp))\\in \\mathrm{gra}\\, M$. This is what lets the algorithm generate half-spaces that contain the solution set. The convergence mechanism is Fej\\'er monotonicity together with condition (ii)[b], the bridge that turns a vanishing inner product into weak convergence of $\\widetilde{x}_n-y_n$ and strong convergence of the kernel difference. In the strong-convergence variant, the extra mechanism is the projection $Q(x_0,x_n,x_{n+1/2})$ onto the intersection of two half-spaces, which preserves Fej\\'er monotonicity and forces convergence to the best approximation of $x_0$ in $Z$.","core_discovery":"The discovery is an abstract convergence principle for zero-finding with maximally monotone operators. On a Hilbert space $X$, take $M$ maximally monotone with nonempty zero set $Z$, a sequence of monotone kernels $K_n$, and auxiliary points $\\widetilde{x}_n$ with $\\widetilde{x}_n-x_n\\to 0$. If each warped resolvent $J^{K_n}_{\\gamma_n M}$ is well defined and the regularity condition $\\langle \\widetilde{x}_n-y_n \\mid (K_n\\widetilde{x}_n-K_n y_n)^\\sharp\\rangle\\to 0$ implies both $\\widetilde{x}_n-y_n\\rightharpoonup 0$ and $K_n\\widetilde{x}_n-K_n y_n\\to 0$, then the iteration (4.5), which evaluates $y_n=J^{K_n}_{\\gamma_n M}\\widetilde{x}_n$ and takes a relaxed projection onto the half-space containing $Z$, converges weakly to a zero of $M$ (Theorem 4.2). With the additional correction step $x_{n+1}=Q(x_0,x_n,x_{n+1/2})$ projecting onto the intersection of two half-spaces, the same hypotheses give strong convergence to $\\mathrm{proj}_Z x_0$ (Theorem 4.8). The proof routes through Fej\\'er monotonicity: the iteration is Fej\\'er-monotone with respect to $Z$, and the regularity condition upgrades the vanishing inner products to enough subsequential information to apply maximal monotonicity.","pith_inferences":["A testable extension is to replace the strong monotonicity plus Lipschitz sufficient condition of Remark 4.3 with a wider sufficient class for condition (ii)[b], for instance kernels built from strongly monotone operators plus compact perturbations; all Section 5 algorithms would carry over to that class.","The static theory already covers reflexive Banach spaces through duality-mapping and distance kernels, but the dynamic convergence theorems are proved only in Hilbert space; a non-Hilbert convergence theorem would be the natural next step and is not established here.","The vanishing-error condition $\\widetilde{x}_n-x_n\\to 0$ without summability suggests that asynchronous or block-coordinate implementations, where the evaluation point lags behind the current iterate by a shrinking error, may still converge under Theorem 4.2.","One could test numerically whether non-symmetric linear kernels of the type used in the primal-dual example give faster convergence on ill-conditioned saddle-point problems, since warped resolvents allow couplings that standard resolvents cannot express."],"forward_implications":["Setting $K_n=\\mathrm{Id}$, $\\widetilde{x}_n=x_n$, and $\\lambda_n=1$ in (4.5) recovers the classical proximal point algorithm, so the theorems contain that method as a special case.","Choosing $K_n=\\mathrm{Id}-\\gamma_n B$ recovers the forward-backward-forward splitting method for $0\\in Ax+Bx$; the proof attributes the required regularity to strong monotonicity and cocoercivity of the kernels.","The new perturbed forward-backward-forward algorithm with memory (Corollary 5.2) converges weakly to a zero of $A+B$ under additive perturbations $e_n$ that only tend to $0$ and under inertial weights supported on a bounded window, conditions milder than summability of the errors.","For the coupled system of primal-dual inclusions in Problem 5.4, Corollary 5.5 yields a parallel splitting algorithm whose primal and dual iterates converge weakly to a point in the primal-dual set $\\mathcal{Z}$; the strong-convergence counterpart follows from Theorem 4.8.","The multi-point extension in Remark 4.10 allows several warped resolvent evaluations per step to be combined through weighted half-spaces, so the principle covers algorithms using more than one graph point per iteration."],"supporting_citations":[{"why":"Introduces the proximal point algorithm whose resolvent the paper generalizes; supplies the baseline method and its convergence question.","marker":"[34]"},{"why":"Supplies standard facts on maximal monotone operators, averaged and firmly nonexpansive operators, resolvents, and half-space projections used throughout the proofs.","marker":"[7]"},{"why":"Supplies the Fej\\'er-monotonicity convergence principle used to prove Proposition 4.1 and hence the weak convergence part of Theorems 4.2 and 4.8.","marker":"[16]"},{"why":"Provides the best-approximation convergence result used in Proposition 4.7 to obtain strong convergence in Theorem 4.8.","marker":"[2]"},{"why":"Establishes the primal-dual set and the maximal monotonicity of the product operator in Lemma 2.2, which underlies the coupled inclusion application.","marker":"[20]"},{"why":"Proposes the forward-backward-forward splitting method that the paper recovers as a corollary, demonstrating that the framework subsumes an existing algorithm.","marker":"[36]"},{"why":"Concurrent work proposing a similar resolvent notion without perturbation; the paper cites it as complementary and using the idea for other splitting schemes.","marker":"[23]"}],"fun_headline_variants":["Warped resolvents unify proximal splitting algorithms","Warped resolvents: a master template for monotone inclusions","New convergence principles from warped resolvents","Warped proximal iterations cover known and new solvers","Abstract convergence for monotone inclusions via warped resolvents"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The load-bearing premise is condition (ii)[b] of Theorems 4.2 and 4.8: each kernel family must convert a vanishing inner product $\\langle \\widetilde{x}_n-y_n \\mid (K_n\\widetilde{x}_n-K_n y_n)^\\sharp\\rangle\\to 0$ into both $\\widetilde{x}_n-y_n\\rightharpoonup 0$ and $K_n\\widetilde{x}_n-K_n y_n\\to 0$. The only concrete sufficient condition the paper supplies for this is strong monotonicity plus Lipschitzianity of the kernels, so a kernel family that violates the implication falls outside the theorems.","fun_headline_variants_meta":{"raw":{"variants":["Warped resolvents unify proximal splitting algorithms","Warped resolvents: a master template for monotone inclusions","New convergence principles from warped resolvents","Warped proximal iterations cover known and new solvers","Abstract convergence for monotone inclusions via warped resolvents"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000645,"raw_usage":{"total_tokens":2970,"prompt_tokens":955,"completion_tokens":2015,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":571,"completion_tokens_details":{"reasoning_tokens":1936}},"tokens_in":571,"tokens_out":2015,"duration_ms":15603,"temperature":1.0,"reasoning_tokens":1936,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-14T12:26:44.209284+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Take a maximally monotone operator with a known zero set, for instance the normal cone $N_C$ of a closed convex set $C$ in $\\ell^2$, and implement iteration (4.5) with $\\widetilde{x}_n=x_n$ and a family of monotone kernels $K_n$ for which the warped resolvents are easy to compute but which are not strongly monotone and Lipschitz. If a family can be found where the inner products in condition (ii)[b] tend to $0$ while the iterates have a weak cluster point outside $C$, or while $K_n\\widetilde{x}_n-K_n y_n$ fails to converge strongly to $0$, then Theorem 4.2 as stated fails. A sharper falsifier is a single example satisfying every hypothesis of Theorem 4.2 except (ii)[b], with the inner products tending to $0$ and the iterates not converging to a zero of $M$.","supporting_citations":[{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Introduces the proximal point algorithm whose resolvent the paper generalizes; supplies the baseline method and its convergence question."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies standard facts on maximal monotone operators, averaged and firmly nonexpansive operators, resolvents, and half-space projections used throughout the proofs."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the Fej\\'er-monotonicity convergence principle used to prove Proposition 4.1 and hence the weak convergence part of Theorems 4.2 and 4.8."},{"cited_title":"Alotaibi, P","cited_arxiv_id":null,"evidence_quote":"Provides the best-approximation convergence result used in Proposition 4.7 to obtain strong convergence in Theorem 4.8."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Establishes the primal-dual set and the maximal monotonicity of the product operator in Lemma 2.2, which underlies the coupled inclusion application."},{"cited_title":"Tseng, A modified forward-backward splitting method for maximal monotone mappings, SIAM J","cited_arxiv_id":null,"evidence_quote":"Proposes the forward-backward-forward splitting method that the paper recovers as a corollary, demonstrating that the framework subsumes an existing algorithm."},{"cited_title":"Nonlinear Forward-Backward Splitting with Projection Correction","cited_arxiv_id":"1908.07449","evidence_quote":"Concurrent work proposing a similar resolvent notion without perturbation; the paper cites it as complementary and using the idea for other splitting schemes."}],"review_version":1}