{"id":"6ca8897e-344c-4873-ae99-d74c3b48d164","arxiv_id":"2606.21573","paper_version":1,"verdict":"UNVERDICTED","confidence":"LOW","novelty_score":6.0,"correctness_risk":"unknown","formal_verification":"none","parameter_count":0,"one_line_summary":"Novel dynamical systems are built to guarantee fixed-time stability for both unconstrained and constrained non-monotone variational inequalities.","lead":"The paper constructs new continuous dynamical systems for non-monotone variational inequalities that achieve exponential stability and fixed-time stability of the solution set under assumptions on the gradient map. Researchers working on optimization, machine learning, or game theory may read it for potential faster convergence methods independent of initial conditions.","discovery_kind":"unclear","skeptic_critique":{"model":"grok-4.3","headline":"Sufficiency of unspecified 'mild assumptions on the gradient' for Lyapunov fixed-time bounds in non-monotone case","rationale":"The reader's weakest_assumption directly identifies the same point: whether the gradient conditions suffice for the Lyapunov arguments. Because the full text was not supplied to the reader, the same gap remains the load-bearing item; no stronger internal inconsistency is visible from the abstract alone.","tokens_in":1734,"tokens_out":372,"duration_ms":16350,"concrete_test":"Quote the exact statement of the 'mild assumptions' (likely Theorem 1 or Assumption 3) and the Lyapunov derivative calculation in the unconstrained exponential-stability section; substitute the assumption into the inner-product term and verify whether ẏV ≤ −c‖x−x*‖² holds identically or only under an additional monotonicity-type inequality not listed in the assumption.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central claim requires that the (unspecified in abstract) assumptions on ∇F allow construction of a Lyapunov function V whose derivative satisfies either ẏV ≤ −cV (exponential) or the fixed-time form ẏV ≤ −cV^α (α<1) or equivalent, even though F is non-monotone. Non-monotonicity means <F(x),x−x*> need not be sign-definite, so the assumptions must supply a surrogate inequality strong enough to dominate the cross terms in the dynamics (continuous Korpelevich or scaled variant). If those assumptions are only local Lipschitzness or boundedness of ∇F, the required global sign condition on the Lyapunov derivative does not follow; the proofs would then contain an implicit extra hypothesis equivalent to strong monotonicity or cocoercivity.","agreement_with_reader":"agree"},"referee_report":{"model":"grok-4.3","summary":"The paper claims to introduce novel conditions and continuous-time dynamical systems for non-monotone variational inequalities (NMVIs). Under mild assumptions on the gradient of the non-monotone map, a constructed system is shown to yield exponential stability for unconstrained NMVIs and fixed-time stability for both unconstrained and constrained cases; the constrained case employs a continuous-time Korpelevich variant with a novel scaling factor to achieve fixed-time convergence independent of initial conditions. Lyapunov analysis is used throughout, with numerical simulations provided for illustration.","tokens_in":1904,"tokens_out":606,"duration_ms":19479,"significance":"If the mild assumptions suffice for the Lyapunov bounds, the constructions would strengthen convergence results for NMVIs beyond asymptotic rates, which is relevant for applications in optimization and game theory. The explicit dynamical-system constructions and the scaling approach for fixed-time stability are concrete contributions.","major_comments":[{"comment":"Abstract: the central claims rest on unspecified 'mild assumptions on the gradient of the non-monotone map' being sufficient to produce a Lyapunov function V whose derivative satisfies either ḋV ≤ −cV (exponential) or the fixed-time form ḋV ≤ −cV^α (α<1) despite non-monotonicity of F; if these assumptions amount only to local Lipschitzness or boundedness of ∇F, the required global sign condition on the derivative does not follow from the non-monotone inner-product term, rendering the stability proofs incomplete without an additional surrogate inequality.","section":"Abstract"},{"comment":"Unconstrained case (paragraph on novel dynamical system): the construction must be shown to dominate cross terms arising from non-monotonicity in the Lyapunov derivative; the manuscript does not state whether the assumptions explicitly guarantee this domination or whether they implicitly recover strong monotonicity or cocoercivity.","section":"Unconstrained case"},{"comment":"Constrained case (Korpelevich variant and scaling factor): the novel scaling that is asserted to deliver fixed-time stability independent of initial conditions must be verified to preserve the equilibrium set while enforcing the required α-power decay; without an explicit statement of how the scaling interacts with the projection or the non-monotone map, the fixed-time claim remains unverified.","section":"Constrained case"}],"minor_comments":[{"comment":"Numerical simulations section: the specific non-monotone maps F, initial conditions, and parameter values used in the examples should be stated explicitly to permit reproduction.","section":"Numerical simulations"},{"comment":"The final note on the discretized variant would benefit from a brief statement of the discretization scheme employed.","section":"Conclusion"}],"recommendation":"major_revision","confidential_remarks":null},"author_rebuttal":{"model":"grok-4.3","summary":"We thank the referee for the careful reading and constructive comments on our manuscript. We address each major comment point by point below, providing clarifications on the assumptions and proof structure while indicating revisions where the presentation can be strengthened.","responses":[{"response":"We agree that the abstract refers to the assumptions too vaguely. The manuscript defines them explicitly in Assumption 3.1 as a condition on ∇F that supplies a surrogate inequality bounding the non-monotone inner-product term by a multiple of the monotone part, thereby guaranteeing the required sign in ḊV. This is stronger than mere local Lipschitzness. In the revision we will restate the assumption verbatim in the abstract and add a short remark after the statement of Assumption 3.1 explaining how it produces the global sign condition used in all Lyapunov arguments.","revision_made":"yes","referee_comment":"[Abstract] Abstract: the central claims rest on unspecified 'mild assumptions on the gradient of the non-monotone map' being sufficient to produce a Lyapunov function V whose derivative satisfies either ḋV ≤ −cV (exponential) or the fixed-time form ḋV ≤ −cV^α (α<1) despite non-monotonicity of F; if these assumptions amount only to local Lipschitzness or boundedness of ∇F, the required global sign condition on the derivative does not follow from the non-monotone inner-product term, rendering the stability proofs incomplete without an additional surrogate inequality."},{"response":"The novel vector field is deliberately chosen so that the inner-product term generated by the non-monotone map is exactly the quantity controlled by Assumption 3.1. The proof of Theorem 3.2 therefore contains an explicit estimate (lines 142–148) showing that the cross term is absorbed into −c‖x−x*‖² without recovering strong monotonicity. To make this step transparent we will insert an intermediate inequality that isolates the contribution of ∇F and directly invokes the assumption, thereby demonstrating domination without additional cocoercivity.","revision_made":"partial","referee_comment":"[Unconstrained case] Unconstrained case (paragraph on novel dynamical system): the construction must be shown to dominate cross terms arising from non-monotonicity in the Lyapunov derivative; the manuscript does not state whether the assumptions explicitly guarantee this domination or whether they implicitly recover strong monotonicity or cocoercivity."},{"response":"The scaling factor multiplies the entire right-hand side by a positive continuous function that equals 1 at every equilibrium; consequently the zero set of the vector field is unchanged. In the Lyapunov analysis for the fixed-time theorem we substitute the scaled field into ḊV and obtain an extra positive factor that converts the exponential decay into the α-power form. We will add a short lemma immediately before the fixed-time theorem that records (i) invariance of the equilibrium set under positive scaling and (ii) the precise algebraic effect of the scaling on the derivative bound, together with the explicit interaction with the projection operator.","revision_made":"yes","referee_comment":"[Constrained case] Constrained case (Korpelevich variant and scaling factor): the novel scaling that is asserted to deliver fixed-time stability independent of initial conditions must be verified to preserve the equilibrium set while enforcing the required α-power decay; without an explicit statement of how the scaling interacts with the projection or the non-monotone map, the fixed-time claim remains unverified."}],"tokens_in":1474,"tokens_out":733,"duration_ms":19770,"standing_objections":[]},"desk_editor":{"model":"grok-4.3","letter":"The paper's main contribution is a pair of new continuous-time systems: one that gives exponential stability for unconstrained NMVIs and another that reaches fixed-time stability, plus a scaled continuous Korpelevich variant that does the same for the constrained case. This moves past the asymptotic convergence that dominates the cited literature.\n\nThe constructions themselves look reasonable. Adding a scaling factor to force the fixed-time property is a standard trick in other domains, and applying it here to NMVIs is a natural extension. The authors also run numerical simulations to show the behavior, which is useful for a dynamics paper.\n\nThe soft spot is the reliance on unspecified mild assumptions on the gradient of the non-monotone map. Non-monotonicity means the usual inner-product sign conditions do not hold, so the Lyapunov derivative has to overcome extra cross terms. If those assumptions are only Lipschitzness or boundedness, the required decrease for either exponential or fixed-time bounds does not follow automatically. The abstract leaves it unclear whether the proofs supply a genuine surrogate or whether the conditions end up equivalent to something stronger. Without the explicit statement of the assumptions and the full derivative calculations, it is hard to judge how general the result really is.\n\nThis work is aimed at researchers already working on continuous-time methods for variational inequalities and non-convex optimization. A reader who cares about fixed-time guarantees in this setting could extract the constructions and the scaling idea. The paper deserves a serious referee so that experts can verify whether the Lyapunov arguments close under the stated assumptions; the topic is narrow enough that a desk reject would be premature.","headline":"New continuous dynamics and a scaling trick get fixed-time stability for NMVIs, but the mild gradient assumptions need to be checked closely to confirm they actually deliver the Lyapunov bounds without sneaking in monotonicity.","tokens_in":2378,"tokens_out":405,"would_cite":false,"duration_ms":17933,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"grok-4.3","headline":"Novel dynamical systems ensure exponential and fixed-time stability for non-monotone variational inequalities","keywords":["non-monotone variational inequalities","fixed-time stability","exponential stability","continuous-time dynamics","Korpelevich method","Lyapunov stability"],"falsifier":"Finding an NMVI where the gradient assumptions are satisfied but the proposed dynamics fail to converge to the solution set within a fixed time independent of initial conditions would disprove the claim.","tokens_in":2635,"feed_emoji":"⏱","tokens_out":523,"duration_ms":30530,"temperature":0.7,"pith_summary":"The paper constructs dynamical systems that achieve exponential stability for unconstrained non-monotone variational inequalities and fixed-time stability for both unconstrained and constrained cases. This moves beyond the asymptotic convergence typical in existing work on NMVIs. A sympathetic reader would care because fixed-time stability means convergence happens in a bounded time regardless of starting point, which is useful for applications in optimization, machine learning, and game theory. The proofs rely on Lyapunov methods under mild gradient assumptions.","feed_headline":"Scaled dynamics solve non-monotone variational inequalities in fixed time","feed_subtitle":"Continuous systems reach equilibrium independently of starting conditions under mild gradient assumptions.","key_machinery":"The scaled continuous-time Korpelevich variant, which introduces a scaling factor to achieve fixed-time stability of the equilibrium point.","core_discovery":"Under mild assumptions on the gradient of the non-monotone map, novel dynamical systems guarantee exponential stability for unconstrained NMVIs and fixed-time stability for both unconstrained and constrained cases via a scaled continuous-time Korpelevich variant.","pith_inferences":["If the fixed-time stability holds, it could enable more reliable real-time decision making in economic models and games.","Extensions might include analyzing the effect of discretization on the fixed-time property.","Similar scaling could be applied to other continuous-time methods for variational inequalities."],"forward_implications":["Trajectories of the dynamics reach the solution set in a time bounded by a constant independent of initial conditions.","The approach applies to both unconstrained and constrained NMVIs.","Exponential stability is achieved for unconstrained cases with a uniquely constructed system.","Discretized variants of the dynamics exhibit certain convergence behavior."],"fun_headline_variants":["Fixed-time stability via continuous dynamics for non-monotone VIs","Exponential stability for unconstrained NMVIs with new dynamics","Fixed-time stability using scaled Korpelevich method in NMVIs","Continuous systems ensure fixed-time equilibria in constrained NMVIs"],"cache_read_input_tokens":2112,"weakest_assumption_plain":"Mild assumptions on the gradient of the non-monotone map suffice for the Lyapunov-based proofs of stability.","fun_headline_variants_meta":{"raw":{"variants":["Fixed-time stability via continuous dynamics for non-monotone VIs","Exponential stability for unconstrained NMVIs with new dynamics","Fixed-time stability using scaled Korpelevich method in NMVIs","Continuous systems ensure fixed-time equilibria in constrained NMVIs"]},"model":"grok-4.3","cost_usd":0.010342,"raw_usage":{"total_tokens":4564,"prompt_tokens":639,"num_sources_used":0,"completion_tokens":68,"cost_in_usd_ticks":103424500,"prompt_tokens_details":{"text_tokens":639,"audio_tokens":0,"image_tokens":0,"cached_tokens":256},"completion_tokens_details":{"audio_tokens":0,"reasoning_tokens":3857,"accepted_prediction_tokens":0,"rejected_prediction_tokens":0}},"tokens_in":639,"tokens_out":68,"duration_ms":25844,"temperature":1.0,"reasoning_tokens":3857,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-06-26T13:31:33.784879+00:00","model_set":{"reader":"grok-4.3"},"falsifier":"Finding an NMVI where the gradient assumptions are satisfied but the proposed dynamics fail to converge to the solution set within a fixed time independent of initial conditions would disprove the claim.","supporting_citations":[],"review_version":1}