{"id":"156a4278-a0e1-42f5-be1a-eff22b7d367d","arxiv_id":"1908.03517","paper_version":5,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":3,"one_line_summary":"A modified proximal dynamical system with a state-dependent scaling is shown to converge to the solution of a mixed variational inequality within a fixed time, uniformly over initial conditions.","lead":"This paper designs a continuous-time dynamical system whose solution reaches the unique solution of a mixed variational inequality within a guaranteed time bound, regardless of the starting point. The result extends fixed-time convergence ideas to proximal and projected dynamics, widely used in optimization and game theory.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Discrete-time guarantee hinges on unpublished [22, Thm. 4]; without it, Corollary 2's fixed-step bound is unproved, and Euler stability for the superlinear field in (5) is not automatic.","rationale":"The continuous-time fixed-time result (Theorem 2) appears internally coherent: the Lyapunov computation yields (17) with exponents γ1<1<γ2, and Lemma 4 supplies the needed parameter interval. I do not see a defect there that would justify rejection. The stress point is the discretization theorem. Theorem 4 is stated as a theorem of this paper, but its proof outsources the decisive (T,ε)-closeness assertion to [22, Thm. 4], an unpublished arXiv preprint by a co-author; the paper only says that the requirements are met. This is not a complete proof from the reader's point of view. Moreover, even if [22] is correct, the application to (5) needs a uniform Euler-closeness estimate over the finite settling interval for a superlinear field (α2>1); such estimates normally require a global Lipschitz or one-sided Lipschitz constant, or a step size depending on the initial condition. Since the abstract promises a fixed number of time steps independent of the initial conditions, the quantifier structure of Corollary 2 (η* after x0) should be clarified. These considerations do not invalidate the continuous-time contribution, but they justify a conditional verdict rather than full acceptance.","tokens_in":19934,"tokens_out":18543,"duration_ms":184540,"concrete_test":"Obtain [22] and independently prove the claimed (T,ε)-closeness for the forward-Euler map of (24) under only the hypotheses of Theorem 3, and determine whether its η* can be chosen independent of x0 without a global Lipschitz condition. Then test the bound (26) on the scalar system ẋ = -sign(x)|x|^{1/2} - sign(x)|x|^{3/2}, which satisfies (21) and (25) with V=x², using fixed η=10⁻⁴ and x0=10⁴, 10⁸, 10¹². If the Euler iterates diverge or the step count for large x0 exceeds the theoretical K, the discretization claim as stated fails.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The fixed-number-of-steps claim (Theorem 4 and Corollary 2) is not self-contained: the proof of Theorem 4 delegates the critical (T,ε)-closeness of Euler solutions to [22, Thm. 4], an unpublished co-authored preprint, and only asserts that its hypotheses are met. Every other hypothesis is checked, but the one result that bridges continuous-time fixed-time stability to discrete-time convergence is not proved in this paper. This matters because the bound (26)/(30) requires the Euler map to remain close to the true flow over the full settling-time interval, for arbitrarily large initial conditions. The field in (5) is superlinear for α2>1: the magnitude of ρ(x)(x−y(x)) is κ1‖x−y(x)‖^α1 + κ2‖x−y(x)‖^α2, which grows faster than linearly when ‖x−y(x)‖ is large. Explicit Euler is not automatically stable for such fields, and the usual closeness estimates require a global Lipschitz or one-sided Lipschitz constant, or a step size that shrinks with the initial condition. If [22, Thm. 4] fails, or if its η* depends on x0 in a way incompatible with the stated bound, the discrete-time contribution of the paper has no support here.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper proposes a modified proximal dynamical system (Eq. (5)) with a state-dependent gain that scales the nominal proximal flow by powers of the residual norm. Under strong monotonicity and Lipschitz continuity of the operator F, it proves existence, uniqueness, and fixed-time convergence of solutions to the unique solution of the mixed variational inequality MVI(F,g) (Theorem 2). It also treats the projected/VI special case, claims an extension to strong pseudomonotonicity, discusses convex optimization connections, and provides a discretization result (Theorem 4 and Corollary 2) asserting that forward-Euler iterates reach any prescribed neighborhood of the solution in a fixed number of steps independent of the initial condition. Two numerical examples illustrate the behavior.","tokens_in":20254,"tokens_out":5402,"duration_ms":59897,"significance":"The continuous-time contribution is potentially valuable: it upgrades known exponential convergence results for proximal/projected dynamical systems to fixed-time convergence for MVIPs under the standard strong-monotonicity and Lipschitz assumptions, and the proof is self-contained given the contraction estimate in Theorem 1. The discrete-time fixed-number-of-steps claim, however, is not self-contained: Theorem 4 and Corollary 2 delegate the central (T,epsilon)-closeness estimate to the unpublished preprint [22]. Since the field in (5) is superlinear for alpha2>1, the Euler discretization guarantee is not an automatic consequence of the continuous-time result. The numerical examples are useful but one of them is justified with an incorrect appeal to Assumption 1. Overall, the central continuous-time derivation appears sound, but the main advertised discrete-time guarantee needs a complete proof before the paper can be accepted.","major_comments":[{"comment":"The proof of Theorem 4 and hence Corollary 2 relies on [22, Theorem 4], an unpublished arXiv preprint by one of the co-authors, for the (T,epsilon)-closeness of forward-Euler solutions to solutions of the differential inclusion (20). The manuscript verifies only that the hypotheses of [22, Theorem 4] are met; it neither states that theorem nor proves it. This is load-bearing because the abstract and contribution (iii) explicitly claim a fixed number of steps independent of the initial condition. Moreover, the vector field in (5) grows superlinearly in the residual for alpha2>1, so an unconditional Euler stability result is not evident. Please include a complete proof of the required (T,epsilon)-closeness theorem, or cite a published peer-reviewed version and spell out its hypotheses, and clarify how the step size eta* is controlled uniformly with respect to the initial condition.","section":"Section VI, Theorem 4 and Corollary 2"},{"comment":"The text states that \"Assumption 1 holds for this example\" and then invokes Corollary 1, but the operator F is described only as strongly pseudomonotone with modulus 11, not strongly monotone. The global Jacobian condition for strong monotonicity fails because entries such as 0.5*x2 are not uniformly positive over all of R^2. The correct route is the extension in Remark 7, which is itself only sketched via reference [8]. Please either correct the attribution to Assumption 1 or prove the strong-pseudomonotonicity contraction estimate needed for Corollary 1, so that the numerical example is fully justified by the stated theory.","section":"Section VII-A, Example 1"},{"comment":"The bound in Theorem 4 contains an unexplained dependence of eta* on the horizon T. In the proof, (T,epsilon)-closeness is used with t = eta k for k up to ceil(xi*pi/(2*eta*sqrt(a*b))), so T must grow as eta decreases. The manuscript does not make explicit that [22, Theorem 4] provides a uniform eta* for all T in a prescribed range, nor does it show how the quantifiers over epsilon, T, and the initial condition are ordered. Please state the precise quantifier structure and ensure eta* does not depend on x0 in a way that invalidates the claimed uniform step bound.","section":"Section VI, Eq. (26)"}],"minor_comments":[{"comment":"There are several typographical errors, including \"Benosm an\" in the author line, \"intial\" in Appendix B, and \"Lypaunov\" in Appendix B. These should be corrected.","section":"Throughout"},{"comment":"The claim that Theorem 1 and Corollary 1 continue to hold under strong pseudomonotonicity is delegated to the proof of [8, Theorem 2]. Please provide more details or a formal proof, since this extension is announced in the abstract and used implicitly in Example 1.","section":"Section V, Remark 7"},{"comment":"The statement that convergence is \"independent of the time-step eta up to numerical tolerance\" is qualitative. Since the paper's discrete-time theory has a specific bound with eta*, it would be helpful to indicate how the observed thresholds compare with the theoretical prediction for the chosen eta values.","section":"Section VII-A, Fig. 2"},{"comment":"The values mu = 0.5 and L = 0.5 for the logistic-regression operator are asserted without derivation. Please provide a short justification or a reference, since the admissible range lambda in (0, 2*mu/L^2) depends on these constants.","section":"Section VII-B, Example 2"}],"recommendation":"major_revision","confidential_remarks":"The main unresolved issue is the dependence of the discrete-time result on the co-authored unpublished preprint [22]. This is not, by itself, a sign of misconduct, but it is a significant completeness problem for a journal submission: the central advertised guarantee is not verified within the manuscript or by an independent published source. I recommend the editor require the authors to either prove the needed (T,epsilon)-closeness theorem in the paper or cite a published version with the proof. The continuous-time part appears sound and should be preserved."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Dear colleague,\n\nHere is my take on arXiv:1908.03517. The core continuous-time result is real and new: the modified proximal dynamical system (5)-(6) gives fixed-time convergence to the solution of a strongly monotone, Lipschitz MVIP, with a uniform settling-time bound. I checked the proof of Theorem 2 carefully, including the algebra around (16) and the use of Lemma 4, and it holds up. The contraction estimate in Theorem 1 is clean, and Proposition 2's existence/uniqueness argument for the rescaled field is careful. The extension to projected dynamics and the connection to nonsmooth convex and saddle-point problems is a nice bonus.\n\nThe soft spot is the discrete-time half. Theorem 4 and Corollary 2 import the critical (T, epsilon)-closeness result for forward Euler from the unpublished preprint [22]. That result is not proved here, and the paper only asserts that its hypotheses are met. More concerning, the quantifier in Corollary 2 allows eta* to depend on the initial condition x0, while the abstract promises a fixed number of steps independent of x0. Because the vector field in (5) is superlinear when alpha2 > 1, explicit Euler is not automatically stable for arbitrarily large initial data, and it is not clear that [22] supplies a uniform eta*. If it does not, the fixed-step claim as stated is unsupported. This should be fixed by either proving the needed closeness bound or citing a published version, or by softening the claim to depend on the initial condition.\n\nMinor items: the numerical examples are illustrative but not rigorous. Example 2's reported L=0.5 is not derived from the data, and the k* values are quoted without showing the constants. None of this affects the continuous-time core.\n\nBottom line: the paper deserves a serious referee. The fixed-time proximal dynamics result is a genuine contribution and the proof is solid. The discrete-time guarantee needs to be made self-contained or explicitly conditional on [22]. I would not cite it for the fixed-step result until that is resolved, but the continuous-time part is worth knowing.","headline":"Solid continuous-time fixed-time proximal dynamics for MVIPs; the discrete-time fixed-step claim rests on an unproved preprint and needs repair.","tokens_in":14,"tokens_out":8195,"would_cite":true,"duration_ms":97663,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["90C33","49J40","34D05"],"pacs":[],"model":"deepseek-v4-flash","headline":"A fixed-time proximal flow solves mixed variational inequalities","keywords":["mixed variational inequality","fixed-time stability","proximal dynamical system","projected dynamical system","forward-Euler discretization","strong monotonicity","strong pseudomonotonicity","convex optimization"],"falsifier":"Take a concrete strongly monotone, Lipschitz MVIP with a known solution x*, choose λ ∈ (0, 2μ/L²), and simulate forward-Euler iterates from a sequence of initial conditions whose distances ‖x0−x*‖ grow without bound. If the number of steps required to enter a fixed ε-neighborhood grows with ‖x0−x*‖, or if the settling time of the continuous flow grows with initial distance, the fixed-time claims are false; the paper's own examples suggest the opposite but are not a proof.","tokens_in":19773,"feed_emoji":"⏱️","tokens_out":7103,"duration_ms":66890,"temperature":0.7,"pith_summary":"The paper seeks to show that a simple modification of the classic proximal dynamical system can solve mixed variational inequalities (MVIPs) with a convergence time that is uniformly bounded for every initial condition, not merely asymptotic or finite-time with unbounded settling time. The modification replaces the constant gain of the nominal proximal flow ẋ = −κ(x − prox_{λg}(x − λF(x))) with a state-dependent gain having two power-law terms, one with exponent below one and one above one. Under strong monotonicity and Lipschitz continuity of F, the paper proves existence, uniqueness, and fixed-time convergence of solutions to the unique MVIP solution for λ in (0, 2μ/L²). It also proves that forward-Euler discretization reaches any ε-neighborhood of the solution in a fixed number of steps regardless of initialization, which is what matters for actual algorithms.","feed_headline":"A fixed-time proximal flow solves mixed variational inequalities","feed_subtitle":"Convergence time is uniformly bounded for all initial conditions, and forward-Euler steps inherit the guarantee.","key_machinery":"The central object is the modified proximal dynamical system $$\\dot{x} = -\\rho(x)\\bigl(x - \\mathrm{prox}_{\\$\\lambda$ g}(x - \\$\\lambda$ F(x))\\bigr),$$ with $\\rho(x) = \\kappa_1\\|x-y(x)\\|^{\\alpha_1-1} + \\kappa_2\\|x-y(x)\\|^{\\alpha_2-1}$ when $x \\notin \\mathrm{Fix}(y)$ and $\\rho=0$ on $\\mathrm{Fix}(y)$, where $y(x)=\\mathrm{prox}_{\\lambda g}(x-\\lambda F(x))$. The state-dependent gain converts the nominal proximal residual into two competing powers: the $\\alpha_1<1$ term drives finite-time convergence, while the $\\alpha_2>1$ term bounds the settling time uniformly over initial conditions. The argument is carried by a contraction inequality (Theorem 1) showing $\\|y(x)-x^*\\| \\le c\\|x-x^*\\|$ for $c = 1/\\sqrt{1+2\\lambda\\mu-\\lambda^2 L^2} \\in (0,1)$, which lets the Lyapunov derivative be bounded below in magnitude by the required two-term form. For the discretization claim, the machinery is a general (T,ε)-closeness result for differential inclusions (Theorem 3 and Theorem 4) applied to the forward-Euler map.","core_discovery":"Theorem 2 is the core claim: for every λ ∈ (0, 2μ/L²) there is an ε > 0 such that the unique solution x* of MVI(F,g) is a fixed-time stable equilibrium of (5) for any α1 ∈ (1−ε,1) and α2 > 1. The proof builds the Lyapunov function V(x)=½‖x−x*‖² and shows its derivative satisfies −(a1 $V^{{γ1}}$ + a2 $V^{{γ2}}$) with γ1<1<γ2, which by the standard fixed-time stability lemma yields an explicit uniform settling-time bound. For the projection special case, the fixed-time guarantee survives if strong monotonicity is relaxed to strong pseudomonotonicity (Corollary 1 and Remark 7). Corollary 2 then transfers the continuous-time guarantee to forward-Euler iterates: for every ε>0 there is a step size such that all iterates enter the ε-neighborhood of x* within k* = ⌈ξπ/(2η√(ab))⌉ steps, independent of x0.","pith_inferences":["Editorial extension: the same state-dependent gain ρ(x) could be applied to other proximal splitting flows, such as forward-backward or Douglas–Rachford dynamics, wherever a contraction inequality analogous to Theorem 1 holds, yielding fixed-time variants.","Editorial extension: the explicit dependence of k* on ξ suggests a trade-off: lowering ξ accelerates convergence but may require a smaller step size to preserve the (T,ε)-closeness bound; a quantitative relation between ξ and η* would be a useful testable extension.","Editorial extension: if the (T,ε)-closeness theorem used for discretization extends to time-varying or non-autonomous fixed-time flows, the fixed-step guarantee would carry over to time-varying optimization and saddle-point problems formulated as MVIPs."],"forward_implications":["For any MVIP with strongly monotone, Lipschitz operator, the modified flow has an explicit settling-time bound $T(x(0)) \\le 1/(a(\\kappa_1,\\alpha_1)(1-\\gamma(\\alpha_1))) + 1/(a(\\kappa_2,\\alpha_2)(\\gamma(\\alpha_2)-1))$, so parameters can be tuned to meet any prescribed time budget.","The projected special case gives a fixed-time projected dynamical system for variational inequalities under strong pseudomonotonicity, extending the existing exponential-convergence results for projected dynamics.","Forward-Euler discretization is a consistent discretization: for every ε > 0 there exists a step size η* such that iterates reach the ε-neighborhood of x* within a fixed number of steps independent of x0, making the method suitable for implementation with a guaranteed worst-case complexity.","Convex optimization problems with nonsmooth terms, such as elastic-net logistic regression, are MVIPs, so the fixed-time flow solves them without smoothing the nonsmooth part.","The flow handles non-smoothness through the proximal operator, so no differentiability of g is required."],"supporting_citations":[{"why":"Supplies the proximal operator characterization used in Lemma 2 and Theorem 1.","marker":"[25]"},{"why":"Provides existence and uniqueness of the MVIP solution under strong monotonicity and Lipschitz continuity.","marker":"[26]"},{"why":"Supplies the fixed-time stability Lyapunov lemma that turns the two-term derivative bound into a uniform settling-time estimate.","marker":"[14]"},{"why":"Contains the (T,ε)-closeness theorem for differential inclusions that underpins the fixed-number-of-steps discretization guarantee.","marker":"[22]"},{"why":"Gives the projected dynamical system exponential-stability result under strong pseudomonotonicity that is extended to fixed-time stability in Remark 7.","marker":"[8]"},{"why":"Introduces the two-exponent scaling idea for fixed-time gradient flows that the modified proximal system adapts.","marker":"[19]"},{"why":"Defines consistent discretization, the framework used to state the discrete-time preservation of fixed-time convergence.","marker":"[30]"}],"fun_headline_variants":["Fixed-time proximal solver for MVIPs","Proximal flow hits MVIP solution in fixed time","Uniform fixed-time convergence for MVIP solutions","Proximal dynamics with fixed settling time for MVIPs","Fixed-time convergence for variational inequalities via proximal flow"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The load-bearing premise is that the (T, ε)-closeness theorem stated as [22, Theorem 4] actually holds; the paper verifies the hypotheses of that external result but does not prove it, so the fixed-number-of-steps guarantee (Theorem 4 and Corollary 2) collapses if that theorem fails.","fun_headline_variants_meta":{"raw":{"variants":["Fixed-time proximal solver for MVIPs","Proximal flow hits MVIP solution in fixed time","Uniform fixed-time convergence for MVIP solutions","Proximal dynamics with fixed settling time for MVIPs","Fixed-time convergence for variational inequalities via proximal flow"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.00105,"raw_usage":{"total_tokens":4402,"prompt_tokens":926,"completion_tokens":3476,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":542,"completion_tokens_details":{"reasoning_tokens":3402}},"tokens_in":542,"tokens_out":3476,"duration_ms":23994,"temperature":1.0,"reasoning_tokens":3402,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-14T14:11:50.939030+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Take a concrete strongly monotone, Lipschitz MVIP with a known solution x*, choose λ ∈ (0, 2μ/L²), and simulate forward-Euler iterates from a sequence of initial conditions whose distances ‖x0−x*‖ grow without bound. If the number of steps required to enter a fixed ε-neighborhood grows with ‖x0−x*‖, or if the settling time of the continuous flow grows with initial distance, the fixed-time claims are false; the paper's own examples suggest the opposite but are not a proof.","supporting_citations":[{"cited_title":"Mixed variational inequalities,","cited_arxiv_id":null,"evidence_quote":"Provides existence and uniqueness of the MVIP solution under strong monotonicity and Lipschitz continuity."},{"cited_title":"Nonlinear feedback design for fixed-tim e stabilization of linear control systems,","cited_arxiv_id":null,"evidence_quote":"Supplies the fixed-time stability Lyapunov lemma that turns the two-term derivative bound into a uniform settling-time estimate."},{"cited_title":"Optimizing dee p neural networks via discretization of finite-time convergent flows ,","cited_arxiv_id":null,"evidence_quote":"Contains the (T,ε)-closeness theorem for differential inclusions that underpins the fixed-number-of-steps discretization guarantee."},{"cited_title":"On the global exponential stability of a projected dynamical system for strongly pseu domonotone variational inequalities,","cited_arxiv_id":null,"evidence_quote":"Gives the projected dynamical system exponential-stability result under strong pseudomonotonicity that is extended to fixed-time stability in Remark 7."},{"cited_title":"Fixed-Time stable gradient flow s: Applica- tions to continuous-time optimization,","cited_arxiv_id":null,"evidence_quote":"Introduces the two-exponent scaling idea for fixed-time gradient flows that the modified proximal system adapts."},{"cited_title":"Consistent di scretization of finite-time and fixed-time stable systems,","cited_arxiv_id":null,"evidence_quote":"Defines consistent discretization, the framework used to state the discrete-time preservation of fixed-time convergence."}],"review_version":1}