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arxiv: 2606.21573 · v1 · pith:VB2UOUBWnew · submitted 2026-06-19 · 🧮 math.OC · math.DS

On Fixed-Time Stability of Continuous Dynamics for Non-Monotone Variational Inequalities

Sina Arefizadeh , Kunal Garg , Angelia Nedi\'c This is my paper

Pith reviewed 2026-06-26 13:31 UTC · model grok-4.3

classification 🧮 math.OC math.DS
keywords non-monotone variational inequalitiesfixed-time stabilityexponential stabilitycontinuous-time dynamicsKorpelevich methodLyapunov stability
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The pith

Novel dynamical systems ensure exponential and fixed-time stability for non-monotone variational inequalities

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

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.

Core claim

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.

What carries the argument

The scaled continuous-time Korpelevich variant, which introduces a scaling factor to achieve fixed-time stability of the equilibrium point.

If this is right

  • 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.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • 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.

Load-bearing premise

Mild assumptions on the gradient of the non-monotone map suffice for the Lyapunov-based proofs of stability.

What would settle it

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.

Figures

Figures reproduced from arXiv: 2606.21573 by Angelia Nedi\'c, Kunal Garg, Sina Arefizadeh.

Figure 1
Figure 1. Figure 1: Log-scale convergence of the Korpelevich-type al [PITH_FULL_IMAGE:figures/full_fig_p020_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Log-scale of the norm ∥x k+1 − x k∥ for different γ values. Although a brief discussion on the discretized dynamics and its convergence behavior was provided based on the existing results, more work needs to be done for finite- or fixed-time convergence of discrete time algorithms for solving VIs. Future work will explore further relaxations of the conditions imposed in this work to accelerate convergence … view at source ↗
read the original abstract

Non-monotone variational inequalities (NMVI) are an important class of problems that generalize non-convex optimization and have various applications in optimization theory, machine learning, game theory, and economics, among others. Most existing work on NMVIs focuses on the asymptotic convergence of algorithms proposed to solve these problems. In this paper, we tackle the problems of exponential and fixed-time stability of the solution set of a class of NMVIs for both unconstrained and constrained problems. We first present novel conditions guaranteeing exponential stability of solutions to unconstrained NMVIs for a uniquely constructed dynamical system under mild assumptions on the gradient of the non-monotone map. Then, under similar assumptions, we construct another novel dynamical system whose equilibrium point is fixed-time stable, i.e., the trajectories reach the equilibrium within a fixed time, independent of the initial conditions. For the case of constrained NMVIs, we employ a continuous-time variant of the Korpelevich method for exponential stability of the solution set, and provide a novel scaling factor in the dynamics to achieve fixed-time stability. We illustrate the efficacy of the proposed modified dynamical systems through numerical simulations and conclude the paper with a brief note on the behavior of the discretized variant of the proposed dynamics and on further work that remains to be done.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

3 major / 2 minor

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.

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 (3)
  1. [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.
  2. [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.
  3. [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.
minor comments (2)
  1. [Numerical simulations] 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.
  2. [Conclusion] The final note on the discretized variant would benefit from a brief statement of the discretization scheme employed.

Simulated Author's Rebuttal

3 responses · 0 unresolved

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.

read point-by-point responses

  1. Referee: [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.

    Authors: 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: yes

  2. Referee: [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.

    Authors: 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: partial

  3. Referee: [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.

    Authors: 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: yes

Circularity Check

0 steps flagged

No circularity; stability claims rest on standard Lyapunov analysis of explicitly constructed dynamics under stated gradient assumptions

full rationale

The paper introduces novel dynamical systems (continuous-time Korpelevich variant and scaled version) and states explicit novel conditions on ∇F for the non-monotone map. It then applies standard Lyapunov-function arguments to show that the constructed vector fields yield Ẇ ≤ −cW or the fixed-time form under those conditions. No equation reduces to a self-definition, no fitted parameter is relabeled as a prediction, and no load-bearing premise is justified solely by self-citation. The derivation chain is therefore self-contained: the assumptions are external to the target stability conclusion, the vector fields are newly defined, and the Lyapunov inequalities are derived directly from the dynamics rather than presupposed.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claims rest on standard stability theorems from dynamical systems theory and domain-specific assumptions about the gradient map of the non-monotone operator; no free parameters or invented entities are introduced in the abstract.

axioms (2)
  • standard math Lyapunov stability theorems for continuous dynamical systems
    Invoked to establish exponential and fixed-time stability of the proposed vector fields.
  • domain assumption Existence of solutions to the NMVI under the stated gradient assumptions
    Required for the equilibrium set to be well-defined and reachable.

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discussion (0)

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