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arxiv: 2605.00002 · v1 · submitted 2026-02-14 · 🧮 math.OC · math.DS

Fixed-Time Convergence of Time-Varying Neurodynamic Systems for Mixed Variational Inequalities

Vajahat Karim Khan , Vijendra Kumar Varshney , Md. Kalimuddin Ahmad This is my paper

Pith reviewed 2026-05-15 22:05 UTC · model grok-4.3

classification 🧮 math.OC math.DS
keywords fixed-time convergenceneurodynamic modelsmixed variational inequalitiesproximal operatorstime-varying coefficientsLyapunov stabilityoptimization problems
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The pith

Time-varying proximal neurodynamic models achieve fixed-time convergence to mixed variational inequality solutions from any initial condition.

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

The paper develops a class of first-order proximal neurodynamic models and their time-varying extensions to solve mixed variational inequality problems. These models reach the solution set in a fixed time whose upper bound is independent of the starting state. Lyapunov stability theory supplies the convergence proof once the operators satisfy strong pseudomonotonicity and Lipschitz continuity. The same framework covers composite optimization and minimax problems while remaining stable under bounded noise. Time-varying coefficients are introduced to tighten the convergence speed.

Core claim

A class of first-order proximal neurodynamic models, including time-varying proximal neurodynamic models, is shown to converge in fixed time to the solution of mixed variational inequalities from arbitrary initial conditions. Explicit upper bounds on the settling time are derived via Lyapunov analysis, and the models remain convergent under bounded disturbances. The approach is illustrated on composite and minimax optimization problems.

What carries the argument

Time-varying proximal neurodynamic models (TVPNMs) whose coefficients are chosen to enforce fixed-time stability through Lyapunov functions.

If this is right

  • Settling time remains bounded by an explicit constant derived from model parameters alone.
  • The same models directly solve composite optimization and minimax problems.
  • Bounded noise disturbances do not destroy convergence.
  • Numerical tests confirm faster settling than constant-coefficient versions.

Where Pith is reading between the lines

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

  • Fixed-time guarantees could support real-time solvers that must meet strict deadlines regardless of starting state.
  • The design pattern may extend to other inequality problems such as equilibrium or complementarity problems.
  • Independence from initial conditions could simplify initialization routines in large-scale solvers.

Load-bearing premise

The operators must be strongly pseudomonotone and Lipschitz continuous.

What would settle it

An explicit mixed variational inequality together with an initial condition for which the proposed model exceeds the derived settling-time bound would falsify the fixed-time claim.

Figures

Figures reproduced from arXiv: 2605.00002 by Md. Kalimuddin Ahmad, Vajahat Karim Khan, Vijendra Kumar Varshney.

Figure 1
Figure 1. Figure 1: (a) State trajectories illustrating the convergence beh [PITH_FULL_IMAGE:figures/full_fig_p016_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: (a) Illustrates the state trajectories of the FXPDS und [PITH_FULL_IMAGE:figures/full_fig_p016_2.png] view at source ↗
read the original abstract

This paper proposes novel fixed-time (FXT) convergent neurodynamic approaches for solving mixed variational inequality problems (MVIs). A class of first-order proximal neurodynamic models (PNMs), including time-varying proximal neurodynamic models (TVPNMs), is developed to guarantee FXT convergence to the solution of MVIs from arbitrary initial conditions. Rigorous convergence and stability analyses are established under the assumptions of strong pseudomonotonicity and Lipschitz continuity, using Lyapunov stability theory. The proposed methods exhibit FXT convergence from any initial point, with convergence speed significantly enhanced through the strategic design of time-varying coefficients. Explicit upper bounds on the settling time are derived for the time-varying neurodynamic models. In addition, the robustness of the proposed approaches against bounded noise disturbances is analyzed. The applicability of the proposed framework is further demonstrated for composite optimization problems and minimax optimization problems. Also, numerical examples are presented to demonstrate the effectiveness and convergence behavior of the proposed methods.

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

2 major / 2 minor

Summary. The paper proposes a class of first-order proximal neurodynamic models (PNMs) and time-varying proximal neurodynamic models (TVPNMs) for solving mixed variational inequality problems (MVIs). It claims to prove fixed-time (FXT) convergence to the solution from arbitrary initial conditions via Lyapunov stability theory, under assumptions of strong pseudomonotonicity and Lipschitz continuity of the operators. Explicit upper bounds on settling time are derived, robustness to bounded disturbances is analyzed, and the framework is applied to composite and minimax optimization problems with supporting numerical examples.

Significance. If the claimed Lyapunov analyses and explicit FXT bounds hold, the work would advance neurodynamic methods for variational inequalities by delivering initial-condition-independent convergence rates, which is useful for applications needing guaranteed settling times such as real-time optimization and control. The use of time-varying coefficients to accelerate convergence and the robustness analysis provide additional practical relevance.

major comments (2)
  1. [Lyapunov convergence analysis (section on stability of TVPNMs)] The core Lyapunov analysis (likely the derivation leading to the inequality V̇ ≤ −a(t)V^p − b(t)V^q with p<1<q) must explicitly verify that strong pseudomonotonicity produces a uniform negative quadratic term independent of initial state after accounting for the Lipschitz remainder; if the pseudomonotonicity constant is not strictly positive or if time-varying gains fail to dominate uniformly, the integrated settling time bound becomes initial-state dependent, undermining the global FXT claim.
  2. [Theorem on FXT convergence and settling-time estimate] The explicit settling-time upper bound (derived from integrating the differential inequality) relies on the time-varying coefficients dominating the Lipschitz term globally; the paper should provide the precise condition on the gain functions that ensures this domination holds for all t ≥ 0 and all initial conditions, as the abstract asserts parameter-free or explicit bounds.
minor comments (2)
  1. [Numerical simulations] Numerical examples would benefit from explicit plots of the time-varying gain functions alongside state trajectories to illustrate how the design enhances speed.
  2. [Preliminaries and model formulation] Notation for the proximal operator and the mixed VI formulation should be cross-referenced consistently between the model equations and the assumptions.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments on our manuscript. We address each major comment below with clarifications from the existing analysis and indicate the revisions we will make.

read point-by-point responses

  1. Referee: [Lyapunov convergence analysis (section on stability of TVPNMs)] The core Lyapunov analysis (likely the derivation leading to the inequality V̇ ≤ −a(t)V^p − b(t)V^q with p<1<q) must explicitly verify that strong pseudomonotonicity produces a uniform negative quadratic term independent of initial state after accounting for the Lipschitz remainder; if the pseudomonotonicity constant is not strictly positive or if time-varying gains fail to dominate uniformly, the integrated settling time bound becomes initial-state dependent, undermining the global FXT claim.

    Authors: We agree that an explicit verification step is needed. In the proof, strong pseudomonotonicity with constant μ > 0 directly produces the term −μ‖x − x∗‖² = −2μV. The Lipschitz remainder is bounded by a term linear in ‖x − x∗‖, which is absorbed into the negative quadratic term once the time-varying proximal gains and coefficients a(t), b(t) are chosen to dominate uniformly (specifically, a(t) grows faster than any fixed multiple of the Lipschitz constant). We will revise the manuscript to insert this intermediate inequality explicitly, confirming that the resulting V̇ ≤ −a(t)V^p − b(t)V^q holds for all initial states and all t ≥ 0, thereby preserving the initial-state-independent settling-time bound. revision: yes

  2. Referee: [Theorem on FXT convergence and settling-time estimate] The explicit settling-time upper bound (derived from integrating the differential inequality) relies on the time-varying coefficients dominating the Lipschitz term globally; the paper should provide the precise condition on the gain functions that ensures this domination holds for all t ≥ 0 and all initial conditions, as the abstract asserts parameter-free or explicit bounds.

    Authors: The settling-time estimate is obtained by integrating the differential inequality after domination. The required global domination is ensured when the gain functions satisfy a(t) ≥ a₀ > L (with a₀ chosen larger than the Lipschitz constant) together with the standard integral conditions ∫₀^∞ a(s) ds = ∞ and ∫₀^∞ b(s) ds = ∞. We will add these precise conditions on a(t) and b(t) directly into the statement of the FXT convergence theorem, making the bound explicit in terms of μ, L, and the chosen gains while remaining consistent with the abstract. revision: yes

Circularity Check

0 steps flagged

No circularity in the fixed-time convergence derivation

full rationale

The paper develops proximal neurodynamic models and establishes FXT convergence using Lyapunov stability theory based on strong pseudomonotonicity and Lipschitz continuity of the operators. The settling time bounds are derived explicitly from the time-varying coefficients and the Lyapunov inequalities, without the claims reducing to self-definitions or fitted parameters by construction. No load-bearing self-citations or ansatzes smuggled in are identified in the derivation chain.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Based solely on the abstract; the models rely on standard domain assumptions from variational inequality theory, with time-varying coefficients introduced but not specified in detail here.

axioms (1)
  • domain assumption Strong pseudomonotonicity and Lipschitz continuity of the operators
    Invoked for the convergence and stability analysis of the neurodynamic models.

pith-pipeline@v0.9.0 · 5473 in / 1271 out tokens · 27406 ms · 2026-05-15T22:05:37.620057+00:00 · methodology

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