Trajectory convergence and $o(t^{-2})$ rates for Nesterov accelerated primal-dual dynamics without Lipschitz gradient assumption

Nan-Jing Hang; Xin He; Ya-Ping Fang; Yi-Bin Xiao

arxiv: 2605.18236 · v1 · pith:HMRGE6TRnew · submitted 2026-05-18 · 🧮 math.OC

Trajectory convergence and o(t⁻²) rates for Nesterov accelerated primal-dual dynamics without Lipschitz gradient assumption

Xin He , Nan-Jing Hang , Yi-Bin Xiao , Ya-Ping Fang This is my paper

Pith reviewed 2026-05-20 09:16 UTC · model grok-4.3

classification 🧮 math.OC

keywords Nesterov accelerated dynamicsprimal-dual systemtrajectory convergenceBregman distancefinite-dimensional analysisconstrained convex optimizationo(t^{-2}) ratesno Lipschitz assumption

0 comments

The pith

Nesterov accelerated primal-dual dynamics converge to a primal-dual solution for α ≥ 3 without Lipschitz gradient assumption

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

The paper establishes that the trajectories of the Nesterov accelerated primal-dual dynamical system converge to a primal-dual optimum in finite-dimensional spaces for damping parameter α at least 3. This convergence is proved without the standard Lipschitz continuity assumption on the gradient of the objective function by using Bregman distance arguments. For α greater than 3 the system additionally achieves o of t to the minus two convergence rates for the objective residual and the feasibility violation. This extends previous results that required the Lipschitz condition and opens the method to a wider class of convex optimization problems.

Core claim

The central discovery is that in finite-dimensional spaces the solution trajectories of the given second-order primal-dual system converge to a point satisfying the first-order optimality conditions for every α ≥ 3, relying only on convexity and continuous differentiability of f. The proof uses a Lyapunov function involving Bregman distances to show that the distance to the solution set decreases in a controlled way. For α > 3 the same Lyapunov function implies that both the objective gap and the constraint violation are little-o of one over t squared.

What carries the argument

Bregman distance arguments in finite-dimensional Euclidean space that substitute for the missing Lipschitz gradient estimates

If this is right

The trajectory converges even at the critical value α = 3.
o(t^{-2}) rates hold for both objective residual and feasibility violation when α > 3.
The strategy extends to time-scaled primal-dual dynamics.
The result applies to convex problems with continuously differentiable but non-Lipschitz gradients.

Where Pith is reading between the lines

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

Numerical simulations on non-Lipschitz gradient functions could verify the rates in practice.
The finite-dimensional requirement indicates that additional tools would be needed for infinite-dimensional extensions.
Similar acceleration techniques might now be analyzed for a broader set of convex programs without the Lipschitz restriction.

Load-bearing premise

The underlying vector space must be finite-dimensional to enable the Bregman-distance technique that bypasses the Lipschitz gradient requirement.

What would settle it

A counterexample consisting of a convex continuously differentiable function with non-Lipschitz gradient in an infinite-dimensional Hilbert space for which the primal-dual trajectory does not converge would disprove the necessity of restricting to finite dimensions.

read the original abstract

We consider the Nesterov accelerated primal-dual dynamical system \[ \begin{cases} \ddot{x}(t)+\dfrac{\alpha}{t}\dot{x}(t) +\nabla f(x(t)) +A^\top\bigl(\lambda(t)+\theta t\dot{\lambda}(t)\bigr)+\beta A^\top(Ax(t)-b)=0,\\[0.6em] \ddot{\lambda}(t)+\dfrac{\alpha}{t}\dot{\lambda}(t) -\bigl(A(x(t)+\theta t\dot{x}(t))-b\bigr)=0, \end{cases} \] which is linked to the linearly constrained optimization problem $ \min_{x\in\mathbb{R}^n} f(x),\ s.t.\ Ax=b, $ where $\alpha\ge 3$ and $f$ is convex and continuously differentiable. In a Hilbert framework, the weak convergence of its trajectory was established by Bo\c{t} and Nguyen (J. Differential Equations, 303:369--406, 2021) under $\alpha>3$ and the Lipschitz continuity assumption on $\nabla f$. In this paper, we prove in finite-dimensional spaces that the trajectory converges to a primal-dual solution for $\alpha\ge3$, without assuming Lipschitz continuity of $\nabla f$. Moreover, when $\alpha>3$, we establish improved $o(t^{-2})$ convergence rates for both the objective residual and the feasibility violation. Our analysis relies on Bregman-distance arguments, instead of the Lipschitz continuity of $\nabla f$. The same strategy can also be extended to time-scaled primal-dual dynamics to obtain analogous convergence results. To the best of our knowledge, this is the first results in this topic without Lipschitz gradient assumption. Our result also present the first work on the convergence of the trajectory of the accelerated primal-dual dynamical system for the critical case $\alpha=3$.

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.

Desk Editor's Note private letter to a colleague

This paper drops the Lipschitz gradient assumption for Nesterov primal-dual dynamics using Bregman distances and handles the alpha=3 case in finite dimensions.

read the letter

The punchline is that they prove convergence of the trajectories to a primal-dual solution for alpha at least 3 without needing the gradient to be Lipschitz, and they get the o(t^{-2}) rates for alpha bigger than 3. This is done in finite dimensions. They build on the 2021 work by switching the proof technique to Bregman distances. That change removes the need for the Lipschitz condition that was in the earlier Hilbert-space analysis. The result covers the critical damping parameter alpha equals 3 for the first time. The same idea is said to work for time-scaled versions too. The paper stays with convex continuously differentiable objectives and linear constraints. The dynamical system is the standard second-order accelerated form with the primal-dual coupling terms. One area that could use more detail is the global-in-time existence of the solutions. The right-hand side is continuous, so Peano gives local existence. Extending it to the whole half-line requires showing the solution cannot explode in finite time. If the bound on the norm comes from the Bregman Lyapunov function that is also used for the convergence rates, there is a risk of circular reasoning. The authors should separate that step clearly. The finite-dimensional setting is a limitation compared to the general Hilbert space results, but it is what allows the Bregman approach to go through without Lipschitz. This work is aimed at researchers studying continuous-time accelerated optimization methods. It would interest people who want to relax smoothness assumptions in these models. The paper shows clear thinking on the technical side and engages honestly with the prior literature. I recommend sending it for peer review. The main claims are specific enough to be checked, and the method looks sound even if the existence argument needs tightening.

Referee Report

1 major / 2 minor

Summary. This manuscript analyzes the convergence of trajectories for a Nesterov-accelerated primal-dual dynamical system associated with linearly constrained convex optimization problems. In finite-dimensional spaces, it establishes convergence to a primal-dual solution for damping parameter α ≥ 3 without assuming Lipschitz continuity of ∇f, relying on Bregman-distance arguments. For α > 3, o(t^{-2}) rates are derived for the objective residual and feasibility gap. The analysis is also extended to time-scaled variants.

Significance. If the results hold, the work is significant because it removes the standard Lipschitz-gradient assumption that limits many continuous-time optimization analyses, thereby broadening applicability to convex problems with non-Lipschitz gradients. The treatment of the critical case α = 3 and the successful use of Bregman distances in finite dimensions constitute a clear technical advance over the cited Hilbert-space results of Boţ and Nguyen.

major comments (1)

[Well-posedness / existence analysis (likely early in the proofs section)] The global existence of solutions on [0, ∞) is load-bearing for the central convergence claim. Local existence follows from Peano’s theorem since the right-hand side is merely continuous, but extension to the whole half-line requires an a priori bound preventing finite-time blow-up of ||x(t)|| + ||λ(t)||. It is unclear whether this bound is obtained independently of the Bregman-distance Lyapunov function constructed later for convergence and rates; any circular dependence would invalidate the statement that “the trajectory converges.” Please identify the precise section or lemma that establishes the bound without reference to the maximal existence interval.

minor comments (2)

[Statement of main theorems] Clarify whether the o(t^{-2}) notation denotes little-o or big-O and whether the hidden constant depends on initial data or problem parameters.
[Introduction] The abstract and introduction cite Boţ and Nguyen (2021) but do not explicitly contrast the new finite-dimensional Bregman argument with the Lipschitz-based Lyapunov function used in that reference; a short comparative paragraph would help readers.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading of our manuscript and for the constructive comment on the well-posedness analysis. We address the point below.

read point-by-point responses

Referee: The global existence of solutions on [0, ∞) is load-bearing for the central convergence claim. Local existence follows from Peano’s theorem since the right-hand side is merely continuous, but extension to the whole half-line requires an a priori bound preventing finite-time blow-up of ||x(t)|| + ||λ(t)||. It is unclear whether this bound is obtained independently of the Bregman-distance Lyapunov function constructed later for convergence and rates; any circular dependence would invalidate the statement that “the trajectory converges.” Please identify the precise section or lemma that establishes the bound without reference to the maximal existence interval.

Authors: The global existence is treated in Lemma 3.1, which appears before the convergence statements. In this lemma we construct the Bregman-distance Lyapunov function V(t) directly from the system equations and the convexity of f. We derive the differential inequality ˙V(t) ≤ 0 (or a suitable negative term when α > 3) that holds on any interval where a solution exists. Because V(t) is nonnegative and controls the Euclidean norms of (x(t), λ(t)) through the properties of the Bregman distance in finite dimensions, the trajectory remains bounded on the maximal interval [0, T_max). Standard ODE continuation theory then implies T_max = ∞. The construction of V and the inequality ˙V ≤ 0 rely only on the vector field and convexity; they do not invoke the limit of the trajectory or any convergence result proved later. We will add a short clarifying remark immediately after Lemma 3.1 to make the logical order and the absence of circularity explicit. revision: partial

Circularity Check

0 steps flagged

No significant circularity; derivation relies on independent mathematical tools

full rationale

The paper's central claims rest on standard local existence via Peano's theorem in finite dimensions followed by Bregman-distance Lyapunov analysis to obtain global convergence and rates for α ≥ 3 without Lipschitz assumptions. No step reduces by construction to a fitted parameter, self-citation chain, or renamed input; the finite-dimensional restriction is explicitly justified as enabling the Bregman arguments, and the derivation chain remains self-contained against external ODE and convex-analysis benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The paper rests on standard domain assumptions for convex optimization and introduces no new entities or free parameters; the key technical shift is the use of Bregman distances to replace the Lipschitz condition.

axioms (2)

domain assumption f is convex and continuously differentiable
Explicitly stated in the abstract as the setting for the optimization problem.
domain assumption The dynamical system admits trajectories in finite-dimensional space
Implicit prerequisite for discussing trajectory convergence.

pith-pipeline@v0.9.0 · 5898 in / 1311 out tokens · 47103 ms · 2026-05-20T09:16:05.538640+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

Our analysis relies on Bregman-distance arguments... Z +∞ t0 t Df(x*, x(t)) dt < +∞ ... trajectory converges to a primal-dual solution for α≥3
IndisputableMonolith/Foundation/RealityFromDistinction.lean reality_from_one_distinction unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

energy function Ez*(t) = θ² t² (Lβ(x(t),λ*)−Lβ(x*,λ(t))) + ½∥v(t)∥² + ξ/2 ∥z(t)−z*∥²

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score.

Convergence of iterates and improved rates for accelerated augmented Lagrangian methods for linearly constrained convex optimization
math.OC 2026-05 unverdicted novelty 7.0

Accelerated augmented Lagrangian methods for convex problems achieve convergence and o(1/k^2) rates for feasibility violation and objective residual under suitable parameters.

Reference graph

Works this paper leans on

41 extracted references · 41 canonical work pages · cited by 1 Pith paper

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and Chbani, Z

Attouch, H. and Chbani, Z. and Fadili, J. and Riahi, H. , title =. J. Optim. Theory Appl. , volume =. 2022 , pages =

work page 2022

[2] [2]

and Chbani, Z

Attouch, H. and Chbani, Z. and Peypouquet, J. and Redont, P. , title =. Math. Program. , volume =. 2018 , pages =

work page 2018

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and Chbani, Z

Attouch, H. and Chbani, Z. and Riahi, H. , title =. SIAM J. Optim. , volume =. 2019 , pages =

work page 2019

[4] [4]

Attouch, H. and Bo. Fast optimization via inertial dynamics with closed-loop damping , journal =. 2023 , pages =

work page 2023

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and Peypouquet, J

Attouch, H. and Peypouquet, J. , title =. SIAM J. Optim. , volume =. 2016 , pages =

work page 2016

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and Dossal, C

Aujol, J.-F. and Dossal, C. , title =. SIAM J. Optim. , volume =. 2015 , pages =

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Alecsa, C. D. and L. Tikhonov regularization of a perturbed heavy ball system with vanishing damping , journal =. 2021 , pages =

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and Jia, L

Bai, J. and Jia, L. and Peng, Z. , title =. J. Sci. Comput. , volume =. 2024 , pages =

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Bai, J. and Chen, Y. and Yu, X. and Zhang, H. , title =. J. Sci. Comput. , volume =. 2025 , pages =

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and Chbani, Z

Battahi, F. and Chbani, Z. and Riahi, H. , title =. Evol. Equ. Control Theory , volume =. 2025 , pages =

work page 2025

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Bo. Improved convergence rates and trajectory convergence for primal-dual dynamical systems with vanishing damping , journal =. 2021 , pages =

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Bo. Tikhonov regularization of a second order dynamical system with Hessian driven damping , journal =. 2021 , pages =

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Fast optimistic gradient descent ascent method in continuous and discrete time , journal =

Bo. Fast optimistic gradient descent ascent method in continuous and discrete time , journal =. 2025 , pages =

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Fast augmented Lagrangian method in the convex regime with convergence guarantees for the iterates , journal =

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A primal-dual dynamical approach to structured convex minimization problems , journal =

Bo. A primal-dual dynamical approach to structured convex minimization problems , journal =. 2020 , pages =

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The iterates of Nesterov's accelerated algorithm converge in the critical regimes , year =

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work page arXiv

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Fast second-order dynamics with slow vanishing damping approaching the zeros of a monotone and continuous operator , year =

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work page arXiv

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and Parikh, N

Boyd, S. and Parikh, N. and Chu, E. and Peleato, B. and Eckstein, J. , title =. Found. Trends Mach. Learn. , volume =. 2010 , pages =

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and Shi, X

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He, X. and Guo, L. and He, D. , title =. Neural Networks , volume =. 2025 , pages =

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Hulett, D. A. and Nguyen, D.-K. , title =. Appl. Math. Optim. , volume =. 2023 , pages =

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Jang, U. and Ryu, E. K. , title =. 2025 , eprint =

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and Wilson, A

Wibisono, A. and Wilson, A. C. and Jordan, M. I. , title =. Proc. Natl. Acad. Sci. USA , volume =. 2016 , pages =

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and Lei, J

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Zeng, X. and Yi, P. and Hong, Y. and Xie, L. , title =. SIAM J. Control Optim. , volume =. 2018 , pages =

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Zhang, H. and Sun, X. and Li, S. and Teo, K. L. , title =. Appl. Math. Optim. , volume =. 2026 , pages =

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and Liao, X

Zhao, Y. and Liao, X. and He, X. and Zhou, M. and Li, C. , title =. J. Mach. Learn. Res. , volume =. 2023 , pages =

work page 2023