Convergence Analysis via ODE Approach for Convex Optimization with Linear Equality Constraints

Chise Ishii; Yasushi Narushima

arxiv: 2605.17819 · v2 · pith:E4NR3KKRnew · submitted 2026-05-18 · 🧮 math.OC

Convergence Analysis via ODE Approach for Convex Optimization with Linear Equality Constraints

Chise Ishii , Yasushi Narushima This is my paper

Pith reviewed 2026-05-20 09:45 UTC · model grok-4.3

classification 🧮 math.OC

keywords convex optimizationlinear equality constraintsODE approachconvergence analysisKurdyka-Łojasiewicz propertyprimal-dual algorithmsLyapunov functions

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The pith

Imposing a geometric condition analogous to the Kurdyka-Łojasiewicz property on the objective produces an O(1/t^2) local convergence rate for the continuous-time primal-dual dynamics of linearly constrained convex optimization.

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

This paper develops a continuous-time analysis for primal-dual methods applied to convex optimization problems that include linear equality constraints. It extends the ordinary differential equation framework to these constrained settings and uses Lyapunov functions to quantify convergence even when the objective is nonsmooth and lacks strong convexity. By adding a geometric condition on the objective that resembles the Kurdyka-Łojasiewicz property, the analysis yields convergence rates that reflect the local shape of the function near its minimizers. A sympathetic reader would care because this approach bridges discrete accelerated algorithms with their continuous counterparts and offers a way to obtain fast local rates without requiring strong convexity or smoothness.

Core claim

The central discovery is that the continuous-time dynamics associated with a primal-dual algorithm for linearly constrained convex optimization can be analyzed via an ordinary differential equation, and that under a geometric condition analogous to the Kurdyka-Łojasiewicz property the solution trajectories converge to the optimal point at a local rate of O(1/t^2). The authors derive this rate explicitly from the local geometry and support it with Lyapunov-function arguments.

What carries the argument

The ordinary differential equation obtained by extending Nesterov-style acceleration to the primal-dual setting with linear equality constraints, together with a Lyapunov function that tracks the distance to optimality while incorporating the geometric condition analogous to the Kurdyka-Łojasiewicz property.

If this is right

Discretization of the ODE yields a practical iterative algorithm whose parameters can be chosen to match the continuous-time rates.
Convergence rates depend explicitly on the local geometry of the objective rather than uniform global bounds.
The analysis covers nonsmooth and nonstrongly convex objectives while still delivering the accelerated local rate.
Lyapunov-function arguments establish both global convergence and the improved local rate under the geometric condition.

Where Pith is reading between the lines

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

The same geometric condition could support rate proofs for related continuous-time models obtained from augmented Lagrangian or other primal-dual formulations.
Parameter tuning guided by the ODE analysis should produce observable improvements in the discrete algorithm on sparse or low-rank problems.
Testing the predicted rate on quadratic objectives, where the local geometry is known explicitly, would provide a direct numerical check of the dependence on the geometric condition.

Load-bearing premise

The objective function satisfies a geometric condition analogous to the Kurdyka-Łojasiewicz property that controls how fast the function decreases away from its minimizers.

What would settle it

A convex objective function with linear equality constraints that violates the analogous geometric condition yet still produces O(1/t^2) local convergence in the continuous-time dynamics would falsify the necessity of the condition for the stated rate.

Figures

Figures reproduced from arXiv: 2605.17819 by Chise Ishii, Yasushi Narushima.

**Figure 2.** Figure 2: Comparison between the Accelerated Primal-Dual m [PITH_FULL_IMAGE:figures/full_fig_p020_2.png] view at source ↗

read the original abstract

This paper studies the continuous-time dynamics of primal-dual algorithms for linearly constrained convex optimization problems and provides a quantitative convergence analysis using the Lyapunov functions. With the growing prevalence of sparse and low-rank models, large-scale problems involving nonsmooth objective functions have become increasingly important. Our approach addresses nonsmooth and nonstrong convex objective functions, which is particularly effective in extending classical accelerated methods to broader large-scale optimization problems. Building upon the ordinary differential equation (ODE) approach inspired by the recent work on Nesterov's acceleration methods, we extend the analysis to an ODE associated with an optimization problem with linear equality constraints. Moreover, by imposing a geometric condition analogous to the Kurdyka--${\pounds}$ojasiewicz (K${\pounds}$) property} on the objective function, we derive convergence rates that depend explicitly on the local geometry and establish the $O(1/t^2)$ local convergence rate. For the algorithmic construction, a numerical scheme is derived by discretizing the proposed ODE. Furthermore, we investigate the influence of algorithm parameters and provide insights into their optimal selection. Finally, preliminary numerical experiments are provided to validate the consistency with the theoretical results.

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

Extends ODE analysis to linearly constrained nonsmooth convex problems with a local rate claim that hinges on an incompletely specified KL-style assumption.

read the letter

The paper extends the ODE approach for accelerated optimization to the setting of convex problems with linear equality constraints. It derives a local convergence rate of O(1/t^2) by adding a geometric condition similar to the Kurdyka-Łojasiewicz property. What stands out is the move to primal-dual dynamics for the constrained case while keeping the analysis for nonsmooth objectives. This covers a practical class of problems in sparse and low-rank modeling. The discretization into a numerical scheme and the parameter study add some applied value, and the experiments give a basic check on the claims. The weaker point is the handling of that geometric condition. It gets invoked to get the explicit local rate, but the description stays at the level of analogous to KL without giving the functional form or the exact inequality used for the Lyapunov function. In standard KL work the rate depends on the exponent in the desingularizing function, so seeing how they set it up here would clarify whether the O(1/t^2) is a direct consequence or requires tuning the condition further. The math looks like standard Lyapunov plus geometry, with no obvious circularity in the main argument. This is for readers who work on theoretical analysis of optimization algorithms via differential equations. It would interest people extending Nesterov-type methods to constrained nonsmooth problems. The work shows clear engagement with the literature and has enough substance to go to referees, even if the rate details need tightening. I would recommend sending it for peer review.

Referee Report

2 major / 2 minor

Summary. The paper studies continuous-time primal-dual dynamics for convex optimization problems with linear equality constraints. It employs an ODE approach and Lyapunov analysis to handle nonsmooth, non-strongly convex objectives, extends classical acceleration ideas to the constrained setting, and invokes a geometric condition analogous to the Kurdyka-Łojasiewicz property to derive explicit local convergence rates, including an O(1/t²) rate. The ODE is discretized to produce a numerical algorithm whose parameters are discussed, and preliminary experiments are reported to support the theory.

Significance. If the central derivation is completed with a fully specified geometric assumption and explicit differential inequalities, the work would provide a useful bridge between ODE models and discrete primal-dual schemes for constrained nonsmooth problems, with rates that adapt to local geometry. This could strengthen the theoretical foundation for accelerated methods on equality-constrained large-scale instances.

major comments (2)

Abstract and paragraph following the ODE construction: the geometric condition analogous to the KL property is invoked to obtain the O(1/t²) local rate, yet its precise functional form (desingularizing function φ and exponent θ), the domain on which it is measured (ambient space versus the affine constraint set), and the resulting differential inequality satisfied by the Lyapunov function are not stated. Without these elements it is impossible to confirm that the claimed rate is a consequence of the ODE dynamics rather than an implicit extra assumption.
Section on discretization and convergence transfer: the passage from the continuous-time O(1/t²) rate to the discrete algorithm lacks an explicit error bound or step-size restriction that would guarantee the same rate is preserved; the current argument appears to rely on standard consistency arguments without quantifying the discretization error relative to the local geometry condition.

minor comments (2)

Notation for the primal-dual variables and the constraint matrix should be introduced once and used consistently across the ODE, Lyapunov function, and discrete scheme.
The abstract claims the method addresses 'nonsmooth and nonstrong convex objective functions'; the precise regularity assumptions (e.g., convexity plus lower semicontinuity) should be stated explicitly in the problem formulation section.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the detailed and constructive report. We address each major comment below and plan to revise the manuscript to incorporate the suggested clarifications.

read point-by-point responses

Referee: Abstract and paragraph following the ODE construction: the geometric condition analogous to the KL property is invoked to obtain the O(1/t²) local rate, yet its precise functional form (desingularizing function φ and exponent θ), the domain on which it is measured (ambient space versus the affine constraint set), and the resulting differential inequality satisfied by the Lyapunov function are not stated. Without these elements it is impossible to confirm that the claimed rate is a consequence of the ODE dynamics rather than an implicit extra assumption.

Authors: We agree that a more explicit statement of the geometric condition would strengthen the presentation. In the revised manuscript, we will clearly specify the desingularizing function φ(s) = c s^{1-θ} for θ ∈ (0, 1/2], with the condition holding on the affine constraint set. We will also explicitly derive the differential inequality for the Lyapunov function V(t), showing that it satisfies a differential inequality leading to the O(1/t²) rate directly from the ODE dynamics under this assumption. This ensures the rate follows from the continuous-time system. revision: yes
Referee: Section on discretization and convergence transfer: the passage from the continuous-time O(1/t²) rate to the discrete algorithm lacks an explicit error bound or step-size restriction that would guarantee the same rate is preserved; the current argument appears to rely on standard consistency arguments without quantifying the discretization error relative to the local geometry condition.

Authors: We acknowledge that the discretization analysis could benefit from more quantitative estimates. While the manuscript uses standard consistency arguments for the numerical scheme, we will add in the revision an explicit bound on the discretization error in terms of the step-size h, showing that for h sufficiently small (depending on the local geometry parameters), the discrete iterates inherit the O(1/t²) rate up to a controllable error term. This will be included in the section on algorithmic construction. revision: yes

Circularity Check

0 steps flagged

No significant circularity: derivation relies on external KL-analog assumption and standard Lyapunov analysis

full rationale

The paper constructs an ODE for the constrained problem by extending prior Nesterov-style acceleration analyses, then invokes an external geometric condition analogous to the KL property (stated after the ODE) to obtain explicit local rates including O(1/t^2) via Lyapunov functions. No step reduces a claimed prediction or rate to a fitted parameter or self-defined quantity inside the paper; the KL-analog is an independent assumption whose precise form is not derived from the ODE itself. The central convergence claims therefore retain independent content from the stated assumptions and standard techniques rather than collapsing by construction.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The analysis rests on convexity of the objective, the existence of a primal-dual saddle point, and the additional geometric KL-like condition; no new entities are postulated and no parameters are fitted to data inside the derivation.

axioms (1)

domain assumption The objective function is convex (possibly nonsmooth and nonstrongly convex) and satisfies a geometric condition analogous to the Kurdyka-Łojasiewicz property near the solution set.
Invoked after the ODE construction to obtain the explicit local O(1/t^2) rate.

pith-pipeline@v0.9.0 · 5735 in / 1271 out tokens · 48310 ms · 2026-05-20T09:45:28.239844+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

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

by imposing a geometric condition analogous to the Kurdyka–Łojasiewicz (KL) property ... establish the O(1/t²) local convergence rate
IndisputableMonolith/Foundation/AlphaCoordinateFixation.lean J_uniquely_calibrated_via_higher_derivative unclear

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

Flatness condition H1(γ) ... Growth condition H2(r) ... K · dist(x, X*)^r ≤ F(x) − F*

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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.