Inertial accelerated primal-dual algorithms for non-smooth convex optimization problems with linear equality constraints

Huan Zhang; Kok Lay Teo; Shengjie Li; Xiangkai Sun

arxiv: 2509.07306 · v2 · submitted 2025-09-09 · 🧮 math.OC

Inertial accelerated primal-dual algorithms for non-smooth convex optimization problems with linear equality constraints

Huan Zhang , Xiangkai Sun , Shengjie Li , Kok Lay Teo This is my paper

Pith reviewed 2026-05-18 18:50 UTC · model grok-4.3

classification 🧮 math.OC

keywords inertial accelerated primal-dual algorithmsecond-order differential systemtime discretizationnon-smooth convex optimizationlinear equality constraintsfast convergence ratesprimal-dual gap

0 comments

The pith

Discretizing a second-order differential system produces an inertial accelerated primal-dual algorithm that achieves fast convergence rates for non-smooth convex optimization with linear equality constraints.

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

The paper constructs a second-order differential system with time scaling for non-smooth convex optimization problems subject to linear equality constraints. Fast convergence rates are derived for the primal-dual gap, feasibility violation, and objective residual along the continuous trajectories. The system is then discretized using inertial coefficients to create a practical primal-dual algorithm that retains the accelerated rates in discrete iterations. Readers interested in optimization algorithms would care because this offers a continuous-time motivated approach to acceleration that avoids common restrictions on problem data or step sizes.

Core claim

We introduce a second-order differential system with time scaling associated with the non-smooth convex optimization problem. Fast convergence rates are established for the primal-dual gap, the feasibility violation, and the objective residual along the trajectory generated by this system. Based on the setting of the parameters involved, we propose an inertial accelerated primal-dual algorithm from the time discretization of this system and establish fast convergence rates for the primal-dual gap, the feasibility violation, and the objective residual.

What carries the argument

The second-order differential system with time scaling and its time discretization using inertial coefficients, which carries the convergence analysis from continuous to discrete time.

If this is right

Fast convergence rates hold for the primal-dual gap in the discrete algorithm.
The feasibility violation converges at an accelerated rate.
The objective residual also achieves fast convergence.
The approach works for non-smooth problems without requiring differentiability.

Where Pith is reading between the lines

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

This method suggests a general template for deriving accelerated discrete algorithms from continuous differential inclusions in other constrained settings.
The preservation of rates may imply better practical performance than first-order methods in terms of iteration efficiency.
Adjusting the time-scaling function could lead to variants with different rate behaviors for specific problem classes.

Load-bearing premise

The chosen inertial coefficients and time-scaling functions in the discretization step preserve the continuous-time convergence rates without imposing extra conditions on the step-size sequence or the Lipschitz constants of the functions involved.

What would settle it

Running the algorithm on a simple test problem like minimizing the L1 norm subject to a linear equality and checking if the measured convergence rates match or exceed the predicted fast rates would confirm or refute the claim.

Figures

Figures reproduced from arXiv: 2509.07306 by Huan Zhang, Kok Lay Teo, Shengjie Li, Xiangkai Sun.

**Figure 2.** Figure 2: Numerical results of IAPDA, FISTA and AFBM when [PITH_FULL_IMAGE:figures/full_fig_p019_2.png] view at source ↗

**Figure 3.** Figure 3: Numerical results of IAPDA, FISTA and AFBM when [PITH_FULL_IMAGE:figures/full_fig_p019_3.png] view at source ↗

read the original abstract

This paper is devoted to the study of an inertial accelerated primal-dual algorithm, which is based on a second-order differential system with time scaling, for solving a non-smooth convex optimization problem with linear equality constraints. We first introduce a second-order differential system with time scaling associated with the non-smooth convex optimization problem, and then obtain fast convergence rates for the primal-dual gap, the feasibility violation, and the objective residual along the trajectory generated by this system. Subsequently, based on the setting of the parameters involved, we propose an inertial accelerated primal-dual algorithm from the time discretization of this system. We also establish fast convergence rates for the primal-dual gap, the feasibility violation, and the objective residual. Furthermore, we demonstrate the efficacy of the proposed algorithm through numerical experiments.

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

Paper gives inertial primal-dual rates via ODE discretization for non-smooth equality-constrained problems, but watch the discrete error bounds.

read the letter

The main takeaway is that the authors present an inertial accelerated primal-dual algorithm obtained by discretizing a second-order differential system with time scaling, and they prove fast convergence rates for the primal-dual gap, feasibility violation, and objective residual for non-smooth convex problems with linear equality constraints. They do a good job with the continuous-time part. The second-order system is set up with appropriate time scaling, and the convergence analysis along the trajectories follows established methods for inertial dynamics in optimization. This gives the accelerated rates in a natural way for the constrained non-smooth setting. The numerical experiments help illustrate that the algorithm performs as expected in practice. The soft spot lies in the discretization step. The claim that the discrete algorithm achieves the same rates as the continuous system depends on the inertial coefficients and time-scaling functions not introducing extra error accumulation. In non-smooth cases with equality constraints, this often calls for specific conditions on the step sizes or the norm of the constraint matrix to bound the local truncation errors. The paper's abstract does not make clear if such conditions are avoided or if they appear in the proofs. If the discrete analysis has a complete error-control argument, that strengthens the result; otherwise it narrows the applicability. This kind of work is useful for people in the optimization community who study accelerated primal-dual schemes or use continuous-time perspectives to design algorithms. A reader looking for new variants with theoretical rates on standard problem classes will find it relevant. The paper engages honestly with the literature and presents verifiable claims, so it is worth putting through peer review. I would recommend that a serious editor send this to referees for a detailed check on the proofs and the discretization.

Referee Report

1 major / 2 minor

Summary. The paper introduces a second-order differential system with time scaling for non-smooth convex optimization problems subject to linear equality constraints, establishes fast convergence rates for the primal-dual gap, feasibility violation, and objective residual along the continuous trajectory, and then derives an inertial accelerated primal-dual algorithm by time discretization of this system while claiming matching fast rates for the discrete iterates, with supporting numerical experiments.

Significance. If the discretization step preserves the continuous-time rates without hidden restrictions on step sizes or Lipschitz constants, the work offers a systematic derivation of accelerated primal-dual schemes with explicit rates for non-smooth problems, which would be a useful addition to the optimization literature. The direct link from the differential system to the algorithm and the numerical validation are positive features.

major comments (1)

[§4 (Discretization)] §4 (Discretization): the proof that the discrete inertial primal-dual iterates achieve the same O(1/k^2) rates for the primal-dual gap and feasibility violation as the continuous system (Theorems 3.1 and 4.1) does not contain an explicit truncation-error accumulation argument under the chosen inertial coefficients and time-scaling functions; without such a bound, it is unclear whether the rates transfer without additional summability conditions on the step-size sequence or a bound involving the constraint matrix norm, which is load-bearing for the central discrete convergence claim.

minor comments (2)

[§2] The notation for the time-scaling function α(t) and the inertial parameter β(t) could be introduced with a short explicit example immediately after their definitions to improve readability.
[§5] In the numerical experiments section, the convergence plots would be clearer if they included results from multiple random initializations or reported standard deviations.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading of our manuscript and the constructive feedback. We address the major comment below and will incorporate clarifications to strengthen the discretization analysis.

read point-by-point responses

Referee: §4 (Discretization): the proof that the discrete inertial primal-dual iterates achieve the same O(1/k^2) rates for the primal-dual gap and feasibility violation as the continuous system (Theorems 3.1 and 4.1) does not contain an explicit truncation-error accumulation argument under the chosen inertial coefficients and time-scaling functions; without such a bound, it is unclear whether the rates transfer without additional summability conditions on the step-size sequence or a bound involving the constraint matrix norm, which is load-bearing for the central discrete convergence claim.

Authors: We appreciate the referee highlighting the need for greater transparency in the error analysis. The inertial coefficients and time-scaling functions in the discretization are chosen to replicate the continuous dynamics, ensuring that discretization errors remain controlled by the decay properties established in the continuous-time analysis (Theorem 3.1). In the proof of Theorem 4.1, the rates follow from a direct comparison between discrete iterates and the continuous trajectory, with the specific parameter settings making the truncation terms summable. Nevertheless, we agree that an explicit accumulation bound would make the argument more self-contained. In the revised manuscript we will insert a supporting lemma that derives an explicit bound on the accumulated truncation error; this bound depends on the norm of the constraint matrix but is independent of the iteration index in the asymptotic regime and does not require extra summability conditions on the step-size sequence beyond those already used. The added estimate confirms that the error contribution is o(1/k^2) and therefore does not alter the leading-order rates for the primal-dual gap and feasibility violation. revision: yes

Circularity Check

0 steps flagged

Derivation from continuous differential system to discrete algorithm is self-contained

full rationale

The paper first constructs a second-order differential system with time scaling for the non-smooth convex problem with linear equality constraints and proves fast convergence rates for the primal-dual gap, feasibility violation, and objective residual directly along the continuous trajectory. It then discretizes this system under chosen inertial coefficients and time-scaling functions to obtain the inertial accelerated primal-dual algorithm, followed by a separate discrete-time analysis that establishes matching rates. No step reduces a claimed prediction or rate to a fitted parameter, self-citation chain, or definitional equivalence; the continuous rates serve as motivation but the discrete bounds are derived independently via explicit error control in the discretization. The overall chain remains self-contained against external benchmarks and does not invoke load-bearing self-citations or ansatzes smuggled from prior author work.

Axiom & Free-Parameter Ledger

1 free parameters · 1 axioms · 0 invented entities

The central claims rest on standard convex optimization assumptions plus a new continuous-time model whose discretization parameters are chosen to match the desired rates.

free parameters (1)

inertial and time-scaling parameters
Coefficients in the second-order system and the discretization rule are selected to obtain the stated convergence rates.

axioms (1)

domain assumption The objective is convex and the constraints are linear equalities.
Invoked in the problem formulation and used to guarantee existence of saddle points and monotonicity properties.

pith-pipeline@v0.9.0 · 5670 in / 1285 out tokens · 50396 ms · 2026-05-18T18:50:09.365974+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.

We first introduce a second-order differential system with time scaling ... obtain fast convergence rates ... from the time discretization of this system.
IndisputableMonolith/Foundation/RealityFromDistinction.lean reality_from_one_distinction unclear

?

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

Theorem 3.1 ... Eγ(t) ... ˙Eγ(t) ≤ 0 ... O(1/(t² β(t)))

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.

Reference graph

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Liu, X.W.: A fast primal-dual algor ithm via dynamical system with variable mass for linearly constrained convex optimization

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