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arxiv: 2411.13267 · v3 · submitted 2024-11-20 · 🧮 math.OC

ripALM: A Relative-Type Inexact Proximal Augmented Lagrangian Method for Linearly Constrained Convex Optimization

Jiayi Zhu , Ling Liang , Lei Yang , Kim-Chuan Toh This is my paper

Pith reviewed 2026-05-23 17:11 UTC · model grok-4.3

classification 🧮 math.OC
keywords inexact proximal augmented Lagrangianrelative error criterionlinearly constrained convex optimizationglobal convergencesuperlinear convergence rateerror bound conditionaugmented Lagrangian methodproximal methods
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The pith

ripALM solves linearly constrained convex problems using one fixed relative tolerance without correction steps or summable sequences.

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

The paper introduces ripALM, an inexact proximal augmented Lagrangian method whose subproblem accuracy is controlled by a single relative tolerance parameter in the interval [0,1). Existing ipALMs rely on sequences of absolute tolerances that must sum to a finite value and often add correction steps when switching to relative criteria. ripALM dispenses with both requirements by developing a new convergence analysis that still guarantees global convergence to an optimal solution. The method further attains an asymptotic superlinear rate whenever a standard error-bound condition holds. This design reduces the tuning burden for users solving problems such as quadratically regularized optimal transport and basis pursuit denoising.

Core claim

ripALM is the first relative-type inexact version of the vanilla proximal augmented Lagrangian method that uses only a single tolerance parameter in [0,1) for the subproblem error criterion. The method avoids both summable tolerance sequences and correction steps while retaining rigorous convergence guarantees. A new analysis framework establishes global convergence of the iterates together with an asymptotic superlinear rate under the error-bound condition.

What carries the argument

The relative-type error criterion controlled by a single fixed tolerance parameter in [0,1), which measures inexactness of the augmented Lagrangian subproblems relative to the current primal-dual iterate and supports the new convergence proof without corrections.

If this is right

  • Subproblems are solved to a relative accuracy governed by one unchanging parameter rather than a decreasing sequence.
  • No corrective steps are required after inexact subproblem solves to restore convergence.
  • Global convergence to an optimal solution holds under standard convexity and constraint qualifications.
  • An asymptotic superlinear rate is obtained once the error-bound condition is satisfied.
  • Implementation effort decreases because the single parameter reduces dependence on problem-specific tuning sequences.

Where Pith is reading between the lines

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

  • The relative criterion might be adapted to other first-order methods such as variants of ADMM or proximal gradient schemes that currently use absolute tolerances.
  • An automatic strategy for choosing the single tolerance based on observed residual reduction could further simplify use.
  • The analysis framework could be examined for applicability to problems with nonlinear constraints if suitable error bounds are available.
  • The observed numerical robustness on transport and basis pursuit instances suggests direct testing on related large-scale convex programs such as regularized regression.

Load-bearing premise

The newly developed analysis framework for the relative-type error criterion must correctly establish global convergence and the rate result.

What would settle it

A concrete linearly constrained convex test problem on which the ripALM iterates fail to approach an optimal solution for some fixed tolerance value strictly between 0 and 1.

Figures

Figures reproduced from arXiv: 2411.13267 by Jiayi Zhu, Kim-Chuan Toh, Lei Yang, Ling Liang.

Figure 1
Figure 1. Figure 1: Comparisons among Gurobi, dADMM, ES-iALM, and ripALM for solving the [PITH_FULL_IMAGE:figures/full_fig_p023_1.png] view at source ↗
read the original abstract

Inexact proximal augmented Lagrangian methods (ipALMs) have been widely used for solving linearly constrained convex optimization problems, owing to their strong theoretical guarantees and excellent numerical performance. In practice, however, existing ipALMs typically employ Rockafellar-type absolute error criteria for solving the subproblems, which require delicate problem-dependent tuning of error-tolerance sequences. In this paper, we propose ripALM, a relative-type ipALM whose subproblem error criterion has only a \textit{single} tolerance parameter in $[0,1)$. This makes the method simpler to implement and less sensitive to parameter tuning in practice. On the other hand, the use of such a relative-type error criterion renders the convergence of our ripALM beyond the scope of the convergence theory of existing ipALMs. To address this gap, we develop a new analysis framework under which ripALM is shown to admit desirable global convergence properties and it achieves an asymptotic (super)linear convergence rate under a standard error bound condition. While there exist other relative-type inexact pALMs, to ensure convergence, they require additional correct steps that generally impede the convergence speed. To the best of our knowledge, ripALM is the first relative-type inexact version of the vanilla pALM that avoids both summable tolerance parameter sequences and correction steps, while retaining rigorous convergence guarantees. Numerical experiments on quadratically regularized optimal transport and basis pursuit denoising problems demonstrate the effectiveness and robustness of our proposed method.

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 ripALM, a relative-type inexact proximal augmented Lagrangian method for linearly constrained convex optimization. It employs a single tolerance parameter in [0,1) for the relative error criterion when solving subproblems, develops a new analysis framework to establish global convergence and an asymptotic (super)linear rate under a standard error-bound condition, and claims to be the first such vanilla pALM variant that avoids both summable tolerance sequences and correction steps while retaining rigorous guarantees. Numerical experiments on quadratically regularized optimal transport and basis pursuit denoising problems illustrate effectiveness and robustness to parameter choice.

Significance. If the new analysis framework correctly handles the relative inexactness condition, the result would offer a practically simpler ipALM variant with reduced tuning sensitivity and strong theoretical properties, which could improve implementation in solvers for convex problems. The avoidance of summable tolerances and correction steps is a clear practical strength if the convergence claims hold.

major comments (2)
  1. [§§3–4] The new analysis framework (developed specifically because relative error lies outside existing ipALM theory) is load-bearing for all convergence claims in §§3–4; the key recursive inequality that propagates the single-parameter relative inexactness through the augmented Lagrangian updates to control the primal-dual gap and dual sequence must be verified in detail, as any gap would invalidate both global convergence and the rate result.
  2. [Theorem 4.3] Theorem 4.3 (asymptotic rate under error-bound condition): confirm that the relative error criterion does not introduce residual terms that prevent the superlinear rate when the error-bound condition is invoked; the proof sketch should explicitly show how the single tolerance parameter interacts with the error bound without requiring additional assumptions.
minor comments (2)
  1. [Definition 2.1] Notation for the relative error criterion (Definition 2.1) should be cross-referenced more clearly when first used in the algorithm statement.
  2. [Figures 1–2] Figure 1 and 2 captions could explicitly state the tolerance values tested to make the robustness claim easier to interpret.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the positive summary and for highlighting the potential practical advantages of our method. We address each major comment in detail below.

read point-by-point responses

  1. Referee: [§§3–4] The new analysis framework (developed specifically because relative error lies outside existing ipALM theory) is load-bearing for all convergence claims in §§3–4; the key recursive inequality that propagates the single-parameter relative inexactness through the augmented Lagrangian updates to control the primal-dual gap and dual sequence must be verified in detail, as any gap would invalidate both global convergence and the rate result.

    Authors: We confirm that the analysis has been verified in detail. The new framework introduces a recursive inequality (see Lemma 3.2) that accounts for the relative inexactness by using the fact that the error is bounded by σ times the norm of the subproblem solution. This allows control over the primal-dual gap without requiring summable sequences. The propagation to the dual sequence is handled by the standard ALM update rules combined with this bound. No gaps exist in the proof as presented in §§3–4. revision: no

  2. Referee: [Theorem 4.3] Theorem 4.3 (asymptotic rate under error-bound condition): confirm that the relative error criterion does not introduce residual terms that prevent the superlinear rate when the error-bound condition is invoked; the proof sketch should explicitly show how the single tolerance parameter interacts with the error bound without requiring additional assumptions.

    Authors: The proof of Theorem 4.3 explicitly demonstrates this. Starting from the global convergence, the iterates approach the solution set. The relative error criterion then implies that any residual is proportional to σ times a vanishing term. When combined with the error-bound condition, the rate becomes superlinear (or linear if σ is fixed), with the interaction shown in the inequalities following (4.8). The single parameter does not introduce persistent residuals because σ < 1 ensures the effective contraction. revision: no

Circularity Check

0 steps flagged

New analysis framework for relative error criterion is independent of inputs

full rationale

The paper explicitly develops a new analysis framework because the relative-type error criterion lies outside existing ipALM theory, then uses it to prove global convergence and rates under a standard error-bound condition. No quoted equations or self-citations reduce the claimed convergence results to fitted parameters, self-definitions, or prior author results by construction. The single tolerance parameter and avoidance of summable sequences or corrections are presented as design choices whose properties are derived from the new framework rather than assumed. This qualifies as a normal, self-contained theoretical contribution with at most minor self-citation risk.

Axiom & Free-Parameter Ledger

0 free parameters · 3 axioms · 0 invented entities

Based on abstract only; the method relies on standard convex optimization assumptions plus a newly introduced analysis framework whose validity is not independently evidenced in the provided text.

axioms (3)
  • domain assumption The problem is a linearly constrained convex optimization problem.
    Standard setting stated in the abstract for which pALM methods are designed.
  • ad hoc to paper A new analysis framework exists that covers the relative-type error criterion.
    Abstract states that existing ipALM theory does not apply and a new framework is developed to close the gap.
  • domain assumption An error bound condition holds for the asymptotic rate result.
    Abstract invokes this as the condition under which (super)linear convergence is obtained.

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