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arxiv: 2604.14368 · v1 · submitted 2026-04-15 · 🧮 math.OC

A Noise Tolerant SQP Algorithm for Inequality Constrained Optimization

Figen Oztoprak , Richard Byrd This is my paper

Pith reviewed 2026-05-10 12:14 UTC · model grok-4.3

classification 🧮 math.OC
keywords sequential quadratic programmingbounded noiseinequality constrained optimizationglobal convergenceline search methodsquasi-Newton updatesnoise tolerant algorithms
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The pith

A line search SQP algorithm for inequality constrained optimization remains globally convergent under bounded noise in all evaluations.

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

The paper develops a sequential quadratic programming algorithm that incorporates relaxations to manage noise in the objective function, constraints, and their derivatives. The analysis shows that the method preserves global convergence properties even when these evaluations are inexact within a known bound. Implementation with modified quasi-Newton updates allows the solution accuracy to scale with the noise level. This matters because many practical optimization problems rely on noisy simulations or measurements rather than exact computations. The result provides a reliable way to solve such problems without assuming perfect data.

Core claim

The paper proposes a noise-tolerant SQP algorithm for inequality constrained optimization problems. It is a line search method that uses relaxations to handle potential inconsistency in the quadratic subproblems due to noise. The theory establishes global convergence to a stationary point, and the achieved accuracy is proportional to the noise level and problem parameters. Numerical experiments with noise-aware quasi-Newton updates support the theoretical predictions.

What carries the argument

Line search sequential quadratic programming with relaxations for handling noisy and inconsistent subproblems, paired with noise-aware quasi-Newton Hessian approximations.

Load-bearing premise

All evaluations of the objective, constraints, and derivatives contain noise bounded by a known constant, and the relaxations do not invalidate the convergence theory of the underlying SQP method.

What would settle it

Observe whether the algorithm fails to converge or exceeds the predicted accuracy when noise in some evaluations surpasses the assumed bound on a test problem.

Figures

Figures reproduced from arXiv: 2604.14368 by Figen Oztoprak, Richard Byrd.

Figure 4.1
Figure 4.1. Figure 4.1: The dependence of v(xf ) (upper subplot) and |f(xf )−f(x∗)| (lower subplot) to the noise level, where xf is the iterate returned by the algorithm. The plots are generated at the logarithmic scale for clarity; the labels display the true (unscaled) values. We next run the algorithm by injecting noise to function and gradient evaluations. We added uniformly distributed random noise to each function evaluat… view at source ↗
Figure 4.2
Figure 4.2. Figure 4.2: The final value of the penalty parameter [PITH_FULL_IMAGE:figures/full_fig_p027_4_2.png] view at source ↗
Figure 4.3
Figure 4.3. Figure 4.3: The iterate the algorithm finds for the first time a solution with an objective [PITH_FULL_IMAGE:figures/full_fig_p028_4_3.png] view at source ↗
read the original abstract

We propose a sequential quadratic programming (SQP) algorithm for inequality constrained optimization that is robust to the presence of bounded noise in function and derivative evaluations. We cover the case where constraint evaluations contain noise as well as the objective. The proposed algorithm is a line search SQP method with relaxations to deal with noise. We study the effect of noise on the global convergence behavior of the algorithm. We implement the algorithm with noise-aware quasi-Newton updates, and numerically observe that the algorithm can achieve accuracy proportional to the noise level and problem-dependent parameters, as suggested by the theory.

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

0 major / 2 minor

Summary. The paper proposes a line-search SQP algorithm for inequality-constrained optimization that incorporates explicit relaxations to accommodate bounded noise in objective, constraint, and derivative evaluations. It analyzes the global convergence of the method to a neighborhood whose radius scales with the noise bound and problem-dependent constants, implements the approach using noise-aware quasi-Newton updates, and reports numerical experiments in which observed accuracy is proportional to the noise level as predicted by the theory.

Significance. If the convergence result holds, the work provides a theoretically grounded extension of SQP to noisy settings that arise in simulation-based or experimental optimization. The explicit proportionality between solution accuracy and noise level, together with the numerical confirmation of this scaling, is a concrete strength that distinguishes the contribution from purely heuristic robustification approaches.

minor comments (2)
  1. The abstract asserts global convergence and noise-proportional accuracy but does not state the precise noise model or give a one-sentence sketch of the key relaxation mechanism; adding these would improve readability without lengthening the abstract appreciably.
  2. The numerical section would benefit from an explicit description of the test problems, the precise error metrics used to measure accuracy, and how the bounded noise was generated in the experiments, to allow independent verification of the observed scaling.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive assessment of our manuscript, accurate summary of the noise-tolerant SQP method with relaxations, and recommendation for minor revision. The referee correctly identifies the global convergence to a noise-dependent neighborhood and the observed proportionality in numerical results as key strengths.

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The paper's central result—that a line-search SQP method with explicit relaxations for bounded noise in function/derivative/constraint evaluations retains global convergence to a neighborhood whose radius scales with the noise bound—is derived from standard SQP convergence analysis once the noise is assumed bounded and known. No equations, fitted parameters, or predictions are shown to reduce by construction to the inputs; the accuracy proportionality is an output of the analysis rather than a definitional renaming or self-citation chain. The provided abstract and context contain no load-bearing self-citations, ansatzes smuggled via prior work, or uniqueness theorems imported from the authors themselves. The derivation chain is therefore self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on the assumption that all noise is bounded by a fixed constant and on standard SQP regularity conditions (constraint qualification, boundedness of iterates) that are not detailed in the abstract.

axioms (1)
  • domain assumption All function, constraint, and derivative evaluations contain additive noise bounded by a known constant.
    Explicitly stated in the abstract as the setting the algorithm is designed to handle.

pith-pipeline@v0.9.0 · 5384 in / 1205 out tokens · 25809 ms · 2026-05-10T12:14:26.050405+00:00 · methodology

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Reference graph

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