Constrained Optimization in the Presence of Noise

Oztoprak, Figen; Byrd, Richard H.; Nocedal, Jorge

doi:10.48550/arxiv.2110.04355

Cited by 6 publications

(9 citation statements)

References 4 publications

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“…The reason for relaxing both the numerator and denominator in (7) is to be consistent with the classical narrative of trust region methods where a ratio close to 1 is an indication that the model is adequate. An alternative approach would be to relax only the numerator and interpret the condition ρ k > c (where c > 0 is a constant) as a relaxed Armijo condition of the type studied in [2,21]. We find the first interpretation to be easier to motivate and to yield tighter bounds in the convergence analysis.…”

Section: Problem Statement and Algorithmmentioning

confidence: 96%

“…Those three papers give conditions under which convergence can be expected, giving careful attention to the behavior of the penalty parameter. Using a relaxed Armijo line search procedure, [21] shows global convergence to a neighborhood of the solution for an SQP method for equality constrained problems.…”

Section: Literature Reviewmentioning

confidence: 99%

“…This raises the question of how to best modify the method to avoid failures. The algorithm proposed here is inspired by work on line search methods for unconstrained optimization [2,27] and equality constrained optimization [21]. In those papers, convergence-to-neighborhood results were derived but the analysis presented here follows different lines, as trust region methods require different proof techniques.…”

Section: Introductionmentioning

confidence: 99%

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A Trust Region Method for the Optimization of Noisy Functions

Sun¹,

Nocedal²

2022

Preprint

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Classical trust region methods were designed to solve problems in which function and gradient information are exact. This paper considers the case when there are bounded errors (or noise) in the above computations and proposes a simple modification of the trust region method to cope with these errors. The new algorithm only requires information about the size of the errors in the function evaluations and incurs no additional computational expense. It is shown that, when applied to a smooth (but not necessarily convex) objective function, the iterates of the algorithm visit a neighborhood of stationarity infinitely often, and that the rest of the sequence cannot stray too far away, as measured by function values. Numerical results illustrate how the classical trust region algorithm may fail in the presence of noise, and how the proposed algorithm ensures steady progress towards stationarity in these cases.

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Section: Problem Statement and Algorithmmentioning

confidence: 96%

Section: Literature Reviewmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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A Trust Region Method for the Optimization of Noisy Functions

Sun¹,

Nocedal²

2022

Preprint

Self Cite

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“…All these works showed global convergence of different StoSQP schemes. In addition to these works, Oztoprak et al (2021); Sun and Nocedal (2022) considered optimization with noisy functions. Those analyses require known, deterministic, and bounded noise, and thus they are not suited for the considered problems in (1).…”

Section: Literature Reviewmentioning

confidence: 99%

Asymptotic Convergence Rate and Statistical Inference for Stochastic Sequential Quadratic Programming

Na¹,

Mahoney²

2022

Preprint

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We apply a stochastic sequential quadratic programming (StoSQP) algorithm to solve constrained nonlinear optimization problems, where the objective is stochastic and the constraints are in equality and deterministic. We study a fully stochastic setup, where only a single sample is available in each iteration for estimating the gradient and Hessian of the objective. We allow StoSQP to select a random stepsize ᾱt adaptively, such that β t ≤ ᾱt ≤ β t + χ t , where β t , χ t = o(β t ) are prespecified deterministic sequences. We also allow StoSQP to solve Newton system inexactly via randomized iterative solvers, e.g., with the sketch-and-project method; and we do not require the approximation error of inexact Newton direction to vanish (thus, the per-iteration computational cost does not blow up). For this general StoSQP framework, we establish the asymptotic convergence rate for its last iterate, with the worst-case iteration complexity as a byproduct; and we perform statistical inference. In particular, under mild assumptions and with proper decaying sequences β t , χ t , we show that: (i) the StoSQP scheme can take at most O(1/ 4 ) iterations to achieve -stationarity; (ii) asymptotically and almost surely, (x t −x , λ t −λ ) = O( β t log(1/β t ))+O(χ t /β t ), where (x t , λ t ) is the primaldual StoSQP iterate; (iii) the sequence 1/ √ β t • (x t − x , λ t − λ ) converges to a mean zero Gaussian distribution with a nontrivial covariance matrix. Furthermore, we establish the Berry-Esseen bound for (x t , λ t ) to measure quantitatively the convergence of its distribution function. We also provide a practical estimator for the covariance matrix, from which the confidence intervals (or regions) of (x , λ ) can be constructed using the iterates {(x t , λ t )} t . All our theorems are validated using nonlinear problems in CUTEst test set.

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“…We note that several algorithm choices in the two papers [7,28], e.g., merit functions and merit parameters, are different. Several other extensions have been proposed [3,6,8,17,27,32], and very few of these works (or others in the literature) derive worst-case iteration complexity (or sample complexity) due to the difficulties that arise because of the constrained setting and the stochasticity. Notable exceptions are, [16] where the authors provide convergence rates (and complexity guarantees) for the algorithm proposed in [7], and [3,29] that provide complexity bounds for variants of the stochastic SQP methods under additional assumptions and in the setting in which the errors can be diminished.…”

Section: Introductionmentioning

confidence: 99%

Sequential Quadratic Optimization for Nonlinear Equality Constrained Stochastic Optimization

Berahas¹,

Curtis²,

Robinson³

et al. 2021

SIAM J. Optim.

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A step-search sequential quadratic programming method is proposed for solving nonlinear equality constrained stochastic optimization problems. It is assumed that constraint function values and derivatives are available, but only stochastic approximations of the objective function and its associated derivatives can be computed via inexact probabilistic zeroth-and first-order oracles. Under reasonable assumptions, a high-probability bound on the iteration complexity of the algorithm to approximate first-order stationarity is derived. Numerical results on standard nonlinear optimization test problems illustrate the advantages and limitations of our proposed method.

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Constrained Optimization in the Presence of Noise

Cited by 6 publications

References 4 publications

A Trust Region Method for the Optimization of Noisy Functions

A Trust Region Method for the Optimization of Noisy Functions

Asymptotic Convergence Rate and Statistical Inference for Stochastic Sequential Quadratic Programming

Sequential Quadratic Optimization for Nonlinear Equality Constrained Stochastic Optimization

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