Model-Based Methods in Derivative-Free Nonsmooth Optimization

Audet, Charles; Hare, Warren

doi:10.1007/978-3-030-34910-3_19

Cited by 10 publications

(11 citation statements)

References 74 publications

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“…Moreover, in the practical implementation of our algorithms, we have included a weight 0 ≤ ω ≤ 1 in the quadratic term of the models, that is we replaced 1 2 s B k s with ω 2 s B k s in Problems (2) and (27). In such a way, the users of our solver can tune it to the degree of nonsmoothness of their problems.…”

Section: Implementation and Numerical Resultsmentioning

confidence: 99%

“…In a way similar to the application of direct-search methods to nonsmooth functions, our convergence results state that the Clarke generalized derivative is nonnegative at any limit point of a subsequence of unsuccessful iterates, along any direction in the unit sphere, assuming some form of asymptotic density of the vectors randomly generated for the linear terms of the models. The Hessian of the quadratic term added to the max-linear model term does not have to be 1 An anonymous Referee has drawn our attention to the recent works [2,22] (the latter one of trust-region type). However, both require the calculation of subgradients of approximate or nearby subdifferentials, therefore only applicable when the nonsmoothness of the objective function is known through some algebraic or composite form.…”

Section: Introductionmentioning

confidence: 99%

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Trust-Region Methods for the Derivative-Free Optimization of Nonsmooth Black-Box Functions

Liuzzi¹,

Lucidi²,

Rinaldi³

et al. 2019

SIAM J. Optim.

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In this paper we study the minimization of a nonsmooth black-box type function, without assuming any access to derivatives or generalized derivatives and without any knowledge about the analytical origin of the function nonsmoothness. Directional methods have been derived for such problems but to our knowledge no model-based method like a trust-region one has yet been proposed. Our main contribution is thus the derivation of derivative-free trust-region methods for black-box type functions. We propose a trust-region model that is the sum of a max-linear term with a quadratic one so that the function nonsmoothness can be properly captured, but at the same time the curvature of the function in smooth subdomains is not neglected. Our trust-region methods enjoy global convergence properties similar to the ones of the directional methods, provided the vectors randomly generated for the max-linear term are asymptotically dense in the unit sphere. The numerical results reported demonstrate that our approach is both efficient and robust for a large class of nonsmooth unconstrained optimization problems. Our software is made available under request.

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Section: Implementation and Numerical Resultsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Trust-Region Methods for the Derivative-Free Optimization of Nonsmooth Black-Box Functions

Liuzzi¹,

Lucidi²,

Rinaldi³

et al. 2019

SIAM J. Optim.

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“…3.1 Sample set Z k , gradient set D k , and generator set G k Manifold sampling is an iterative method that builds component models m Fi of each F i around the current point x k . We place the first-order terms of each model in column i of the matrix ∇M (x k ) ∈ R n×p as in (2).…”

Section: Manifold Sampling For Piecewise-smooth Compositionsmentioning

confidence: 99%

Manifold Sampling for Optimizing Nonsmooth Nonconvex Compositions

Larson,

Menickelly,

Zhou

2020

Preprint

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We propose a manifold sampling algorithm for minimizing a nonsmooth composition f = h • F , where we assume h is nonsmooth and may be inexpensively computed in closed form and F is smooth but its Jacobian may not be available. We additionally assume that the composition h • F defines a continuous selection. Manifold sampling algorithms can be classified as model-based derivative-free methods, in that models of F are combined with particularly sampled information about h to yield local models for use within a trust-region framework. We demonstrate that cluster points of the sequence of iterates generated by the manifold sampling algorithm are Clarke stationary. We consider the tractability of three particular subproblems generated by the manifold sampling algorithm and the extent to which inexact solutions to these subproblems may be tolerated. Numerical results demonstrate that manifold sampling as a derivative-free algorithm is competitive with state-of-the-art algorithms for nonsmooth optimization that utilize first-order information about f .

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“…Derivative-Free Optimization (DFO) is the mathematical study of algorithms for continuous optimization that do not use first-order information [1]. In general, DFO methods can be categorized into model-based methods and direct search methods.…”

Section: Introductionmentioning

confidence: 99%

“…1 , then it is Lipschitz continuous on any convex compact set K with constant max Lemma 29 Suppose the mapping g = (g 1 , ..., g m…”

mentioning

confidence: 99%

Error Analysis of Surrogate Models Constructed through Operations on Sub-models

Chen¹,

Jarry-Bolduc²,

Hare³

2021

Preprint

Self Cite

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Model-based methods are popular in derivative-free optimization (DFO). In most of them, a single model function is built to approximate the objective function. This is generally based on the assumption that the objective function is one blackbox. However, some real-life and theoretical problems show that the objective function may consist of several blackboxes. In those problems, the information provided by each blackbox may not be equal. In this situation, one could build multiple sub-models that are then combined to become a final model. In this paper, we analyze the relation between the accuracy of those sub-models and the model constructed through their operations. We develop a broad framework that can be used as a theoretical tool in model error analysis and future research in DFO algorithms design.

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Model-Based Methods in Derivative-Free Nonsmooth Optimization

Cited by 10 publications

References 74 publications

Trust-Region Methods for the Derivative-Free Optimization of Nonsmooth Black-Box Functions

Trust-Region Methods for the Derivative-Free Optimization of Nonsmooth Black-Box Functions

Manifold Sampling for Optimizing Nonsmooth Nonconvex Compositions

Error Analysis of Surrogate Models Constructed through Operations on Sub-models

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