On the construction of quadratic models for derivative-free trust-region algorithms

We consider derivative-free trust-region algorithms based on sampling approaches for convex constrained problems and discuss two conditions on the quadratic models for ensuring their global convergence. The first condition requires the poisedness of the sample sets, as usual in this context, while t...

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Autores principales:	Adriano Verdério, ElizabethW. Karas, LucasG. Pedroso, Katya Scheinberg
Formato:	article
Lenguaje:	EN
Publicado:	Elsevier 2017
Materias:	90C56 65K05 49M37 Applied mathematics. Quantitative methods T57-57.97 Electronic computers. Computer science QA75.5-76.95
Acceso en línea:	https://doaj.org/article/cac5197b999a4be48e4bcfd4100b7410
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spelling	oai:doaj.org-article:cac5197b999a4be48e4bcfd4100b74102021-12-02T05:01:04ZOn the construction of quadratic models for derivative-free trust-region algorithms2192-440610.1007/s13675-017-0081-7https://doaj.org/article/cac5197b999a4be48e4bcfd4100b74102017-12-01T00:00:00Zhttp://www.sciencedirect.com/science/article/pii/S2192440621000915https://doaj.org/toc/2192-4406We consider derivative-free trust-region algorithms based on sampling approaches for convex constrained problems and discuss two conditions on the quadratic models for ensuring their global convergence. The first condition requires the poisedness of the sample sets, as usual in this context, while the other one is related to the error between the model and the objective function at the sample points. Although the second condition trivially holds if the model is constructed by polynomial interpolation, since in this case the model coincides with the objective function at the sample set, we show that it also holds for models constructed by support vector regression. These two conditions imply that the error between the gradient of the trust-region model and the objective function is of the order of δk, where δk controls the diameter of the sample set. This allows proving the global convergence of a trust-region algorithm that uses two radii, δk and the trust-region radius. Preliminary numerical experiments are presented for minimizing functions with and without noise.Adriano VerdérioElizabethW. KarasLucasG. PedrosoKatya ScheinbergElsevierarticle90C5665K0549M37Applied mathematics. Quantitative methodsT57-57.97Electronic computers. Computer scienceQA75.5-76.95ENEURO Journal on Computational Optimization, Vol 5, Iss 4, Pp 501-527 (2017)
institution	DOAJ
collection	DOAJ
language	EN
topic	90C56 65K05 49M37 Applied mathematics. Quantitative methods T57-57.97 Electronic computers. Computer science QA75.5-76.95
spellingShingle	90C56 65K05 49M37 Applied mathematics. Quantitative methods T57-57.97 Electronic computers. Computer science QA75.5-76.95 Adriano Verdério ElizabethW. Karas LucasG. Pedroso Katya Scheinberg On the construction of quadratic models for derivative-free trust-region algorithms
description	We consider derivative-free trust-region algorithms based on sampling approaches for convex constrained problems and discuss two conditions on the quadratic models for ensuring their global convergence. The first condition requires the poisedness of the sample sets, as usual in this context, while the other one is related to the error between the model and the objective function at the sample points. Although the second condition trivially holds if the model is constructed by polynomial interpolation, since in this case the model coincides with the objective function at the sample set, we show that it also holds for models constructed by support vector regression. These two conditions imply that the error between the gradient of the trust-region model and the objective function is of the order of δk, where δk controls the diameter of the sample set. This allows proving the global convergence of a trust-region algorithm that uses two radii, δk and the trust-region radius. Preliminary numerical experiments are presented for minimizing functions with and without noise.
format	article
author	Adriano Verdério ElizabethW. Karas LucasG. Pedroso Katya Scheinberg
author_facet	Adriano Verdério ElizabethW. Karas LucasG. Pedroso Katya Scheinberg
author_sort	Adriano Verdério
title	On the construction of quadratic models for derivative-free trust-region algorithms
title_short	On the construction of quadratic models for derivative-free trust-region algorithms
title_full	On the construction of quadratic models for derivative-free trust-region algorithms
title_fullStr	On the construction of quadratic models for derivative-free trust-region algorithms
title_full_unstemmed	On the construction of quadratic models for derivative-free trust-region algorithms
title_sort	on the construction of quadratic models for derivative-free trust-region algorithms
publisher	Elsevier
publishDate	2017
url	https://doaj.org/article/cac5197b999a4be48e4bcfd4100b7410
work_keys_str_mv	AT adrianoverderio ontheconstructionofquadraticmodelsforderivativefreetrustregionalgorithms AT elizabethwkaras ontheconstructionofquadraticmodelsforderivativefreetrustregionalgorithms AT lucasgpedroso ontheconstructionofquadraticmodelsforderivativefreetrustregionalgorithms AT katyascheinberg ontheconstructionofquadraticmodelsforderivativefreetrustregionalgorithms
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On the construction of quadratic models for derivative-free trust-region algorithms

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