Nonsmooth spectral gradient methods for unconstrained optimization

To solve nonsmooth unconstrained minimization problems, we combine the spectral choice of step length with two well-established subdifferential-type schemes: the gradient sampling method and the simplex gradient method. We focus on the interesting case in which the objective function is continuously...

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Autores principales:	Milagros Loreto, Hugo Aponte, Debora Cores, Marcos Raydan
Formato:	article
Lenguaje:	EN
Publicado:	Elsevier 2017
Materias:	49M05 90C56 90C26 65K10 Applied mathematics. Quantitative methods T57-57.97 Electronic computers. Computer science QA75.5-76.95
Acceso en línea:	https://doaj.org/article/09769c1ca2fc49daa0950264b364bf55
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spelling	oai:doaj.org-article:09769c1ca2fc49daa0950264b364bf552021-12-02T05:01:05ZNonsmooth spectral gradient methods for unconstrained optimization2192-440610.1007/s13675-017-0080-8https://doaj.org/article/09769c1ca2fc49daa0950264b364bf552017-12-01T00:00:00Zhttp://www.sciencedirect.com/science/article/pii/S2192440621000927https://doaj.org/toc/2192-4406To solve nonsmooth unconstrained minimization problems, we combine the spectral choice of step length with two well-established subdifferential-type schemes: the gradient sampling method and the simplex gradient method. We focus on the interesting case in which the objective function is continuously differentiable almost everywhere, and it is often not differentiable at minimizers. In the case of the gradient sampling method, we also present a simple differentiability test that allows us to use the exact gradient direction as frequently as possible, and to build a stochastic subdifferential direction only if the test fails. The proposed spectral gradient sampling method is combined with a monotone line search globalization strategy. On the other hand, the simplex gradient method is a direct search method that only requires function evaluations to build an approximation to the gradient direction. In this case, the proposed spectral simplex gradient method is combined with a suitable nonmonotone line search strategy. For both scenarios, we discuss convergence properties and present numerical results on a set of nonsmooth test functions. These numerical results indicate that using a spectral step length can improve the practical performance of both methods.Milagros LoretoHugo AponteDebora CoresMarcos RaydanElsevierarticle49M0590C5690C2665K10Applied mathematics. Quantitative methodsT57-57.97Electronic computers. Computer scienceQA75.5-76.95ENEURO Journal on Computational Optimization, Vol 5, Iss 4, Pp 529-553 (2017)
institution	DOAJ
collection	DOAJ
language	EN
topic	49M05 90C56 90C26 65K10 Applied mathematics. Quantitative methods T57-57.97 Electronic computers. Computer science QA75.5-76.95
spellingShingle	49M05 90C56 90C26 65K10 Applied mathematics. Quantitative methods T57-57.97 Electronic computers. Computer science QA75.5-76.95 Milagros Loreto Hugo Aponte Debora Cores Marcos Raydan Nonsmooth spectral gradient methods for unconstrained optimization
description	To solve nonsmooth unconstrained minimization problems, we combine the spectral choice of step length with two well-established subdifferential-type schemes: the gradient sampling method and the simplex gradient method. We focus on the interesting case in which the objective function is continuously differentiable almost everywhere, and it is often not differentiable at minimizers. In the case of the gradient sampling method, we also present a simple differentiability test that allows us to use the exact gradient direction as frequently as possible, and to build a stochastic subdifferential direction only if the test fails. The proposed spectral gradient sampling method is combined with a monotone line search globalization strategy. On the other hand, the simplex gradient method is a direct search method that only requires function evaluations to build an approximation to the gradient direction. In this case, the proposed spectral simplex gradient method is combined with a suitable nonmonotone line search strategy. For both scenarios, we discuss convergence properties and present numerical results on a set of nonsmooth test functions. These numerical results indicate that using a spectral step length can improve the practical performance of both methods.
format	article
author	Milagros Loreto Hugo Aponte Debora Cores Marcos Raydan
author_facet	Milagros Loreto Hugo Aponte Debora Cores Marcos Raydan
author_sort	Milagros Loreto
title	Nonsmooth spectral gradient methods for unconstrained optimization
title_short	Nonsmooth spectral gradient methods for unconstrained optimization
title_full	Nonsmooth spectral gradient methods for unconstrained optimization
title_fullStr	Nonsmooth spectral gradient methods for unconstrained optimization
title_full_unstemmed	Nonsmooth spectral gradient methods for unconstrained optimization
title_sort	nonsmooth spectral gradient methods for unconstrained optimization
publisher	Elsevier
publishDate	2017
url	https://doaj.org/article/09769c1ca2fc49daa0950264b364bf55
work_keys_str_mv	AT milagrosloreto nonsmoothspectralgradientmethodsforunconstrainedoptimization AT hugoaponte nonsmoothspectralgradientmethodsforunconstrainedoptimization AT deboracores nonsmoothspectralgradientmethodsforunconstrainedoptimization AT marcosraydan nonsmoothspectralgradientmethodsforunconstrainedoptimization
_version_	1718400811357175808

Nonsmooth spectral gradient methods for unconstrained optimization

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