On the optimal order of worst case complexity of direct search
Autor(a) principal: | |
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Data de Publicação: | 2016 |
Outros Autores: | , |
Tipo de documento: | Artigo |
Idioma: | eng |
Título da fonte: | Repositório Científico de Acesso Aberto de Portugal (Repositórios Cientìficos) |
Texto Completo: | http://hdl.handle.net/10316/45237 https://doi.org/10.1007/s11590-015-0908-1 |
Resumo: | The worst case complexity of direct-search methods has been recently analyzed when they use positive spanning sets and impose a sufficient decrease condition to accept new iterates. For smooth unconstrained optimization, it is now known that such methods require at most \mathcal {O}(n^2\epsilon ^{-2}) function evaluations to compute a gradient of norm below \epsilon \in (0,1), where n is the dimension of the problem. Such a maximal effort is reduced to \mathcal {O}(n^2\epsilon ^{-1}) if the function is convex. The factor n^2 has been derived using the positive spanning set formed by the coordinate vectors and their negatives at all iterations. In this paper, we prove that such a factor of n^2 is optimal in these worst case complexity bounds, in the sense that no other positive spanning set will yield a better order of n. The proof is based on an observation that reveals the connection between cosine measure in positive spanning and sphere covering. |
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Repositório Científico de Acesso Aberto de Portugal (Repositórios Cientìficos) |
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7160 |
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On the optimal order of worst case complexity of direct searchThe worst case complexity of direct-search methods has been recently analyzed when they use positive spanning sets and impose a sufficient decrease condition to accept new iterates. For smooth unconstrained optimization, it is now known that such methods require at most \mathcal {O}(n^2\epsilon ^{-2}) function evaluations to compute a gradient of norm below \epsilon \in (0,1), where n is the dimension of the problem. Such a maximal effort is reduced to \mathcal {O}(n^2\epsilon ^{-1}) if the function is convex. The factor n^2 has been derived using the positive spanning set formed by the coordinate vectors and their negatives at all iterations. In this paper, we prove that such a factor of n^2 is optimal in these worst case complexity bounds, in the sense that no other positive spanning set will yield a better order of n. The proof is based on an observation that reveals the connection between cosine measure in positive spanning and sphere covering.Springer Berlin Heidelberg2016info:eu-repo/semantics/publishedVersioninfo:eu-repo/semantics/articlehttp://hdl.handle.net/10316/45237http://hdl.handle.net/10316/45237https://doi.org/10.1007/s11590-015-0908-1https://doi.org/10.1007/s11590-015-0908-1enghttps://link.springer.com/article/10.1007/s11590-015-0908-1Dodangeh, MahdiVicente, Luís NunesZhang, Zaikuninfo:eu-repo/semantics/openAccessreponame:Repositório Científico de Acesso Aberto de Portugal (Repositórios Cientìficos)instname:Agência para a Sociedade do Conhecimento (UMIC) - FCT - Sociedade da Informaçãoinstacron:RCAAP2021-06-29T10:02:55Zoai:estudogeral.uc.pt:10316/45237Portal AgregadorONGhttps://www.rcaap.pt/oai/openaireopendoar:71602024-03-19T20:53:21.636313Repositório Científico de Acesso Aberto de Portugal (Repositórios Cientìficos) - Agência para a Sociedade do Conhecimento (UMIC) - FCT - Sociedade da Informaçãofalse |
dc.title.none.fl_str_mv |
On the optimal order of worst case complexity of direct search |
title |
On the optimal order of worst case complexity of direct search |
spellingShingle |
On the optimal order of worst case complexity of direct search Dodangeh, Mahdi |
title_short |
On the optimal order of worst case complexity of direct search |
title_full |
On the optimal order of worst case complexity of direct search |
title_fullStr |
On the optimal order of worst case complexity of direct search |
title_full_unstemmed |
On the optimal order of worst case complexity of direct search |
title_sort |
On the optimal order of worst case complexity of direct search |
author |
Dodangeh, Mahdi |
author_facet |
Dodangeh, Mahdi Vicente, Luís Nunes Zhang, Zaikun |
author_role |
author |
author2 |
Vicente, Luís Nunes Zhang, Zaikun |
author2_role |
author author |
dc.contributor.author.fl_str_mv |
Dodangeh, Mahdi Vicente, Luís Nunes Zhang, Zaikun |
description |
The worst case complexity of direct-search methods has been recently analyzed when they use positive spanning sets and impose a sufficient decrease condition to accept new iterates. For smooth unconstrained optimization, it is now known that such methods require at most \mathcal {O}(n^2\epsilon ^{-2}) function evaluations to compute a gradient of norm below \epsilon \in (0,1), where n is the dimension of the problem. Such a maximal effort is reduced to \mathcal {O}(n^2\epsilon ^{-1}) if the function is convex. The factor n^2 has been derived using the positive spanning set formed by the coordinate vectors and their negatives at all iterations. In this paper, we prove that such a factor of n^2 is optimal in these worst case complexity bounds, in the sense that no other positive spanning set will yield a better order of n. The proof is based on an observation that reveals the connection between cosine measure in positive spanning and sphere covering. |
publishDate |
2016 |
dc.date.none.fl_str_mv |
2016 |
dc.type.status.fl_str_mv |
info:eu-repo/semantics/publishedVersion |
dc.type.driver.fl_str_mv |
info:eu-repo/semantics/article |
format |
article |
status_str |
publishedVersion |
dc.identifier.uri.fl_str_mv |
http://hdl.handle.net/10316/45237 http://hdl.handle.net/10316/45237 https://doi.org/10.1007/s11590-015-0908-1 https://doi.org/10.1007/s11590-015-0908-1 |
url |
http://hdl.handle.net/10316/45237 https://doi.org/10.1007/s11590-015-0908-1 |
dc.language.iso.fl_str_mv |
eng |
language |
eng |
dc.relation.none.fl_str_mv |
https://link.springer.com/article/10.1007/s11590-015-0908-1 |
dc.rights.driver.fl_str_mv |
info:eu-repo/semantics/openAccess |
eu_rights_str_mv |
openAccess |
dc.publisher.none.fl_str_mv |
Springer Berlin Heidelberg |
publisher.none.fl_str_mv |
Springer Berlin Heidelberg |
dc.source.none.fl_str_mv |
reponame:Repositório Científico de Acesso Aberto de Portugal (Repositórios Cientìficos) instname:Agência para a Sociedade do Conhecimento (UMIC) - FCT - Sociedade da Informação instacron:RCAAP |
instname_str |
Agência para a Sociedade do Conhecimento (UMIC) - FCT - Sociedade da Informação |
instacron_str |
RCAAP |
institution |
RCAAP |
reponame_str |
Repositório Científico de Acesso Aberto de Portugal (Repositórios Cientìficos) |
collection |
Repositório Científico de Acesso Aberto de Portugal (Repositórios Cientìficos) |
repository.name.fl_str_mv |
Repositório Científico de Acesso Aberto de Portugal (Repositórios Cientìficos) - Agência para a Sociedade do Conhecimento (UMIC) - FCT - Sociedade da Informação |
repository.mail.fl_str_mv |
|
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1799133820682764288 |