A New Hybrid Preconditioner for the Interior Point Method
Autor(a) principal: | |
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Data de Publicação: | 2019 |
Outros Autores: | , |
Tipo de documento: | Artigo |
Idioma: | eng |
Título da fonte: | TEMA (Sociedade Brasileira de Matemática Aplicada e Computacional. Online) |
Texto Completo: | http://old.scielo.br/scielo.php?script=sci_arttext&pid=S2179-84512019000200359 |
Resumo: | ABSTRACT This study aims to improve the computation of the search direction in the primal-dual Interior Point Method through preconditioned iterative methods. It is about a hybrid approach that combines the Controlled Cholesky Factorization preconditioner and the Splitting preconditioner. This approach has shown good results, however, in these preconditioners there are factors that reduce their efficiency, such as faults on the diagonal when performing the Cholesky factorization, as well as a demand for excessive memory, among others. Thus, some modifications are proposed in these preconditioners, as well as a new phase change, in order to improve the performance of the hybrid preconditioner. In the Controlled Cholesky Factorization, the parameters that control the fill-in and the correction of the faults which occur on the diagonal are modified. It considers the relationship between the components from Controlled Cholesky Factorization obtained before and after the fault on the diagonal. In the Splitting preconditioner, in turn, a sparse base is constructed through an appropriate ordering of the columns from constrained matrix optimization problem. In addition, a theoretical result is presented, which shows that, with the proposed ordering, the condition number of the preconditioned Normal Equation matrix with the Splitting preconditioner is uniformly limited by an amount that depends only on the original data of the problem and not on the iteration of the Interior Point Method. Numerical experiments with large scale problems, corroborate the robustness and computational efficiency from this approach. |
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A New Hybrid Preconditioner for the Interior Point MethodInterior Point MethodControlled Cholesky FactorizationSplitting preconditionerABSTRACT This study aims to improve the computation of the search direction in the primal-dual Interior Point Method through preconditioned iterative methods. It is about a hybrid approach that combines the Controlled Cholesky Factorization preconditioner and the Splitting preconditioner. This approach has shown good results, however, in these preconditioners there are factors that reduce their efficiency, such as faults on the diagonal when performing the Cholesky factorization, as well as a demand for excessive memory, among others. Thus, some modifications are proposed in these preconditioners, as well as a new phase change, in order to improve the performance of the hybrid preconditioner. In the Controlled Cholesky Factorization, the parameters that control the fill-in and the correction of the faults which occur on the diagonal are modified. It considers the relationship between the components from Controlled Cholesky Factorization obtained before and after the fault on the diagonal. In the Splitting preconditioner, in turn, a sparse base is constructed through an appropriate ordering of the columns from constrained matrix optimization problem. In addition, a theoretical result is presented, which shows that, with the proposed ordering, the condition number of the preconditioned Normal Equation matrix with the Splitting preconditioner is uniformly limited by an amount that depends only on the original data of the problem and not on the iteration of the Interior Point Method. Numerical experiments with large scale problems, corroborate the robustness and computational efficiency from this approach.Sociedade Brasileira de Matemática Aplicada e Computacional2019-08-01info:eu-repo/semantics/articleinfo:eu-repo/semantics/publishedVersiontext/htmlhttp://old.scielo.br/scielo.php?script=sci_arttext&pid=S2179-84512019000200359TEMA (São Carlos) v.20 n.2 2019reponame:TEMA (Sociedade Brasileira de Matemática Aplicada e Computacional. Online)instname:Sociedade Brasileira de Matemática Aplicada e Computacionalinstacron:SBMAC10.5540/tema.2019.020.02.0359info:eu-repo/semantics/openAccessHEREDIA,M.R.CASTRO,C.O.OLIVEIRA,A.R.L.eng2019-09-12T00:00:00Zoai:scielo:S2179-84512019000200359Revistahttp://www.scielo.br/temaPUBhttps://old.scielo.br/oai/scielo-oai.phpcastelo@icmc.usp.br2179-84511677-1966opendoar:2019-09-12T00:00TEMA (Sociedade Brasileira de Matemática Aplicada e Computacional. Online) - Sociedade Brasileira de Matemática Aplicada e Computacionalfalse |
dc.title.none.fl_str_mv |
A New Hybrid Preconditioner for the Interior Point Method |
title |
A New Hybrid Preconditioner for the Interior Point Method |
spellingShingle |
A New Hybrid Preconditioner for the Interior Point Method HEREDIA,M.R. Interior Point Method Controlled Cholesky Factorization Splitting preconditioner |
title_short |
A New Hybrid Preconditioner for the Interior Point Method |
title_full |
A New Hybrid Preconditioner for the Interior Point Method |
title_fullStr |
A New Hybrid Preconditioner for the Interior Point Method |
title_full_unstemmed |
A New Hybrid Preconditioner for the Interior Point Method |
title_sort |
A New Hybrid Preconditioner for the Interior Point Method |
author |
HEREDIA,M.R. |
author_facet |
HEREDIA,M.R. CASTRO,C.O. OLIVEIRA,A.R.L. |
author_role |
author |
author2 |
CASTRO,C.O. OLIVEIRA,A.R.L. |
author2_role |
author author |
dc.contributor.author.fl_str_mv |
HEREDIA,M.R. CASTRO,C.O. OLIVEIRA,A.R.L. |
dc.subject.por.fl_str_mv |
Interior Point Method Controlled Cholesky Factorization Splitting preconditioner |
topic |
Interior Point Method Controlled Cholesky Factorization Splitting preconditioner |
description |
ABSTRACT This study aims to improve the computation of the search direction in the primal-dual Interior Point Method through preconditioned iterative methods. It is about a hybrid approach that combines the Controlled Cholesky Factorization preconditioner and the Splitting preconditioner. This approach has shown good results, however, in these preconditioners there are factors that reduce their efficiency, such as faults on the diagonal when performing the Cholesky factorization, as well as a demand for excessive memory, among others. Thus, some modifications are proposed in these preconditioners, as well as a new phase change, in order to improve the performance of the hybrid preconditioner. In the Controlled Cholesky Factorization, the parameters that control the fill-in and the correction of the faults which occur on the diagonal are modified. It considers the relationship between the components from Controlled Cholesky Factorization obtained before and after the fault on the diagonal. In the Splitting preconditioner, in turn, a sparse base is constructed through an appropriate ordering of the columns from constrained matrix optimization problem. In addition, a theoretical result is presented, which shows that, with the proposed ordering, the condition number of the preconditioned Normal Equation matrix with the Splitting preconditioner is uniformly limited by an amount that depends only on the original data of the problem and not on the iteration of the Interior Point Method. Numerical experiments with large scale problems, corroborate the robustness and computational efficiency from this approach. |
publishDate |
2019 |
dc.date.none.fl_str_mv |
2019-08-01 |
dc.type.driver.fl_str_mv |
info:eu-repo/semantics/article |
dc.type.status.fl_str_mv |
info:eu-repo/semantics/publishedVersion |
format |
article |
status_str |
publishedVersion |
dc.identifier.uri.fl_str_mv |
http://old.scielo.br/scielo.php?script=sci_arttext&pid=S2179-84512019000200359 |
url |
http://old.scielo.br/scielo.php?script=sci_arttext&pid=S2179-84512019000200359 |
dc.language.iso.fl_str_mv |
eng |
language |
eng |
dc.relation.none.fl_str_mv |
10.5540/tema.2019.020.02.0359 |
dc.rights.driver.fl_str_mv |
info:eu-repo/semantics/openAccess |
eu_rights_str_mv |
openAccess |
dc.format.none.fl_str_mv |
text/html |
dc.publisher.none.fl_str_mv |
Sociedade Brasileira de Matemática Aplicada e Computacional |
publisher.none.fl_str_mv |
Sociedade Brasileira de Matemática Aplicada e Computacional |
dc.source.none.fl_str_mv |
TEMA (São Carlos) v.20 n.2 2019 reponame:TEMA (Sociedade Brasileira de Matemática Aplicada e Computacional. Online) instname:Sociedade Brasileira de Matemática Aplicada e Computacional instacron:SBMAC |
instname_str |
Sociedade Brasileira de Matemática Aplicada e Computacional |
instacron_str |
SBMAC |
institution |
SBMAC |
reponame_str |
TEMA (Sociedade Brasileira de Matemática Aplicada e Computacional. Online) |
collection |
TEMA (Sociedade Brasileira de Matemática Aplicada e Computacional. Online) |
repository.name.fl_str_mv |
TEMA (Sociedade Brasileira de Matemática Aplicada e Computacional. Online) - Sociedade Brasileira de Matemática Aplicada e Computacional |
repository.mail.fl_str_mv |
castelo@icmc.usp.br |
_version_ |
1752122220587515904 |