Alternative SDP and SOCP approximations for polynomial optimization

In theory, hierarchies of semidefinite programming (SDP) relaxations based on sum of squares (SOS) polynomials have been shown to provide arbitrarily close approximations for a general polynomial optimization problem (POP). However, due to the computational challenge of solving SDPs, it becomes diff...

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Autores principales:	Xiaolong Kuang, Bissan Ghaddar, Joe Naoum-Sawaya, LuisF. Zuluaga
Formato:	article
Lenguaje:	EN
Publicado:	Elsevier 2019
Materias:	90C22 90C26 Applied mathematics. Quantitative methods T57-57.97 Electronic computers. Computer science QA75.5-76.95
Acceso en línea:	https://doaj.org/article/3e521043222c4f23ad2ab9727edb1587
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spelling	oai:doaj.org-article:3e521043222c4f23ad2ab9727edb15872021-12-02T05:01:11ZAlternative SDP and SOCP approximations for polynomial optimization2192-440610.1007/s13675-018-0101-2https://doaj.org/article/3e521043222c4f23ad2ab9727edb15872019-06-01T00:00:00Zhttp://www.sciencedirect.com/science/article/pii/S2192440621001143https://doaj.org/toc/2192-4406In theory, hierarchies of semidefinite programming (SDP) relaxations based on sum of squares (SOS) polynomials have been shown to provide arbitrarily close approximations for a general polynomial optimization problem (POP). However, due to the computational challenge of solving SDPs, it becomes difficult to use SDP hierarchies for large-scale problems. To address this, hierarchies of second-order cone programming (SOCP) relaxations resulting from a restriction of the SOS polynomial condition have been recently proposed to approximate POPs. Here, we consider alternative ways to use the SOCP restrictions of the SOS condition. In particular, we show that SOCP hierarchies can be effectively used to strengthen hierarchies of linear programming relaxations for POPs. Specifically, we show that this solution approach is substantially more effective in finding solutions of certain POPs for which the more common hierarchies of SDP relaxations are known to perform poorly. Furthermore, when the feasible set of the POP is compact, these SOCP hierarchies converge to the POP’s optimal value.Xiaolong KuangBissan GhaddarJoe Naoum-SawayaLuisF. ZuluagaElsevierarticle90C2290C26Applied mathematics. Quantitative methodsT57-57.97Electronic computers. Computer scienceQA75.5-76.95ENEURO Journal on Computational Optimization, Vol 7, Iss 2, Pp 153-175 (2019)
institution	DOAJ
collection	DOAJ
language	EN
topic	90C22 90C26 Applied mathematics. Quantitative methods T57-57.97 Electronic computers. Computer science QA75.5-76.95
spellingShingle	90C22 90C26 Applied mathematics. Quantitative methods T57-57.97 Electronic computers. Computer science QA75.5-76.95 Xiaolong Kuang Bissan Ghaddar Joe Naoum-Sawaya LuisF. Zuluaga Alternative SDP and SOCP approximations for polynomial optimization
description	In theory, hierarchies of semidefinite programming (SDP) relaxations based on sum of squares (SOS) polynomials have been shown to provide arbitrarily close approximations for a general polynomial optimization problem (POP). However, due to the computational challenge of solving SDPs, it becomes difficult to use SDP hierarchies for large-scale problems. To address this, hierarchies of second-order cone programming (SOCP) relaxations resulting from a restriction of the SOS polynomial condition have been recently proposed to approximate POPs. Here, we consider alternative ways to use the SOCP restrictions of the SOS condition. In particular, we show that SOCP hierarchies can be effectively used to strengthen hierarchies of linear programming relaxations for POPs. Specifically, we show that this solution approach is substantially more effective in finding solutions of certain POPs for which the more common hierarchies of SDP relaxations are known to perform poorly. Furthermore, when the feasible set of the POP is compact, these SOCP hierarchies converge to the POP’s optimal value.
format	article
author	Xiaolong Kuang Bissan Ghaddar Joe Naoum-Sawaya LuisF. Zuluaga
author_facet	Xiaolong Kuang Bissan Ghaddar Joe Naoum-Sawaya LuisF. Zuluaga
author_sort	Xiaolong Kuang
title	Alternative SDP and SOCP approximations for polynomial optimization
title_short	Alternative SDP and SOCP approximations for polynomial optimization
title_full	Alternative SDP and SOCP approximations for polynomial optimization
title_fullStr	Alternative SDP and SOCP approximations for polynomial optimization
title_full_unstemmed	Alternative SDP and SOCP approximations for polynomial optimization
title_sort	alternative sdp and socp approximations for polynomial optimization
publisher	Elsevier
publishDate	2019
url	https://doaj.org/article/3e521043222c4f23ad2ab9727edb1587
work_keys_str_mv	AT xiaolongkuang alternativesdpandsocpapproximationsforpolynomialoptimization AT bissanghaddar alternativesdpandsocpapproximationsforpolynomialoptimization AT joenaoumsawaya alternativesdpandsocpapproximationsforpolynomialoptimization AT luisfzuluaga alternativesdpandsocpapproximationsforpolynomialoptimization
_version_	1718400823489200128

Alternative SDP and SOCP approximations for polynomial optimization

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