Exact SDP relaxations of quadratically constrained quadratic programs with forest structures

We study the exactness of the semidefinite programming (SDP) relaxation of quadratically constrained quadratic programs (QCQPs). With the aggregate sparsity matrix from the data matrices of a QCQP with n variables, the rank and positive semidefiniteness of the matrix are examined. We prove that if t...

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Published in:	Journal of global optimization Vol. 82; no. 2; pp. 243 - 262
Main Authors:	Azuma, Godai, Fukuda, Mituhiro, Kim, Sunyoung, Yamashita, Makoto
Format:	Journal Article
Language:	English
Published:	New York Springer US 01.02.2022 Springer Springer Nature B.V
Subjects:	Computer Science Constraints Feasibility Mathematical analysis Mathematics Mathematics and Statistics Matrices (mathematics) Operations Research/Decision Theory Optimization Real Functions Semidefinite programming Sparsity Exact semidefinite relaxations Quadratically constrained quadratic programs The rank of aggregated sparsity matrix Forest graph
ISSN:	0925-5001, 1573-2916
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Abstract	We study the exactness of the semidefinite programming (SDP) relaxation of quadratically constrained quadratic programs (QCQPs). With the aggregate sparsity matrix from the data matrices of a QCQP with n variables, the rank and positive semidefiniteness of the matrix are examined. We prove that if the rank of the aggregate sparsity matrix is not less than n - 1 and the matrix remains positive semidefinite after replacing some off-diagonal nonzero elements with zeros, then the standard SDP relaxation provides an exact optimal solution for the QCQP under feasibility assumptions. In particular, we demonstrate that QCQPs with forest-structured aggregate sparsity matrix, such as the tridiagonal or arrow-type matrix, satisfy the exactness condition on the rank. The exactness is attained by considering the feasibility of the dual SDP relaxation, the strong duality of SDPs, and a sequence of QCQPs with perturbed objective functions, under the assumption that the feasible region is compact. We generalize our result for a wider class of QCQPs by applying simultaneous tridiagonalization on the data matrices. Moreover, simultaneous tridiagonalization is applied to a matrix pencil so that QCQPs with two constraints can be solved exactly by the SDP relaxation.
AbstractList	We study the exactness of the semidefinite programming (SDP) relaxation of quadratically constrained quadratic programs (QCQPs). With the aggregate sparsity matrix from the data matrices of a QCQP with n variables, the rank and positive semidefiniteness of the matrix are examined. We prove that if the rank of the aggregate sparsity matrix is not less than n-1 and the matrix remains positive semidefinite after replacing some off-diagonal nonzero elements with zeros, then the standard SDP relaxation provides an exact optimal solution for the QCQP under feasibility assumptions. In particular, we demonstrate that QCQPs with forest-structured aggregate sparsity matrix, such as the tridiagonal or arrow-type matrix, satisfy the exactness condition on the rank. The exactness is attained by considering the feasibility of the dual SDP relaxation, the strong duality of SDPs, and a sequence of QCQPs with perturbed objective functions, under the assumption that the feasible region is compact. We generalize our result for a wider class of QCQPs by applying simultaneous tridiagonalization on the data matrices. Moreover, simultaneous tridiagonalization is applied to a matrix pencil so that QCQPs with two constraints can be solved exactly by the SDP relaxation. We study the exactness of the semidefinite programming (SDP) relaxation of quadratically constrained quadratic programs (QCQPs). With the aggregate sparsity matrix from the data matrices of a QCQP with n variables, the rank and positive semidefiniteness of the matrix are examined. We prove that if the rank of the aggregate sparsity matrix is not less than n - 1 and the matrix remains positive semidefinite after replacing some off-diagonal nonzero elements with zeros, then the standard SDP relaxation provides an exact optimal solution for the QCQP under feasibility assumptions. In particular, we demonstrate that QCQPs with forest-structured aggregate sparsity matrix, such as the tridiagonal or arrow-type matrix, satisfy the exactness condition on the rank. The exactness is attained by considering the feasibility of the dual SDP relaxation, the strong duality of SDPs, and a sequence of QCQPs with perturbed objective functions, under the assumption that the feasible region is compact. We generalize our result for a wider class of QCQPs by applying simultaneous tridiagonalization on the data matrices. Moreover, simultaneous tridiagonalization is applied to a matrix pencil so that QCQPs with two constraints can be solved exactly by the SDP relaxation. We study the exactness of the semidefinite programming (SDP) relaxation of quadratically constrained quadratic programs (QCQPs). With the aggregate sparsity matrix from the data matrices of a QCQP with n variables, the rank and positive semidefiniteness of the matrix are examined. We prove that if the rank of the aggregate sparsity matrix is not less than [Formula omitted] and the matrix remains positive semidefinite after replacing some off-diagonal nonzero elements with zeros, then the standard SDP relaxation provides an exact optimal solution for the QCQP under feasibility assumptions. In particular, we demonstrate that QCQPs with forest-structured aggregate sparsity matrix, such as the tridiagonal or arrow-type matrix, satisfy the exactness condition on the rank. The exactness is attained by considering the feasibility of the dual SDP relaxation, the strong duality of SDPs, and a sequence of QCQPs with perturbed objective functions, under the assumption that the feasible region is compact. We generalize our result for a wider class of QCQPs by applying simultaneous tridiagonalization on the data matrices. Moreover, simultaneous tridiagonalization is applied to a matrix pencil so that QCQPs with two constraints can be solved exactly by the SDP relaxation.
Audience	Academic
Author	Fukuda, Mituhiro Yamashita, Makoto Kim, Sunyoung Azuma, Godai
Author_xml	– sequence: 1 givenname: Godai surname: Azuma fullname: Azuma, Godai organization: Department of Mathematical and Computing Science, Tokyo Institute of Technology – sequence: 2 givenname: Mituhiro surname: Fukuda fullname: Fukuda, Mituhiro organization: Department of Mathematical and Computing Science, Tokyo Institute of Technology – sequence: 3 givenname: Sunyoung surname: Kim fullname: Kim, Sunyoung email: skim@ewha.ac.kr organization: Department of Mathematics, Ewha W. University – sequence: 4 givenname: Makoto surname: Yamashita fullname: Yamashita, Makoto organization: Department of Mathematical and Computing Science, Tokyo Institute of Technology
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Copyright	The Author(s), under exclusive licence to Springer Science+Business Media, LLC, part of Springer Nature 2021 COPYRIGHT 2022 Springer The Author(s), under exclusive licence to Springer Science+Business Media, LLC, part of Springer Nature 2021.
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Keywords	Exact semidefinite relaxations Quadratically constrained quadratic programs The rank of aggregated sparsity matrix Forest graph
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Snippet	We study the exactness of the semidefinite programming (SDP) relaxation of quadratically constrained quadratic programs (QCQPs). With the aggregate sparsity...
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SubjectTerms	Computer Science Constraints Feasibility Mathematical analysis Mathematics Mathematics and Statistics Matrices (mathematics) Operations Research/Decision Theory Optimization Real Functions Semidefinite programming Sparsity
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Title	Exact SDP relaxations of quadratically constrained quadratic programs with forest structures
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