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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Bibliographic Details
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
Online Access:Get full text
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Summary: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.
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ISSN:0925-5001
1573-2916
DOI:10.1007/s10898-021-01071-6