Paper Info
Title
Exact SDP relaxations of quadratically constrained quadratic programs with forest structures
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 foreststructured 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.
Year
DOI
Venue
2022
10.1007/s10898-021-01071-6
JOURNAL OF GLOBAL OPTIMIZATION
Keywords
DocType
Volume
Quadratically constrained quadratic programs, Exact semidefinite relaxations, Forest graph, The rank of aggregated sparsity matrix
Journal
82
Issue
ISSN
Citations 
2
0925-5001
0
PageRank 
References 
Authors
0.34
0
4
Name
Order
Citations
PageRank
Godai Azuma100.34
Mituhiro Fukuda200.34
Sunyoung Kim346138.82
Makoto Yamashita413613.74