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Exact SDP relaxations of quadratically constrained quadratic programs with forest structures

Godai Azuma (azuma.a.gg***at***m.is.titech.ac.jp)
Mituhiro Fukuda (mituhiro***at***is.titech.ac.jp)
Sunyoung Kim (skim***at***ewha.ac.kr)
Makoto Yamashita (Makoto.Yamashita***at***c.titech.ac.jp)

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.

Keywords: Quadratically constrained quadratic programs, exact semidefinite relaxations, forest graph, rank of aggregated sparsity matrix

Category 1: Global Optimization

Category 2: Linear, Cone and Semidefinite Programming (Semi-definite Programming )


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Entry Submitted: 09/08/2020
Entry Accepted: 09/08/2020
Entry Last Modified: 09/19/2020

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