Exact SDP relaxations of quadratically constrained quadratic programs with forest structures

TL;DR

This paper establishes conditions under which semidefinite programming relaxations exactly solve quadratically constrained quadratic programs, especially those with forest-structured sparsity matrices, by analyzing matrix rank and positive semidefiniteness.

Contribution

It provides new theoretical conditions for the exactness of SDP relaxations in QCQPs with forest structures, including tridiagonal and arrow-type matrices, and extends results via simultaneous tridiagonalization.

Findings

SDP relaxation is exact for QCQPs with forest-structured sparsity matrices.

Exactness is guaranteed when the aggregate sparsity matrix has rank at least n-1 and remains positive semidefinite after zeroing certain off-diagonal elements.

Simultaneous tridiagonalization enables exact solutions for QCQPs with two constraints.

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.