Exact SDP relaxations of quadratically constrained quadratic programs with forest structures

Godai Azuma, Mituhiro Fukuda, Sunyoung Kim, Makoto Yamashita

Research output: Contribution to journalArticlepeer-review

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.

Original languageEnglish
Pages (from-to)243-262
Number of pages20
JournalJournal of Global Optimization
Volume82
Issue number2
DOIs
StatePublished - Feb 2022

Keywords

  • Exact semidefinite relaxations
  • Forest graph
  • Quadratically constrained quadratic programs
  • The rank of aggregated sparsity matrix

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