Exact SDP relaxations of quadratically constrained quadratic programs with forest structures

Godai Azuma, Mituhiro Fukuda, Sunyoung Kim, Makoto Yamashita

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5 Scopus citations


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
Issue number2
StatePublished - Feb 2022

Bibliographical note

Funding Information:
S. Kim: The research was supported by NRF 2021-R1A2C1003810. M. Yamashita: This research was partially supported by JSPS KAKENHI (Grant No. 20H04145).

Publisher Copyright:
© 2021, The Author(s), under exclusive licence to Springer Science+Business Media, LLC, part of Springer Nature.


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


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