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Exact SDP relaxations of quadratically constrained quadratic programs with forest structures

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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.

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Authors and Affiliations

  1. Department of Mathematical and Computing Science, Tokyo Institute of Technology, 2-12-1-W8-29 Oh-Okayama, Tokyo, Meguro-ku, 152-8552, Japan

    Godai Azuma & Makoto Yamashita

  2. Department of Mathematical and Computing Science, Tokyo Institute of Technology, 2-12-1-W8-41 Oh-Okayama, Tokyo, Meguro-ku, 152-8552, Japan

    Mituhiro Fukuda

  3. Department of Mathematics, Ewha W. University, 52 Ewhayeodae-gil, Sudaemoon-gu, Seoul, 03760, Korea

    Sunyoung Kim

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  1. Godai Azuma
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  2. Mituhiro Fukuda
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  3. Sunyoung Kim
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  4. Makoto Yamashita
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Correspondence to Sunyoung Kim.

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

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Azuma, G., Fukuda, M., Kim, S. et al. Exact SDP relaxations of quadratically constrained quadratic programs with forest structures. J Glob Optim 82, 243–262 (2022). https://doi.org/10.1007/s10898-021-01071-6

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  • DOI: https://doi.org/10.1007/s10898-021-01071-6

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