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Exact SDP relaxations of quadratically constrained quadratic programs with forest structures

Author

Listed:
  • Godai Azuma

    (Tokyo Institute of Technology)

  • Mituhiro Fukuda

    (Tokyo Institute of Technology)

  • Sunyoung Kim

    (Ewha W. University)

  • Makoto Yamashita

    (Tokyo Institute of Technology)

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 $$n-1$$ 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.

Suggested Citation

  • Godai Azuma & Mituhiro Fukuda & Sunyoung Kim & Makoto Yamashita, 2022. "Exact SDP relaxations of quadratically constrained quadratic programs with forest structures," Journal of Global Optimization, Springer, vol. 82(2), pages 243-262, February.
    • Handle: RePEc:spr:jglopt:v:82:y:2022:i:2:d:10.1007_s10898-021-01071-6
    DOI: 10.1007/s10898-021-01071-6
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    References listed on IDEAS

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    1. NESTEROV, Yu. & WOLKOWICZ, Henry & YE, Yinyu, 2000. "Semidefinite programming relaxations of nonconvex quadratic optimization," LIDAM Reprints CORE 1471, Université catholique de Louvain, Center for Operations Research and Econometrics (CORE).
    2. Gábor Pataki, 1998. "On the Rank of Extreme Matrices in Semidefinite Programs and the Multiplicity of Optimal Eigenvalues," Mathematics of Operations Research, INFORMS, vol. 23(2), pages 339-358, May.
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    Cited by:

    1. Yuzhou Qiu & E. Alper Yıldırım, 2024. "On exact and inexact RLT and SDP-RLT relaxations of quadratic programs with box constraints," Journal of Global Optimization, Springer, vol. 90(2), pages 293-322, October.
    2. Godai Azuma & Mituhiro Fukuda & Sunyoung Kim & Makoto Yamashita, 2023. "Exact SDP relaxations for quadratic programs with bipartite graph structures," Journal of Global Optimization, Springer, vol. 86(3), pages 671-691, July.
    3. Godai Azuma & Sunyoung Kim & Makoto Yamashita, 2025. "Rank-one matrix completion via high-rank matrices in sum-of-squares relaxations," Journal of Global Optimization, Springer, vol. 92(2), pages 321-343, June.
    4. Cheng Lu & Jitao Ma & Zhibin Deng & Wenxun Xing, 2024. "A graphic structure based branch-and-bound algorithm for complex quadratic optimization and applications to magnitude least-square problem," Journal of Global Optimization, Springer, vol. 88(1), pages 115-137, January.
    5. Bo Zhang & YueLin Gao & Xia Liu & XiaoLi Huang, 2023. "Outcome-space branch-and-bound outer approximation algorithm for a class of non-convex quadratic programming problems," Journal of Global Optimization, Springer, vol. 86(1), pages 61-92, May.

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