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Rank-one matrix completion via high-rank matrices in sum-of-squares relaxations

Author

Listed:
  • Godai Azuma

    (Institute of Science Tokyo
    Aoyama Gakuin University)

  • Sunyoung Kim

    (Ewha W. University)

  • Makoto Yamashita

    (Institute of Science Tokyo)

Abstract

The standard semidefinite programming (SDP) relaxation for quadratically constrained quadratic programming (QCQP) problems generally cannot obtain an exact optimal solution. However, if the optimal solution of the SDP relaxation is of rank-1, then that of QCQP can be constructed. Cosse and Demanet (Found Comput Math 21:891–940, 2021) employed this condition for a rank-one matrix completion problem using the sum-of-squares (SOS) relaxation, which is the dual of the Lasserre’s relaxation. In this paper, we analyze the conditions under which the SOS relaxation provides an exact solution to the rank-one matrix completion problem. In particular, our focus is on obtaining the rank- $$(N-1)$$ ( N - 1 ) dual variable matrix of dimension N, a condition satisfied when the coefficient matrix of the objective function in the SOS relaxation problem exhibits an arrowhead structure. We relax the assumption of the explicit chain structure in Cosse and Demanet (Found Comput Math 21:891–940, 2021), and derive a weaker condition for the SDP relaxation to yield an exact solution compared to the explicit chain structure. We also present a numerical algorithm to find the coefficient matrix with the arrowhead structure, and numerical experiments illustrate the validity of the proposed algorithm.

Suggested Citation

  • 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.
    • Handle: RePEc:spr:jglopt:v:92:y:2025:i:2:d:10.1007_s10898-025-01478-5
    DOI: 10.1007/s10898-025-01478-5
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    References listed on IDEAS

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    1. 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.
    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.
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