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

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

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  • NETL
  • NSTL
  • Springer Nature
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) 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.

Exact sum of squares of relaxationsRank-one matrix completionArrowhead pattern matrixLinear combinations of given dataAlgorithms for matrix completion

Godai Azuma、Sunyoung Kim、Makoto Yamashita

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Department of Mathematical and Computing Science, Institute of Science Tokyo, 2-12-1-W8-29 Oh-Okayama, Meguro-ku, Tokyo 152-8552, Japan||Department of Industrial and Systems Engineering, Aoyama Gakuin University, 5-10-1-O-410b Fuchinobe, Chuo-ku, Sagamihara-shi, Kanagawa 252-5258, Japan

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

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

2025

10.1007/s10898-025-01478-5

Journal of global optimization

Journal of global optimization

ISSN：0925-5001
年,卷(期)：2025.92(2)
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