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

Godai Azuma (), Sunyoung Kim () and Makoto Yamashita ()
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Godai Azuma: Institute of Science Tokyo
Sunyoung Kim: Ewha W. University
Makoto Yamashita: Institute of Science Tokyo

Journal of Global Optimization, 2025, vol. 92, issue 2, No 4, 343 pages

Abstract: 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.

Keywords: Exact sum of squares of relaxations; Rank-one matrix completion; Arrowhead pattern matrix; Linear combinations of given data; Algorithms for matrix completion; 90C22; 90C25; 90C26 (search for similar items in EconPapers)
Date: 2025
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DOI: 10.1007/s10898-025-01478-5

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