Scholarlink

<p>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-<InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10898_2025_1478_Article_IEq1.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="58" /> </InlineMediaObject> <EquationSource Format="TEX">\((N-1)\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo stretchy="false">(</mo> <mi>N</mi> <mo>-</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> dual variable matrix of dimension <i>N</i>, 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.</p>

错误：搜索内容不能为空，请输入英文关键词
错误：关键词超出字数限制，请精简
高级检索

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

  • 期刊论文

  • 发表日期：

    2025年3月21日

查看全文
  • Godai Azuma,
  • Sunyoung Kim,
  • Makoto Yamashita

摘要

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.