秩一非负双矩阵完备化的紧锥松弛
Tight Conic Relaxations for Rank-one Doubly Nonnegative Matrix Completion
浏览论文内容
中文总结 AI 辅助
研究秩一非负双矩阵完备化的QCQP公式的紧锥松弛，证明特定稀疏模式下对偶公式等价，推导循环型稀疏模式下SDP和DNN松弛为紧的充分条件，表明添加边可放宽条件，为锥松弛恢复最优解提供证书。
中文摘要 AI 辅助
我们研究了秩一非负双矩阵完备化的二次约束二次规划（QCQP）公式的紧锥松弛。受稀疏QCQP的启发，其提升矩阵变量包含目标或约束未直接指定的元素，我们将紧性解释为未指定元素的秩一完备性属性。对于块由循环和边组成的稀疏模式，我们证明了与非负双（DNN）和完全正（CP）松弛相关的对偶公式是等价的。对于循环型稀疏模式，我们推导了半定规划（SDP）和DNN松弛为紧的显式充分条件。这些充分条件根据秩一最优解的局部比率界和累积差条件明确给出。我们还表明，在稀疏模式中添加合适的边会放宽紧性所需的比率条件。结果为锥松弛何时恢复基础QCQP的秩一最优解提供了易于处理的证书。
英文摘要
We study tight conic relaxations for a quadratically constrained quadratic programming (QCQP) formulation of rank-one doubly nonnegative (DNN) matrix completion. Motivated by sparse QCQPs whose lifted matrix variables include elements not directly specified by the objective or constraints, we interpret tightness as a rank-one completion property for the unspecified elements. For sparsity patterns whose blocks consist of cycles and edges, we prove that the dual formulations associated with the DNN and completely positive (CP) relaxations are equivalent. For cycle-type sparsity patterns, we derive explicit sufficient conditions under which the semidefinite programming (SDP) and DNN relaxations are tight. These sufficient conditions are stated explicitly in terms of local ratio bounds and cumulative-difference conditions on a rank-one optimal solution. We also show that adding suitable edges to the sparsity pattern relaxes the ratio conditions required for tightness. The results provide tractable certificates for when conic relaxations recover a rank-one optimal solution of the underlying QCQP.
