基于二元规划多面体松弛的人脸识别：与因子宽度部分人脸约简的比较
Face Identification via Polyhedral Relaxations for Binary Programs: A Comparison with Factor-Width Partial Facial Reduction
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中文总结 AI 辅助
该研究针对二元规划的SDP松弛，用基于LP的多面体松弛人脸识别方法替代辅助SDP，构建保留所有可行二元点的多项式规模多面体，实现不劣于因子宽度k辅助SDP的人脸识别。
中文摘要 AI 辅助
人脸约简（FR）是一种用于恢复半定规划（SDP）中斯莱特条件的预处理技术，但每一步通常需要求解一个辅助SDP。因子宽度k的部分FR限制了暴露矩阵的类别，增大k会扩大该类别，但当k≥3时，辅助问题仍为包含k阶半定块的SDP。针对二元规划的SDP松弛，我们用基于线性规划（LP）的人脸识别方法替代该辅助SDP，构建了一个以子集索引矩变量表示的多面体，其在固定k时具有多项式规模。所得人脸包含所有可行二元提升点，且被同一辅助系统中因子宽度k的暴露矩阵所暴露的每个人脸包含。因此，线性规划识别出的人脸不大于从因子宽度k辅助SDP得到的人脸，同时保留所有可行二元点。为迭代该构建过程，我们用原始矩阵变量中的线性方程表示每个人脸，从而在每次迭代中保留矩矩阵索引和因子宽度比较。
英文摘要
Facial reduction (FR) is a preprocessing technique used to restore Slater's condition in semidefinite programming (SDP), but each step generally requires solving an auxiliary SDP. Factor-width-\(k\) partial FR restricts the class of exposing matrices. Increasing \(k\) enlarges this class, but, for \(k\geq3\), the auxiliary problem remains an SDP involving positive semidefinite blocks of order \(k\). For SDP relaxations of binary programs, we replace this auxiliary SDP by an LP-based face-identification method. We construct a polyhedron in subset-indexed moment variables that has polynomial size for fixed \(k\). The resulting face contains every feasible binary lift and is contained in every face exposed by a factor-width-\(k\) exposing matrix from the same auxiliary system. Thus, linear programming identifies a face no larger than those obtained from the factor-width-\(k\) auxiliary SDP while preserving all feasible binary points. To iterate the construction, we represent each identified face by linear equations in the original matrix variable, thereby retaining the moment-matrix indexing and the factor-width comparison at every iteration.
