二元规划半定松弛的部分面约简中的面结构
Face Structure in Partial Facial Reduction of Semidefinite Relaxations of Binary Programs
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中文总结 AI 辅助
研究二元规划半定松弛中基于半正定锥内逼近的部分面约简，证明SDD部分面约简确定的终端面结构，给出多面体内逼近刻画方法，构造奇异性为二的族说明限制，刻画了约简结构及搜索代价。
中文摘要 AI 辅助
二元优化问题的半定松弛常常不满足斯莱特条件。完全面约简可恢复严格可行性，但可能需要与原始松弛一样昂贵的辅助问题。我们基于半正定锥的易处理内逼近进行部分面约简分析，包括非负对角、对角占优（DD）和缩放对角占优（SDD）锥。对于这类半定规划松弛，我们证明由SDD部分面约简确定的每个终端面由固定为零或一的变量以及被迫相等的变量组描述。该面允许一个具有不相交列支撑的满列秩{0,1}基矩阵，产生一个能保持稀疏性的自然约简形式。对于多面体内逼近，我们通过在相关外松弛上活动的不等式来刻画每一步暴露的面。我们还确定了严格的限制。我们构造了一个奇异性为二的族，对于它，DD和SDD部分面约简需要最大可能步数。我们进一步表明因子宽度限制可防止检测到密集有效仿射等式。这些结果刻画了由易处理部分面约简恢复的结构以及限制暴露矩阵搜索的代价。
英文摘要
Semidefinite relaxations of binary optimization problems often fail Slater's condition. Full facial reduction restores strict feasibility but may require auxiliary problems as costly as the original relaxation. We analyze partial facial reduction based on tractable inner approximations of the positive semidefinite cone: the nonnegative diagonal, diagonally dominant (DD), and scaled diagonally dominant (SDD) cones. For this class of SDP relaxations, we prove that every terminal face identified by SDD-partial FR is described by variables fixed to zero or one and groups of variables forced to be equal. The face admits a full-column-rank \(\{0,1\}\) basis matrix with disjoint column supports, yielding a natural reduced formulation that can preserve sparsity. For polyhedral inner approximations, we characterize the face exposed at each step through the inequalities active on the associated outer relaxation. We also establish sharp limitations. We construct a family with singularity degree two for which DD- and SDD-partial FR require the maximum possible number of steps. We further show that factor-width restrictions can prevent detection of dense valid affine equalities. These results characterize both the structure recovered by tractable partial FR and the cost of restricting the exposing-matrix search.
