发表机构
School of Mathematical and Statistical Sciences, Clemson University（克莱姆森大学数学与统计科学学院）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文针对二元规划的等式生成SDP-RLT松弛，确定其最坏情况奇异度的紧界为⌊n/2⌋，该界小于SDP通用界，可预先估计奇异度与误差界的Hölder指数。
AI 中文摘要
奇异度是衡量半定规划（SDP）退化性的重要指标，但通常无法从问题数据中预先获得。我们对非空二元集{x∈{0,1}^n:Ax=b}的Shor松弛进行扩充，加入由定义线性等式生成的一阶重构-线性化技术（RLT）方程。针对由此得到的等式生成SDP-RLT松弛，我们确定了其精确的最坏情况奇异度：若rank(A)=m且0<m<n，则相关松弛的奇异度至多为min{m,n−m}，且该秩-零度界对该范围内的每个可能秩均可达到。因此，当n≥2时，该类问题的最坏情况奇异度为⌊n/2⌋，这远小于含n+1阶矩阵变量的可行SDP系统的紧通用界n（Sturm，2000，例2）。由此，对于单个松弛问题，秩与零度可预先对原本无法获取的奇异度，以及用于从约束残差估计可行性距离的误差界中的Hölder指数提供界。
英文摘要
Singularity degree is an important measure of semidefinite programming (SDP) degeneracy, but it is generally unavailable a priori from the problem data. We augment the Shor relaxation of nonempty binary sets $\{x\in\{0,1\}^n:Ax=b\}$ with the first-level Reformulation-Linearization Technique (RLT) equations generated by the defining linear equalities. For the resulting equality-generated SDP-RLT relaxation, we determine the exact worst-case singularity degree. If $\operatorname{rank}(A)=m$ and $0<m<n$, then the associated relaxation has singularity degree at most $\min\{m,n-m\}$, and this rank-nullity bound is attained for every possible rank in this range. Consequently, the worst-case singularity degree over this class is $\lfloor n/2\rfloor$ for $n\geq 2$. This is strikingly smaller than the sharp general bound $n$ for feasible SDP systems with matrix variables of order $n+1$ (Sturm, 2000, Example 2). Thus, for individual relaxations, rank and nullity provide an a priori bound on the otherwise inaccessible singularity degree and on the Hölder exponent in error bounds estimating distance to feasibility from constraint residuals.
