半定规划中的严格互补性、奇异性程度以及前向和后向误差的（不）关联
Strict complementarity in semidefinite programming, singularity degree, and the (dis)connection of forward and backward errors
AI总结：
研究半定规划中严格互补性缺失问题，通过简单范式刻画，给出构造算法，精确刻画奇异性程度与约束数量关系，构建相关SDP并计算研究，发现前向 - 后向误差差距常见且存在负相关。
AI中文摘要：
原对偶最优解对的严格互补性在半定规划（SDP）的数值分析中至关重要。它推动内点算法的收敛行为，不成立时近似解的两种误差度量存在显著差距，即前向或“真实”误差（到最优解集的距离）和由约束违反度量的不太有用的后向误差。首先通过简单范式刻画SDP中严格互补性的缺失，其具有三个关键特征。还给出了生成算法的变体来构造不满足严格互补性但满足原对偶Slater条件的SDP。接着精确刻画SDP的奇异性程度何时等于约束数量。构建并共享了一组不满足严格互补性的SDP并进行详细计算研究，发现前向 - 后向误差差距很常见，在许多小SDP中，前向（“真实”）误差比后向误差大达七个数量级，且在几个数据集中两者呈负相关。
英文摘要：
Strict complementarity of a primal-dual pair of optimal solutions is fundamental in the numerical analysis of semidefinite programs (SDPs). Strict complementarity drives the convergence behavior of interior point algorithms. When it fails, pathological examples show a striking gap between two error measures of approximate solutions. The first of these is the forward or "true" error, i.e., the distance to the optimal solution set. The second is the less useful backward error, measured by the constraint violation. We first characterize the lack of strict complementarity in SDPs via a simple normal form. Our normal form has three key features: (i) it is obtained using elementary row operations and rotations; (ii) it makes the lack of strict complementarity evident; and (iii) it lets us construct any such SDP by a simple algorithm. A variant of our generating algorithm allows us to construct any SDP that fails strict complementarity but satisfies Slater's condition on both the primal and dual sides. Thus, we {\em parametrize} the data of all SDPs that lack strict complementarity in a manner similar to how the Jordan normal form parametrizes square matrices with given eigenvalue structure. Next, we precisely characterize when the singularity degree of an SDP equals the number of constraints -- a result that underlies our generating algorithms and that we believe is of independent interest. We construct and share a set of SDPs that lack strict complementarity and present a detailed computational study. We find that forward-backward error gaps are quite common: in many small SDPs (with matrix order $\leq 20$), the forward ("true") error exceeds the backward error by up to seven orders of magnitude. Further, in several data sets the forward and backward errors are {\em inversely} correlated. In other words, the worse the "true" forward error is, the harder it is to detect.
