关于Ben-Tal Nemirovski近似的指数级回路不平衡性

中文摘要 AI 辅助

Dadush等人（2024）近期提出了一种用于线性规划的缩放不变分层最小二乘算法，其复杂度取决于最优条件测度$\arχ_A^*$。该工作建立在Vavasis和Ye（1996）的算法基础之上，后者的运行时间仅通过条件数$\arχ_A$依赖于约束矩阵$A$。Monteiro-Tsuchiya（2003）将最优条件数$\arχ_A^*$定义为在所有正对角列缩放矩阵$D$下可达到的最大$\arχ_{AD}$。Dadush等人（2024）引入了最优回路不平衡测度$κ_W^*$，它是$\arχ_A^*$的一个下界。人为构造具有大最优回路不平衡测度$κ_W^*$的实例很容易，但找到该最优缩放不变测度呈指数增长的自然存在实例本身具有独立研究价值。本文中，我们证明单位圆盘的Ben-Tal Nemirovski（BN）线性规划近似恰好提供了这样的实例。通过显式构造BN公式核中的回路，我们证明最优回路不平衡测度$κ_W^*$随近似步数呈指数增长。由于$κ_W^*$是$\arχ_A^*$的下界，我们的结果表明BN近似产生了一族指数级病态的约束矩阵。

英文摘要

Dadush et al.\ (2024) recently developed a scaling-invariant layered least squares algorithm for linear programming whose complexity depends on the optimal condition measure $\barχ_A^*$. Their work builds on Vavasis and Ye's (1996) algorithm whose running time depends only on the constraint matrix $A$ through the condition number $\barχ_A$. Monteiro-Tsuchiya (2003) defined the optimal condition number $\barχ_A^*$ as the maximum $\barχ_{AD}$ achievable over all positive diagonal column rescalings $D$. Dadush et al.\ (2024) introduced the optimal circuit imbalance measure $κ_W^*$, which serves as a lower bound for $\barχ^*_A$. Instances with artificially large optimal circuit imbalance measures $κ_W^*$ can be easily constructed; however, finding naturally occurring examples where this optimal scaling-invariant measure grows exponentially is of independent interest. In this paper, we show that the Ben-Tal Nemirovski (BN) linear programming approximation of the unit disk provides such an example. By explicitly constructing circuits in the kernel of the BN formulation, we prove that the optimal circuit imbalance measure $κ_W^*$ grows exponentially in the number of approximation steps. Since $κ_W^*$ lower bounds $\barχ_A^*$, our result demonstrates that the BN approximation yields an exponentially ill-conditioned family of constraint matrices.

发表机构

• University of Waterloo（滑铁卢大学）

机构由 AI 辅助整理，请以论文原文为准。