On the Exponential Circuit Imbalance of the Ben-Tal Nemirovski Approximation

Abstract

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.