The development of practically well-behaved integer programming formulations is an important aspect of solving linear optimization problems over a set $X\subseteq {\{0,1\}}^n$. In practice, one is often interested in strong integer formulations with additional properties, e.g., bounded coefficients to avoid numerical instabilities. This article presents a lower bound on the size of coefficients in any strong integer formulation of $X$ and demonstrates that certain integer sets $X$ require (exponentially) large coefficients in any strong integer formulation. We also show that strong integer formulations of $X\subseteq {\{0,1\}}^n$ may require exponentially many inequalities while linearly many inequalities may suffice in weak formulations.

开发行为良好的整数规划公式是解决集合上线性优化问题的重要方面 $X\subseteq {\{0，\mathrm{1个}\}}^{ñ}$。在实践中，人们通常对具有附加属性的强整数公式感兴趣，例如，有界系数以避免数值不稳定性。本文提出了在任何强整数公式中的系数大小的下界 $X$ 并证明某些整数集 $X$在任何强整数公式中都需要（指数）大系数。我们还证明了的强整数公式 $X\subseteq {\{0，\mathrm{1个}\}}^{ñ}$ 在弱公式中，可能需要成倍的指数不等式，而在线性上可能需要许多不等式。