It is well-known that the second-order cone can be outer-approximated to an arbitrary accuracy $ϵ$ by a polyhedral cone of compact size defined by irrational data. In this paper, we propose two rational polyhedral outer-approximations of compact size retaining the same guaranteed accuracy $ϵ$. The first outer-approximation has the same size as the optimal but irrational outer-approximation from the literature. In this case, we provide a practical approach to obtain such an approximation defined by the smallest integer coefficients possible, which requires solving a few, small-size integer quadratic programs. The second outer-approximation has a size larger than the optimal irrational outer-approximation by a linear additive factor in the dimension of the second-order cone. However, in this case, the construction is explicit, and it is possible to derive an upper bound on the largest coefficient, which is sublinear in $ϵ$ and logarithmic in the dimension. We also propose a third outer-approximation, which yields the best possible approximation accuracy given an upper bound on the size of its coefficients. Finally, we discuss two theoretical applications in which having a rational polyhedral outer-approximation is crucial, and run some experiments which explore the benefits of the formulations proposed in this paper from a computational perspective.

众所周知，二阶圆锥可以在外部近似到任意精度 $ϵ$由无理数据定义的紧凑尺寸的多面圆锥体。在本文中，我们提出了两个紧凑尺寸的有理多面体外逼近，并且保持了相同的保证精度 $ϵ$。第一外部近似具有与文献中的最佳但不合理的外部近似相同的大小。在这种情况下，我们提供了一种实用的方法来获得由可能的最小整数系数定义的近似值，这需要解决一些小尺寸的整数二次程序。第二外部近似值的大小比最佳无理外部近似值大二阶圆锥尺寸的线性累加因子。但是，在这种情况下，结构是明确的，并且有可能得出最大系数的上限，该上限在$ϵ$在维度上为对数。我们还提出了第三种外部近似方法，在给定其系数大小的上限的情况下，该方法可获得最佳的近似精度。最后，我们讨论了两个理论应用，其中具有合理的多面体外部逼近至关重要，并进行了一些实验，这些实验从计算的角度探讨了本文提出的公式的优点。