Rational polyhedral outer-approximations of the second-order cone

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