Rational Polyhedral Outer-Approximations of the Second-Order Cone

TL;DR

This paper introduces two rational polyhedral outer-approximations of the second-order cone that maintain arbitrary accuracy, with one being practically optimal and the other explicit with bounded coefficients, enhancing computational applications.

Contribution

It presents novel rational outer-approximations of the second-order cone with guaranteed accuracy, including practical methods for minimal coefficients and explicit constructions with bounds.

Findings

The first approximation matches the size of the optimal irrational one.

An explicit second approximation with bounded coefficients is proposed.

Experimental results show computational benefits of the new formulations.

Abstract

It is well-known that the second-order cone can be outer-approximated to an arbitrary accuracy $\epsilon$ 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 $\epsilon$. 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 $\epsilon$ 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.