Valid linear inequalities are substantial in linear and convex mixed-integer programming. This article deals with the computation of valid linear inequalities for nonlinear programs. Given a point in the feasible set, we consider the task of computing a tight valid inequality. We reformulate this geometrically as the problem of finding a hyperplane which minimizes the distance to the given point. A characterization of the existence of optimal solutions is given. If the constraints are given by polynomial functions, we show that it is possible to approximate the minimal distance by solving a hierarchy of sum of squares programs. Furthermore, using a result from real algebraic geometry, we show that the hierarchy converges if the relaxed feasible set is bounded. We have implemented our approach, showing that our ideas work in practice.

中文翻译：

---

通过平方和为非线性程序生成有效的线性不等式

在线性和凸混合整数规划中，有效的线性不等式很重要。本文处理非线性程序的有效线性不等式的计算。给定可行集中的一个点，我们考虑计算严格的有效不等式的任务。我们在几何上将其重新表述为寻找一个超平面的问题，该超平面最小化到给定点的距离。给出了最优解存在的特征。如果约束由多项式函数给出，我们表明可以通过求解平方和程序的层次结构来近似最小距离。此外，使用实代数几何的结果，我们表明如果松弛可行集是有界的，则层次会收敛。我们已经实施了我们的方法，表明我们的想法在实践中有效。