Different versions of polyhedral outer approximation are used by many algorithms for mixed-integer nonlinear programming (MINLP). While it has been demonstrated that such methods work well for convex MINLP, extending them to solve nonconvex problems has traditionally been challenging. The Supporting Hyperplane Optimization Toolkit (SHOT) is a solver based on polyhedral approximations of the nonlinear feasible set of MINLP problems. SHOT is an open source COIN-OR project, and is currently one of the most efficient global solvers for convex MINLP. In this paper, we discuss some extensions to SHOT that significantly extend its applicability to nonconvex problems. The functionality include utilizing convexity detection for selecting the nonlinearities to linearize, lifting reformulations for special classes of functions, feasibility relaxations for infeasible subproblems and adding objective cuts to force the search for better feasible solutions. This functionality is not unique to SHOT, but can be implemented in other similar methods as well. In addition to discussing the new nonconvex functionality of SHOT, an extensive benchmark of deterministic solvers for nonconvex MINLP is performed that provides a snapshot of the current state of nonconvex MINLP.

混合整数非线性规划（MINLP）的许多算法都使用了不同版本的多面体外逼近。虽然已经证明了这种方法对于凸MINLP很好用，但是将其扩展为解决非凸问题在传统上一直具有挑战性。支持超平面优化工具包（SHOT）是基于MINLP问题的非线性可行集的多面体近似的求解器。SHOT是一个开源的COIN-OR项目，目前是凸MINLP的最有效的全局求解器之一。在本文中，我们讨论了SHOT的一些扩展，这些扩展显着将其适用性扩展到非凸问题。该功能包括利用凸度检测来选择要线性化的非线性，取消特殊功能类别的重新公式化，解决不可行的子问题的可行性，并增加客观削减以迫使人们寻求更好的可行解决方案。此功能不是SHOT独有的，但也可以用其他类似方法来实现。除了讨论SHOT的新的非凸功能之外，还对非凸MINLP的确定性求解器进行了广泛的基准测试，提供了非凸MINLP当前状态的快照。