The most important ingredient for solving mixed-integer nonlinear programs (MINLPs) to global \(\epsilon \)-optimality with spatial branch and bound is a tight, computationally tractable relaxation. Due to both theoretical and practical considerations, relaxations of MINLPs are usually required to be convex. Nonetheless, current optimization solvers can often successfully handle a moderate presence of nonconvexities, which opens the door for the use of potentially tighter nonconvex relaxations. In this work, we exploit this fact and make use of a nonconvex relaxation obtained via aggregation of constraints: a surrogate relaxation. These relaxations were actively studied for linear integer programs in the 70s and 80s, but they have been scarcely considered since. We revisit these relaxations in an MINLP setting and show the computational benefits and challenges they can have. Additionally, we study a generalization of such relaxation that allows for multiple aggregations simultaneously and present the first algorithm that is capable of computing the best set of aggregations. We propose a multitude of computational enhancements for improving its practical performance and evaluate the algorithm’s ability to generate strong dual bounds through extensive computational experiments.

解决混合整数非线性程序 (MINLP) 到全局\(\epsilon \) -空间分支和边界最优性的最重要因素是紧密的、计算上易于处理的松弛。由于理论和实践上的考虑，通常要求 MINLP 的松弛是凸的。尽管如此，当前的优化求解器通常可以成功处理适度存在的非凸性，这为使用可能更紧的非凸性松弛打开了大门。在这项工作中，我们利用这一事实并利用通过约束聚合获得的非凸松弛：代理松弛。在 70 年代和 80 年代，针对线性整数规划积极研究了这些松弛，但此后几乎没有考虑过。我们在 MINLP 设置中重新审视这些松弛，并展示它们可以带来的计算优势和挑战。此外，我们研究了允许同时进行多个聚合的这种松弛的泛化，并提出了第一个能够计算最佳聚合集的算法。我们提出了大量计算增强以提高其实际性能，并通过广泛的计算实验评估算法生成强对偶边界的能力。