Solving mixed-integer nonlinear optimization problems (MINLPs) to global optimality is extremely challenging. An important step for enabling their solution consists in the design of convex relaxations of the feasible set. Known solution approaches based on spatial branch-and-bound become more effective the tighter the used relaxations are. Relaxations are commonly established by convex underestimators, where each constraint function is considered separately. Instead, a considerably tighter relaxation can be found via so-called simultaneous convexification, where convex underestimators are derived for more than one constraint function at a time. In this work, we present a global solution approach for solving mixed-integer nonlinear problems that uses simultaneous convexification. We introduce a separation method that relies on determining the convex envelope of linear combinations of the constraint functions and on solving a nonsmooth convex problem. In particular, we apply the method to quadratic absolute value functions and derive their convex envelopes. The practicality of the proposed solution approach is demonstrated on several test instances from gas network optimization, where the method outperforms standard approaches that use separate convex relaxations.

将混合整数非线性优化问题（MINLP）求解为全局最优性是极富挑战性的。实现其解决方案的重要步骤在于设计可行集的凸松弛。使用的松弛越紧密，基于空间分支定界的已知解决方案将变得更加有效。松弛通常由凸低估器建立，其中每个约束函数都被单独考虑。取而代之的是，可以通过所谓的同时凸化来发现相当紧密的松弛，其中一次针对多个约束函数导出凸低估量。在这项工作中，我们提出了一种解决同时使用凸化的整数整数非线性问题的整体解决方案。我们介绍一种分离方法，该方法依赖于确定约束函数的线性组合的凸包络并解决非光滑凸问题。特别是，我们将该方法应用于二次函数的绝对值函数，并得出它们的凸包络。天然气网络优化的几个测试实例证明了所提出的解决方案方法的实用性，其中该方法的性能优于使用单独的凸松弛的标准方法。