Many approaches for addressing Global Optimization problems typically rely on relaxations of nonlinear constraints over specific mathematical primitives. This is restricting in applications with constraints that are black-box, implicit or consist of more general primitives. Trying to address such limitations, Bertsimas and Ozturk (2023) proposed OCTHaGOn as a way of solving black-box global optimization problems by approximating the nonlinear constraints using hyperplane-based Decision-Trees and then using those trees to construct a unified mixed integer optimization (MIO) approximation of the original problem. We provide extensions to this approach, by (i) approximating the original problem using other MIO-representable ML models besides Decision Trees, such as Gradient Boosted Trees, Multi Layer Perceptrons and Suport Vector Machines, (ii) proposing adaptive sampling procedures for more accurate machine learning-based constraint approximations, (iii) utilizing robust optimization to account for the uncertainty of the sample-dependent training of the ML models, and (iv) leveraging a family of relaxations to address the infeasibilities of the final MIO approximation. We then test the enhanced framework in 81 Global Optimization instances. We show improvements in solution feasibility and optimality in the majority of instances. We also compare against BARON, showing improved optimality gaps or solution times in 11 instances.

翻译：解决全局优化问题的多种方法通常依赖于对特定数学原语的非线性约束进行松弛。这在约束条件为黑箱、隐式或包含更一般原语的应用中具有局限性。为应对这些局限，Bertsimas 与 Ozturk（2023）提出了OCTHaGOn方法，通过使用基于超平面的决策树对非线性约束进行近似，进而构建原问题的统一混合整数优化（MIO）近似。我们对该方法进行了扩展，具体包括：（i）除决策树外，使用其他MIO可表示的机器学习模型（如梯度提升树、多层感知机和支持向量机）来近似原问题；（ii）提出自适应采样程序，以实现基于机器学习的更精确约束近似；（iii）利用鲁棒优化处理ML模型因样本依赖训练带来的不确定性；（iv）采用一系列松弛方法解决最终MIO近似中的不可行性问题。随后，我们在81个全局优化实例上测试了增强框架。结果表明，在大多数实例中，解的可行性和最优性均有提升。我们还与BARON进行了对比，在11个实例中展现了更优的最优性间隙或求解时间。