  The global optimization literature places large emphasis on reducing
intractable optimization problems into more tractable structured optimization
forms. In order to achieve this goal, many existing methods are restricted to
optimization over explicit constraints and objectives that use a subset of
possible mathematical primitives. These are limiting in real-world contexts
where more general explicit and black box constraints appear. Leveraging the
dramatic speed improvements in mixed-integer optimization (MIO) and recent
research in machine learning, we propose a new method to learn MIO-compatible
approximations of global optimization problems using optimal decision trees
with hyperplanes (OCT-Hs). This constraint learning approach only requires a
bounded variable domain, and can address both explicit and inexplicit
constraints. We solve the MIO approximation efficiently to find a near-optimal,
near-feasible solution to the global optimization problem. We further improve
the solution using a series of projected gradient descent iterations. We test
the method on a number of numerical benchmarks from the literature as well as
real-world design problems, demonstrating its promise in finding global optima
efficiently.
