Global Optimization via Optimal Decision Trees

TL;DR

This paper introduces a novel method that uses optimal decision trees with hyperplanes to approximate and solve complex global optimization problems efficiently, even with black box constraints.

Contribution

It presents a new approach leveraging OCT-Hs to learn MIO-compatible approximations for global optimization, handling both explicit and black box constraints.

Findings

Effective in solving benchmark global optimization problems

Demonstrates efficiency on real-world design problems

Achieves near-optimal solutions with iterative refinement

Abstract

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.