Constrained Discrete Black-Box Optimization using Mixed-Integer Programming

Theodore Papalexopoulos, Christian Tjandraatmadja, Ross Anderson, Juan Pablo Vielma, David Benjamin Belanger

Keywords: discrete blackbox optimization, mixed integer programming

Abstract: Discrete black-box optimization problems are challenging for model-based optimization (MBO) algorithms, such as Bayesian optimization, due to the size of the search space and the need to satisfy combinatorial constraints. In particular, these methods require repeatedly solving a complex discrete global optimization problem in the inner loop, where popular heuristic inner-loop solvers introduce approximations and are difficult to adapt to combinatorial constraints. In response, we propose NN+MILP, a general discrete MBO framework using piecewise-linear neural networks as surrogate models and mixed-integer linear programming (MILP) to optimize the acquisition function. MILP provides optimality guarantees and a versatile declarative language for domain-specific constraints. We test our approach on a range of unconstrained and constrained problems, including DNA binding and the NAS-Bench-101 neural architecture search benchmark. NN+MILP surpasses or matches the performance of algorithms tailored to the domain at hand, with global optimization of the acquisition problem running in a few minutes using only standard software packages and hardware.

One-sentence Summary: We use mixed-integer programming to solve the inner loop acquisition problem in model-based blackbox optimization, which allows us to flexibly address general combinatorial and constrained domains.