Black-box Combinatorial Optimization using Models with Integer-valued Minima
This work addresses the problem of efficiently optimizing combinatorial objectives with costly evaluations for researchers and practitioners in operations research and machine learning, representing an incremental improvement by adapting existing surrogate modeling techniques to enforce integer solutions.
The paper tackles the challenge of applying surrogate models to black-box combinatorial optimization by designing basis functions that ensure integer-valued minima, enabling effective optimization without continuous relaxations. It demonstrates superior performance over random search, simulated annealing, and one Bayesian optimization algorithm on a noise-perturbed traveling salesman problem, and outperforms all compared methods on a large-scale convex binary optimization problem.
When a black-box optimization objective can only be evaluated with costly or noisy measurements, most standard optimization algorithms are unsuited to find the optimal solution. Specialized algorithms that deal with exactly this situation make use of surrogate models. These models are usually continuous and smooth, which is beneficial for continuous optimization problems, but not necessarily for combinatorial problems. However, by choosing the basis functions of the surrogate model in a certain way, we show that it can be guaranteed that the optimal solution of the surrogate model is integer. This approach outperforms random search, simulated annealing and one Bayesian optimization algorithm on the problem of finding robust routes for a noise-perturbed traveling salesman benchmark problem, with similar performance as another Bayesian optimization algorithm, and outperforms all compared algorithms on a convex binary optimization problem with a large number of variables.