Abstract
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 a 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.
Highlights
Traditional optimization techniques such as first order methods or branch and bound make use of a known mathematical formulation of the objective function, for example by calculating the derivative or a lower bound
A model fits the relation between decision variables and objective function, and standard optimization techniques are used on the model instead of the original objective
The main idea put forward is to use a model that guarantees integer-valued minima. This idea is evaluated with two different models: a basic and an advanced model. We evaluate both models on two different benchmark problems: finding a robust route for a noise-perturbed asymmetric traveling salesman benchmark problem with 17 cities, and an artificial convex binary optimization problem with up to 150 integer variables
Summary
A model fits the relation between decision variables and objective function, and standard optimization techniques are used on the model instead of the original objective These so-called surrogate modeling techniques have been applied successfully to continuous optimization problems in signal processing [4], optics [4], machine learning [5], robotics [6], and more. A common approach is to round to the nearest integer, a method that is known to be sub-optimal in traditional optimization, and in black-box optimization [7] Another option is to use discrete surrogate models from machine learning such as regression trees [8] or linear model trees [9].
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