Distance Measures for Permutations in Combinatorial Efficient Global Optimization

For expensive black-box optimization problems, surrogatemodel based approaches like Efficient Global Optimization are frequently used in continuous optimization. Their main advantage is the reduction of function evaluations by exploiting cheaper, data-driven models of the actual target function. The utilization of such methods in combinatorial or mixed search spaces is less common. Efficient Global Optimization and related methods were recently extended to such spaces, by replacing continuous distance (or similarity) measures with measures suited for the respective problem representations. This article investigates a large set of distance measures for their applicability to various permutation problems. The main purpose is to identify, how a distance measure can be chosen, either a-priori or online. In detail, we show that the choice of distance measure can be integrated into the Maximum Likelihood Estimation process of the underlying Kriging model. This approach has robust, good performance, thus providing a very nice tool towards selection of a distance measure.