Geometric Generalisation of Surrogate Model Based Optimisation to Combinatorial Spaces

Abstract

In continuous optimisation, Surrogate Models (SMs) are often indispensable components of optimisation algorithms aimed at tackling real-world problems whose candidate solutions are very expensive to evaluate. Because of the inherent spatial intuition behind these models, they are naturally suited to continuous problems but they do not seem applicable to combinatorial problems except for the special case when solutions are naturally encoded as integer vectors. In this paper, we show that SMs can be naturally generalised to encompass combinatorial spaces based in principle on any arbitrarily complex underlying solution representation by generalising their geometric interpretation from continuous to general metric spaces. As an initial illustrative example, we show how Radial Basis Function Networks (RBFNs) can be used successfully as surrogate models to optimise combinatorial problems defined on the Hamming space associated with binary strings.