Geometric Generalisation of Surrogate Model Based Optimisation to Combinatorial Spaces

Abstract

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.

Ant colony optimization on a limited budget of evaluations

“…Recently, these surrogate modeling approaches have been applied to the solution of combinatorial optimization problems (Moraglio and Kattan 2011) and of permutation problems Zaefferer et al 2014a, b). This is done by using an appropriate definition of a distance measure between solutions in order to calculate the correlation between data points.…”

“…This approach is widely used in continuous optimization in the form of, for example, the efficient global optimization (EGO) approach (Jones et al 1998). Recently, both Zaefferer et al (2014b and Moraglio and Kattan (2011) have proposed adaptations of the EGO method to tackle combinatorial problems. In this paper, we combine the best performing of these adaptations of EGO (Zaefferer et al 2014b) with various ACO algorithms for optimizing the surrogate model.…”

Model-based methods for continuous and discrete global optimization

“…A promising approach that recently gained more traction is similarity-based modeling (STR-6 Moraglio and Kattan [78] adapted an RBFN to arbitrary distance measures to model arbitrary combinatorial optimization problems. Their approach has also been applied to quadratic assignment problems [79].…”

Time-series event-based prediction: An unsupervised learning framework based on genetic programming

“…The same procedure described in Algorithm 2 is followed. For RBFN, we used the standard notation described in [22].…”