A hard optimisation test function with symbolic solution visualisation for fast interpretation by the human eye

We propose a class of test problems for evaluating the performance of global function optimisers based on finding an optimal spatial distribution of nonidentical particles interacting with two different potential fields. Because of the possibility of intuitive solution visualisation it can be of particular benefit during development of optimisation algorithms. An ensemble of N particles is constrained to a low-dimensional space and each particle contributes in two ways to the total potential energy: by its position on a hilly track and through repulsive neighbour potentials. The task of minimising the ensemble's total potential energy corresponds to searching an N-dimensional space with many local minima separated through higher and lower barriers; hence, it can serve as a performance measure for evolutionary algorithms (EA). The search difficulty is scalable through the number of particles and the hilliness of the track. In particular, if the particles are made nonidentical by giving them different masses or charges, the search will become very challenging because of the introduced combinatorial aspect and the “curse of dimensionality”. Among many similarly challenging optimisation problems this test function class has the advantage that solution candidates can be plotted in ways which allow humans to estimate not only relative objective function values but also DNA vector relations upon a quick glance. For the EA developer this allows a fast feedback cycle between a modification to the EA and the observed change in optimisation history behaviour. This makes experimentation with EA elements at a fundamental level easier. Furthermore, this class of real-domain search offers a wide range of difficulty and complexity levels and can be split up into a two-objective optimisation.