The Pareto-Box problem for the modelling of evolutionary multiobjective optimization algorithms

This paper presents the Pareto-Box problem for modelling evolutionary multi-objective search. The problem is to find the Pareto set of randomly selected points in the unit hypercube. While the Pareto set itself is only comprised of the point 0, this problem allows for a complete analysis of random search and demonstrates the fact that with increasing number of objectives, the probability of finding a dominated vector is decreasing exponentially. Since most nowadays evolutionary multi-objective optimization algorithms rely on the existence of dominated individuals, they show poor performance on this problem. However, the fuzzification of the Pareto-dominance is an example for an approach that does not need dominated individuals, thus it is able to solve the Pareto-Box problem even for a higher number of objectives.