A performance comparison indicator for Pareto front approximations in many-objective optimization

Increasing interest in simultaneously optimizing many objectives (typically more than three objectives) of problems leads to the emergence of various many-objective algorithms in the evolutionary multi-objective optimization field. However, in contrast to the development of algorithm design, how to assess many-objective algorithms has received scant concern. Many performance indicators are designed in principle for any number of objectives, but in practice are invalid or infeasible to be used in many-objective optimization. In this paper, we explain the difficulties that popular performance indicators face and propose a performance comparison indicator (PCI) to assess Pareto front approximations obtained by many-objective algorithms. PCI evaluates the quality of approximation sets with the aid of a reference set constructed by themselves. The points in the reference set are divided into many clusters, and the proposed indicator estimates the minimum moves of solutions in the approximation sets to weakly dominate these clusters. PCI has been verified both by an analytic comparison with several wellknown indicators and by an empirical test on four groups of Pareto front approximations with different numbers of objectives and problem characteristics.