Optimized approximation sets for Low-Dimensional benchmark pareto fronts

Abstract

The problem of finding sets of bounded cardinality maximizing dominated hypervolume is considered for explicitly parameterized Pareto fronts of multi-objective optimization problems. A parameterization of the Pareto front it often known (by construction) for synthetic benchmark functions. For the widely used ZDT and DTLZ families of benchmarks close-to-optimal sets have been obtained only for two objectives, although the three-objective variants of the DTLZ problems are frequently applied. Knowledge of the dominated hypervolume theoretically achievable with an approximation set of fixed cardinality facilitates judgment of (differences in) optimization results and the choice of stopping criteria, two important design decisions of empirical studies. The present paper aims to close this gap. An efficient optimization strategy is presented for two and three objectives. Optimized sets are provided for standard benchmarks.