A benchmark for equality constrained multi-objective optimization

Evolutionary multi-objective optimization (EMO) is certainly a story of great success considering the numerous contributions and their applications to different problems and fields during the last two decades. One issue, however, that has been almost neglected so far is the consideration of multi-objective optimization problems (MOPs) that contain equality constraints. Such constraints play a special role as the inclusion of each equality constraint typically reduces the dimension of the search space by one. Consequently, the probability for a randomly chosen candidate solution of an equality constrained MOP to be feasible is zero, which makes the treatment of such problems very hard for EMO algorithms. In this paper, we propose a new benchmark of equality constrained MOPs. The problems are derived from the well-known DTLZ and IDTLZ problems and hence inherit their properties. The new benchmarks, Eq-DTLZ and Eq-IDTLZ, are scalable both in decision and objective space as well as in the number of equality constraints. Furthermore, all Pareto sets differ from the solution sets of the unconstrained problems and can be expressed analytically which make them good candidates for testing EMO algorithms on this important problem class. Based on the new benchmark, we investigate the performance of some state-of-the-art evolutionary algorithms. The results show that the new problems are indeed hard to solve for all considered algorithms and that further investigation has to be done for the reliable treatment of equality constrained MOPs.