Test Problems Based on Lamé Superspheres

Pareto optimization methods are usually expected to find well-distributed approximations of Pareto fronts with basic geometry, such as smooth, convex and concave surfaces. In this contribution, test-problems are proposed for which the Pareto front is the intersection of a Lame supersphere with the positive Rn-orthant. Besides scalability in the number of objectives and decision variables, the proposed test problems are also scalable in a characteristic we introduce as resolvability of conflict, which is closely related to convexity/concavity, curvature and the position of knee-points of the Pareto fronts. As a very basic bi-objective problem we propose a generalization of Schaffer's problem. We derive closed-form expressions for the efficient sets and the Pareto fronts, which are arcs of Lame supercircles. Adopting the bottom-up approach of test problem construction, as used for the DTLZ test-problem suite, we derive test problems of higher dimension that result in Pareto fronts of superspherical geometry. Geometrical properties of these test-problems, such as concavity and convexity and the position of knee-points are studied. Our focus is on geometrical properties that are useful for performance assessment, such as the dominated hypervolume measure of the Pareto fronts. The use of these test problems is exemplified with a case-study using the SMSEMOA, for which we study the distribution of solution points on different 3-D Pareto fronts.