A set of new multi- and many-objective test problems for continuous optimization and a comprehensive experimental evaluation.

Multi- and many-objective optimization problems have wide applications in the real world, and they have received growing attention from the evolutionary computation community. To promote the algorithm development in this area, numerous studies have been devoted to designing multi- and many-objective test problems. Most of these studies focus on handling complicated Pareto fronts (PFs), and the impact of the Pareto sets (PSs) has not been well-studied, although complicated PSs are prevalent in the real world. This paper presents a set of scalable test problems according to a new principle, which considers the geometrical properties of both PF and PS. A position function with a spherical form is proposed to introduce non-linear variable dependences to the PS, so as to simulate the variable dependencies in the real-world problems. According to the proposed principle, the first m (i.e., the number of objectives) decision variables are used to form the surface of a unit hypersphere, while the rest variables are designed to optimize a certain distance function. A set of test problems are generated by the proposed principle, which are then used to investigate six representative algorithms. The experimental results indicate that the proposed test problems pose considerable difficulties to existing algorithms, calling for the need for designing new algorithms capable of handling complicated PF and PS simultaneously.