Multiobjective test problems with complicated Pareto fronts: Difficulties in degeneracy

It is well-established that the shapes of Pareto-optimal fronts (POFs) can affect the performance of some multiobjective optimization methods. The most well-known characteristics on the shape of POFs are convexity and discontinuity. In this paper, we investigate the construction of multiobjective test problems with complicated POFs, of which its local parts could have mixed dimensionalities. For example, in the case of 3 objectives, some parts of POFs can be 1-D curves while others could be 2-D surfaces. We formulate eight test problems, called CPFT1-8, with such a feature. To study the difficulties of these test problems, we conducted some experiments with two state-of-the-art algorithms MOEA/D and NSGA-II, and analyzed their performances.