Common properties of scalable multiobjective problems and a new framework of test problems

A multiobjective test problem is called “scalable” when the number of its objectives can be arbitrarily specified. Evolutionary many-objective optimization algorithms are usually evaluated using scalable test problems. However, their design is not easy due to the difficulty in formulating a Pareto front and a feasible region in a high-dimensional objective space. As a result, a wide variety of scalable test problems have not been proposed. First, in this paper, existing scalable test problems are examined from some new viewpoints such as the uniqueness of the optimal solution for an objective and the presence of an optimal solution for multiple objectives. It is shown that most existing scalable test problems have some common properties. Next, a new framework for test problem design is proposed. The proposed framework enables us to design the Pareto front in a highly flexible manner. More specifically, we can specify not only its curvature property (e.g., convex, concave and linear) but also its shape (e.g., triangle, rotated triangle, pentagon and hexagon). The feasible region of the objective space can be also designed flexibly. Finally, some scalable test problems are generated by the proposed framework. It is clearly shown that the generated test problems have totally different properties from existing scalable test problems.