Characteristics of many-objective test problems and penalty parameter specification in MOEA/D

Recently a number of evolutionary many-objective algorithms have been proposed using uniformly generated weight vectors. Those algorithms can be viewed as improved versions of MOEA/D with the PBI (penalty-based boundary intersection) function. Reference lines are uniformly specified using the weight vectors in the normalized objective space. The basic idea of those algorithms is to find a single solution along each reference line. Whereas a different search mechanism has been devised in each algorithm, solution assignment to reference lines is commonly based on the distance to the nearest reference line. This solution assignment can be interpreted as using a larger penalty value in MOEA/D with PBI. Actually, MOEA/D with PBI works well on frequently-used many-objective test problems DTLZ1-4 when a large penalty value is used. However, the shape of the contour lines of the PBI function suggests the use of a small penalty value for many-objective problems. Moreover, good results have been reported in the literature for many-objective knapsack problems when a small penalty value was used. In this paper, we discuss why good results are obtained from a large penalty value from a viewpoint of characteristics of DTLZ1-4 as test problems. Our discussions on the use of a large penalty value also explain why good results are obtained by recently-proposed weight vector-based evolutionary many-objective algorithms.