Hypervolume approximation using achievement scalarizing functions for evolutionary many-objective optimization

This paper proposes an idea of approximating the hypervolume of a non-dominated solution set using a number of achievement scalarizing functions with uniformly distributed weight vectors. Each achievement scalarizing function with a different weight vector is used to measure the distance from the reference point of the hypervolume to the attainment surface of the non-dominated solution set along its own search direction specified by its weight vector. Our idea is to approximate the hypervolume by the average distance from the reference point to the attainment surface over a large number of uniformly distributed weight vectors (i.e., over various search directions). We examine the effect of the number of weight vectors (i.e., the number of search directions) on the approximation accuracy and the computation time of the proposed approach. As expected, experimental results show that the approximation accuracy is improved by increasing the number of weight vectors. It is also shown that the proposed approach needs much less computation time than the exact hypervolume calculation for a six-objective knapsack problem even when we use about 100,000 weight vectors.