Truncated expected hypervolume improvement: Exact computation and application

In optimization with expensive black box evaluations, the expected improvement algorithm (also called efficient global optimization) is a commonly applied method. It uses Gaussian Processes (or Kriging) to build a model of the objective function and uses the expected improvement as an infill criterion, taking into account both — predictive mean and variance. It has been generalized to multi-objective optimization using the expected hypervolume improvement, which measures the expected gain in the hypervolume indicator of a Pareto front approximation. However, this criterion assumes an unbounded objective space even if it is often known a-priori that the objective function values are within a prescribed range, e.g., lower bounded by zero. To take advantage of such a-priori knowledge, this paper introduces the truncated expected hypervolume improvement and a multiobjective efficient global optimization method that is based on it. In this paper it is shown how to compute the truncated expected hypervolume improvement exactly and efficiently. Then it is tested as an infill criterion in efficient global optimization. It is shown that it can effectively make use of a-priori knowledge and achieve better results in cases where such knowledge is given. The usefulness of the new approach is demonstrated in benchmark examples and applications from robust PID (proportionalintegral-derivative) controller optimization. The empirical studies in this paper are confined to the bi-objective case.