Faster Computation of Expected Hypervolume Improvement

The expected improvement algorithm (or efficient global optimization) aims for global continuous optimization with a limited budget of black-box function evaluations. It is based on a statistical model of the function learned from previous evaluations and an infill criterion - the expected improvement - used to find a promising point for a new evaluation. The `expected improvement' infill criterion takes into account the mean and variance of a predictive multivariate Gaussian distribution. The expected improvement algorithm has recently been generalized to multiobjective optimization. In order to measure the improvement of a Pareto front quantitatively the gain in dominated (hyper-)volume is used. The computation of the expected hypervolume improvement (EHVI) is a multidimensional integration of a step-wise defined non-linear function related to the Gaussian probability density function over an intersection of boxes. This paper provides a new algorithm for the exact computation of the expected improvement to more than two objective functions. For the bicriteria case it has a time complexity in $O(n^2)$ with $n$ denoting the number of points in the current best Pareto front approximation. It improves previously known algorithms with time complexity $O(n^3 \log n)$. For tricriteria optimization we devise an algorithm with time complexity of $O(n^3)$. Besides discussing the new time complexity bounds the speed of the new algorithm is also tested empirically on test data. It is shown that further improvements in speed can be achieved by reusing data structures built up in previous iterations. The resulting numerical algorithms can be readily used in existing implementations of hypervolume-based expected improvement algorithms.