Complexity Reduction and Validation of Computing the Expected Hypervolume Improvement

Expected improvement algorithms are commonly used in global optimization problems where evaluating the objective function is costly. The Expected Hypervolume Improvement (EHVI) is a recent generalization of these algorithms to multiobjective optimization. The computation of the EHVI is based on a multidimensional integration of a piecewise defined nonlinear function. Exact calculation of the EHVI has so far only been possible in 2-D, and even there it is slow. In higher dimensions it can so far only be approximated, for instance by Monte Carlo integration, and no expression/algorithm for direct integration is available. In this thesis, a new algorithm is devised for the exact calculation of the EHVI in higher dimensions, and its correctness is experimentally verified in three dimensions. Additionally, fast computation schemes are proposed for the exact calculation of the EHVI in two and three dimensions, with time complexity in O(n) (previously O(n log n)) and in O(n), respectively. Empirical tests show that for Pareto front approximations of modest size (< 100 points) with the new algorithm computation times in the order of a second or less are required to perform exact calculations in three dimensions and imprecise Monte Carlo integration is no longer required. The algorithms have been implemented in C++ and can be readily used in global multiobjective optimization algorithms which use the EHVI.