Approximate Hypervolume Calculation with Guaranteed or Confidence Bounds

Abstract

We present a new version of the Quick Hypervolume algorithm allowing calculation of guaranteed lower and upper bounds for the value of hypervolume, which is one of the most often used and recommended quality indicators in multiobjective optimization. To ensure fast convergence of these bounds, we use a priority queue of subproblems instead of the depth-first search applied in the original recursive Quick Hypervolume algorithm. We also combine this new algorithm with the Monte Carlo sampling approach, which allows obtaining better confidence intervals than the standard Monte Carlo sampling. The performance of the two proposed methods is compared with that of a straightforward adaptation of recursive Quick Hypervolume algorithm and the standard Monte Carlo sampling in a comprehensive computational experiment.