In the field of evolutionary multiobjective optimization, the hypervolume indicator is the only single set quality measure that is known to be strictly monotonic with regard to Pareto dominance. This property is of high interest and relevance for multiobjective search involving a large number of objective functions. However, the high computational effort required for calculating the indicator values has so far prevented to fully exploit the potential of hypervolume-based multiobjective optimization. This paper addresses this issue and proposes a fast search algorithm that uses Monte Carlo sampling to approximate the exact hypervolume values. In detail, we present HypE (Hypervolume Estimation Algorithm for Multiobjective Optimization), by which the accuracy of the estimates and the available computing resources can be traded off; thereby, not only many-objective problems become feasible with hypervolume-based search, but also the runtime can be flexibly adapted. The experimental results indicate that HypE is highly effective for many-objective problems in comparison to existing multiobjective evolutionary algorithms.