## Evolutionary Computation

The hypervolume indicator serves as a sorting criterion in many recent multi-objective evolutionary algorithms (MOEAs). Typical algorithms remove the solution with the smallest loss with respect to the dominated hypervolume from the population. We present a new algorithm which determines for a population of size *n* with *d* objectives, a solution with minimal hypervolume contribution in time *n*^{d/2} log *n*) for *d* > 2. This improves all previously published algorithms by a factor of *n* for all *d* > 3 and by a factor of *d* = 3.

We also analyze hypervolume indicator based optimization algorithms which remove λ > 1 solutions from a population of size *n* = μ + λ. We show that there are populations such that the hypervolume contribution of iteratively chosen λ solutions is much larger than the hypervolume contribution of an optimal set of λ solutions. Selecting the optimal set of λ solutions implies calculating *n* with *d* objectives, our algorithm can calculate a set of λ solutions with minimal hypervolume contribution in time *n*^{d/2} log *n* + *n*^{λ}) for *d* > 2. This improves all previously published algorithms by a factor of *n*^{min{λ,d/2}} for *d* > 3 and by a factor of *n* for *d* = 3.