An efficient algorithm for computing hypervolume contributions**

• Authors: Karl Bringmann;Tobias Friedrich
• Venue: Evolutionary Computation
• Year: 2010

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 (nd//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 for d == 3. We also analyze hypervolume indicator based optimization algorithms which remove λλ 1 solutions from a population of size n == μµ ++ λλ. We show that there are populations such that the hypervolume contribution of iteratively chosen λλ solutions is much larger than the hypervolume contribution of an optimal set of λλ solutions. Selecting the optimal set of λλ solutions implies calculating conventional hypervolume contributions, which is considered to be computationally too expensive. We present the first hypervolume algorithm which directly calculates the contribution of every set of λλ solutions. This gives an additive term of in the runtime of the calculation instead of a multiplicative factor of . More precisely, for a population of size n with d objectives, our algorithm can calculate a set of λλ solutions with minimal hypervolume contribution in time (nd//2 log n ++ nλλ) for d 2. This improves all previously published algorithms by a factor of nmin{λλ,d//2} for d 3 and by a factor of n for d == 3.