This page presents a collection of links and papers related to the hypervolume quality indicator, a measure often used to assess the performance of multi-objective evolutionary algorithms. It is also used directly inside EAs to provide a selection pressure rewarding convergence as well as objective-space diversity of the candidate solutions. Hypervolume has the drawback that it takes exponential time in the number of dimensions to compute. For this reason, there is an active community developing faster algorithms (in theory and in practice) for this important problem.
Examples of the dominated hypervolume in two and three dimensions. The left figure shows a minimization case whereas the right one shows maximization.
This is a reimplementation of the code by Fonseca et al. (Variant 3, Version 1.2) in pure Python. Some small modifications have been applied to obtain more performance on the Python interpreter.
# usage example: from hv import HyperVolume referencePoint = [2, 2, 2] hyperVolume = HyperVolume(referencePoint) front = [[1, 0, 1], [0, 1, 0]] result = hyperVolume.compute(front)
The exclusive hypervolume contribution of each point in the same examples as above.
The S-Metric Selection Evolutionary Multiobjective Optmization Algorithm uses the hypervolume indicator to compute the exclusive hypervolume contribution of solutions.
-SIBEA on two pseudo-boolean test functions, see D. Brockhoff, T. Friedrich, and F. Neumann. Analyzing Hypervolume Indicator Based Algorithms. In G. Rudolph et al., editors, Conference on Parallel Problem Solving From Nature (PPSN X), volume 5199 of LNCS, pages 651–660. Springer, 2008
in 2-D and 3-D, see Emmerich and Fonseca.
, see Auger, Bader, Brockhoff and Zitzler, 2009. 
: Bringmann and Friedrich.
points maximizing the (weighted) hypervolume indicator, see Auger, Bader, Brockhoff and Zitzler, 2012.