Non‐monotonicity of Observed Hypervolume in 1‐Greedy S‐Metric Selection
Leonard Judt, Olaf Mersmann, Boris Naujoks · Journal of Multi-Criteria Decision Analysis · 2012
ABSTRACT The progression of the dominated hypervolume in the course of the optimization process, with respect to a global reference point, is thought to be monotonically increasing. This intuition is based on the observation that in each iteration, the solution that contributes the least to the dominated hypervolume is eliminated. Derived from results of multiple optimization runs with incorporated reference point adaptation, we show that this does not always hold for two‐dimensional and three‐dimensional objective spaces. For the two‐dimensional case, this is because the two boundary solutions are always retained in the population regardless of their hypervolume contribution. For the three‐dimensional case, we are able to show that the cause of the drop in dominated hypervolume is the continuous adaption of the reference point during selection. Copyright © 2012 John Wiley & Sons, Ltd.