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Recent theoretical works have shown that the NSGA-II efficiently computes thefull Pareto front when the population size is large enough. In this work, westudy how well it approximates the Pareto front when the population size issmaller.  For the OneMinMax benchmark, we point out situations in which the parents andoffspring cover well the Pareto front, but the next population has large gapson the Pareto front. Our mathematical proofs suggest as reason for thisundesirable behavior that the NSGA-II in the selection stage computes thecrowding distance once and then removes individuals with smallest crowdingdistance without considering that a removal increases the crowding distance ofsome individuals.  We then analyze two variants not prone to this problem. For the NSGA-II thatupdates the crowding distance after each removal (Kukkonen and Deb (2006)) andthe steady-state NSGA-II (Nebro and Durillo (2009)), we prove that the gaps inthe Pareto front are never more than a small constant factor larger than thetheoretical minimum. This is the first mathematical work on the approximationability of the NSGA-II and the first runtime analysis for the steady-stateNSGA-II. Experiments also show the superior approximation ability of the twoNSGA-II variants.

Approximation Guarantees for the Non-Dominated Sorting Genetic Algorithm II (NSGA-II)