On set-based multiobjective optimization
IEEE Transactions on Evolutionary Computation
Convergence of hypervolume-based archiving algorithms I: effectiveness
Proceedings of the 13th annual conference on Genetic and evolutionary computation
Convergence of hypervolume-based archiving algorithms ii: competitiveness
Proceedings of the 14th annual conference on Genetic and evolutionary computation
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In this paper, we study bounds for the α-approximate effectiveness of non-decreasing (μ+λ)-archiving algorithms that optimize the hypervolume. A (μ+λ)-archiving algorithm defines how μ individuals are to be selected from a population of μ parents and λ offspring. It is non-decreasing if the μ new individuals never have a lower hypervolume than the μ original parents. An algorithm is α-approximate if for any optimization problem and for any initial population, there exists a sequence of offspring populations for which the algorithm achieves a hypervolume of at least 1/α times the maximum hypervolume. Bringmann and Friedrich (GECCO 2011, pp. 745---752) have proven that all non-decreasing, locally optimal (μ+1)-archiving algorithms are (2+ε)-approximate for any ε0. We extend this work and substantially improve the approximation factor by generalizing and tightening it for any choice of λ to α=2−(λ−p)/μ with μ=q·λ−p and 0≤p≤λ−1. In addition, we show that $1 + \frac{1}{2 \lambda}-\delta$, for λμ and for any δ0, is a lower bound on α, i.e. there are optimization problems where one can not get closer than a factor of 1/α to the optimal hypervolume.