Logarithmic-Time Updates in SMS-EMOA and Hypervolume-Based Archiving
The hypervolume indicator is frequently used in selection procedures of evolutionary multi-criterion optimization algorithms (EMOA) and in bounded size archivers for Pareto non-dominated points. We propose and study an algorithm that updates all hypervolume contributions and identifies a minimal hypervolume contributor after the removal or insertion of a single point in ℝ2 in amortized time complexity O(logn). This algorithm will be tested for the efficient update of bounded-size archives and for a fast implementation of the steady state selection in the bi-criterion SMS-EMOA. To achieve an amortized time complexity of O(logn) for SMS-EMOA iterations a constant-time update method for establishing a ranking among dominated solutions is suggested as an alternative to non-dominated sorting. Besides the asymptotical analysis, we discuss empirical results on several test problems and discuss the impact of the overhead caused by maintaining additional AVL tree data structures, including scalability studies with very large population size that will yield high resolution approximations.
Unable to display preview. Download preview PDF.
- 7.Deb, K., Pratap, A., Agarwal, S., Meyarivan, T.: A fast and elitist multi-objective genetic algorithm, Nsga-ii (2000)Google Scholar
- 8.Emmerich, M., Beume, N., Naujoks, B.: An EMO algorithm using the hypervolume measure as selection criterion. In: 2005 Intl. Conference, pp. 62–76. Springer (March 2005)Google Scholar
- 10.Fonseca, C.M., Paquete, L., Lopez-Ibanez, M.: An improved dimension-sweep algorithm for the hypervolume indicator, pp. 1157–1163 (July 2006)Google Scholar
- 11.Landis, E.M., Adelson-Velskii, G.: An algorithm for the organization of information. Proceedings of the USSR Academy of Sciences 146, 263–266Google Scholar
- 12.Guerreiro, A.P., Fonseca, C.M., Emmerich, M.T.M.: A fast dimension-sweep algorithm for the hypervolume indicator in four dimensions. In: CCCG, pp. 77–82 (2012)Google Scholar
- 14.Knowles, J.D., Corne, D.W., Fleischer, M.: Bounded archiving using the Lebesgue measure. In: Proceedings of the IEEE Congress on Evolutionary Computation, pp. 2490–2497. IEEE Press (2003)Google Scholar
- 15.Naujoks, B., Beume, N., Emmerich, M.T.M.: Multi-objective optimisation using S-metric selection: application to three-dimensional solution spaces, vol. 2, pp. 1282–1289 (September 2005)Google Scholar