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Convergence of Hypervolume-Based Archiving Algorithms I: Effectiveness

MPS-Authors
http://pubman.mpdl.mpg.de/cone/persons/resource/persons44182

Bringmann,  Karl
Algorithms and Complexity, MPI for Informatics, Max Planck Society;

http://pubman.mpdl.mpg.de/cone/persons/resource/persons44447

Friedrich,  Tobias
Algorithms and Complexity, MPI for Informatics, Max Planck Society;

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Citation

Bringmann, K., & Friedrich, T. (2011). Convergence of Hypervolume-Based Archiving Algorithms I: Effectiveness. In N. Krasnogor, & P. L. Lanzim (Eds.), GECCO 2011 (pp. 745-752). New York, NY: ACM. doi:10.1145/2001576.2001678.


Cite as: http://hdl.handle.net/11858/00-001M-0000-0010-120B-F
Abstract
The core of hypervolume-based multi-objective evolutionary algorithms is an archiving algorithm which performs the environmental selection. A (μ+λ)- archiving algorithm defines how to choose μ children from μ parents and λ offspring together. We study theoretically (μ+λ)-archiving algorithms which never decrease the hypervolume from one generation to the next. Zitzler, Thiele, and Bader (IEEE Trans. Evolutionary Computation, 14:58-79, 2010) proved that all (μ+1)-archiving algorithms are ineffective, which means there is an initial population such that independent of the used reproduction rule, a set with maximum hypervolume cannot be reached. We extend this and prove that for λ<μ all archiving algorithms are ineffective. On the other hand, locally optimal algorithms, which maximize the hypervolume in each step, are effective for λ=μ and can always find a population with hypervolume at least half the optimum for λ<μ. We also prove that there is no hypervolume-based archiving algorithm which can always find a population with hypervolume greater than 1/(1+0.1338(1/λ-1/μ)) times the optimum.