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Black-Box Search by Unbiased Variation

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Abstract

The complexity theory for black-box algorithms, introduced by Droste, Jansen, and Wegener (Theory Comput. Syst. 39:525–544, 2006), describes common limits on the efficiency of a broad class of randomised search heuristics. There is an obvious trade-off between the generality of the black-box model and the strength of the bounds that can be proven in such a model. In particular, the original black-box model provides for well-known benchmark problems relatively small lower bounds, which seem unrealistic in certain cases and are typically not met by popular search heuristics.

In this paper, we introduce a more restricted black-box model for optimisation of pseudo-Boolean functions which we claim captures the working principles of many randomised search heuristics including simulated annealing, evolutionary algorithms, randomised local search, and others. The key concept worked out is an unbiased variation operator. Considering this class of algorithms, significantly better lower bounds on the black-box complexity are proved, amongst them an Ω(nlogn) bound for functions with unique optimum. Moreover, a simple unimodal function and plateau functions are considered. We show that a simple (1+1) EA is able to match the runtime bounds in several cases.

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Authors and Affiliations

  1. School of Computer Science, University of Nottingham, Jubilee Campus, Wollaton Road, Nottingham, NG8 1BB, UK

    Per Kristian Lehre

  2. Technical University of Denmark, Kgs. Lyngby, Denmark

    Carsten Witt

Authors
  1. Per Kristian Lehre
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  2. Carsten Witt
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Corresponding author

Correspondence to Per Kristian Lehre.

Additional information

This work is based on earlier work in [21].

P.K. Lehre was supported by EPSRC under grant no. EP/D052785/1, and Deutsche Forschungsgemeinschaft (DFG) under grant no. WI 3552/1-1.

C. Witt was supported by Deutsche Forschungsgemeinschaft (DFG) under grant no. WI 3552/1-1.

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Lehre, P.K., Witt, C. Black-Box Search by Unbiased Variation. Algorithmica 64, 623–642 (2012). https://doi.org/10.1007/s00453-012-9616-8

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  • DOI: https://doi.org/10.1007/s00453-012-9616-8

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