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How to Detect all Maxima of a Function

  • J. Garnier
  • L. Kallel
Part of the Natural Computing Series book series (NCS)

Abstract

The first contribution of this paper is a theoretical investigation of a family of landscapes characterized by the number of their local optima N and the distribution of the sizes (α j ) of their attraction basins. For each landscape case, we give precise estimates of the size of the random sample that ensures that at least one point lies in each basin of attraction. A practical methodology is then proposed for identifying these quantities (N and (α j ) distribution) for an unknown landscape, given a random sample on that landscape and a local steepest ascent search. This methodology can be applied to any landscape specified with a modification operator and provides bounds on search complexity to detect all local optima. Experiments demonstrate the efficiency of this methodology for guiding the choice of modification operators, eventually leading to the design of problem-dependent optimnization heuristics.

Keywords

Combinatorial complexity local search neighborhood graph randomized starting solution 

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Copyright information

© Springer-Verlag Berlin Heidelberg 2001

Authors and Affiliations

  • J. Garnier
    • 1
  • L. Kallel
    • 1
  1. 1.Centre de Mathématiques AppliquéesUMR CNRS 7641, Ecole PolytechniquePalaiseau CedexFrance

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