On Confidence Intervals for the Number of Local Optima

  • Anton V. Eremeev
  • Colin R. Reeves
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 2611)


The number of local optima is an important indicator of optimization problem difficulty for local search algorithms. Here we will discuss some methods of finding the confidence intervals for this parameter in problems where the large cardinality of the search space does not allow exhaustive investigation of solutions. First results are reported that were obtained by using these methods for NK landscapes, and for the low autocorrelation binary sequence and vertex cover problems.


Local Search Local Optimum Repetition Time Maximum Clique Local Search Algorithm 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Tovey, C.A.: Local Improvement on Discrete Structures. In: Aarts, E., Lenstra, J. K. (eds.): Local Search in Combinatorial Optimization. John Wiley éamp; Sons (1997) 57–90Google Scholar
  2. 2.
    Reeves, C.R.: The Crossover Landscape and the Hamming Landscape for Binary Search Spaces. To appear in Proc. of Foundations of Genetic Algorithms-VII. (2003)Google Scholar
  3. 3.
    Reeves C.R.: Fitness Landscapes and Evolutionary Algorithms. In: Fonlupt, C., Hao, J-K., Lutton, E., Ronald, E. and Schoenauer, M. (eds.): Artificial Evolution: 4th European Conference. Lecture Notes in Computer Science, Vol. 1829. Springer-Verlag, Berlin (2000) 3–20Google Scholar
  4. 4.
    Stadler, P.F.: Towards a Theory of Landscapes. In: Lopéz-Peña, R., Capovilla, R., García-Pelayo, R., Waelbroeck, H. and Zertuche, F. (eds.): Complex Systems and Binary Networks. Springer-Verlag, Berlin (1995) 77–163Google Scholar
  5. 5.
    Reeves, C.R.: Landscapes, Operators and Heuristic Search. Annals of Operations Research. 86 (1999) 473–490zbMATHCrossRefMathSciNetGoogle Scholar
  6. 6.
    Boese, K.D., Kahng, A.B., Muddu S.: A New Adaptive Multi-Start Technique for Combinatorial Global Optimizations. Operations Research Letters. 16 (1994) 101–113zbMATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    Reeves, C.R.: Estimating the Number of Optima in a Landscape, Part I: Statistical Principles. Coventry University Technical Report SOR#01-03 (2001)Google Scholar
  8. 8.
    Garnier, J., Kallel, L.: How to Detect All Maxima of a Function? In: Proceedings of the Second EVONET Summer School on Theoretical Aspects of Evolutionary Computing (Anvers, 1999). Springer, Berlin (2001) 343–370Google Scholar
  9. 9.
    Reeves, C.R.: Estimating the Number of Optima in a Landscape, Part II: Experimental Investigations. Coventry University Technical Report SOR#01-04 (2001)Google Scholar
  10. 10.
    Eremeev, A.V., Reeves C.R.: Non-Parametric Estimation of Properties of Combinatorial Landscapes. In: Cagnoni, S., Gottlieb, J., Hart, E., Middendorf, M. and Raidl, G. (eds.): Applications of Evolutionary Computing: Proceedings of EvoWorkshops 2002. Lecture Notes in Computer Science, Vol. 2279. Springer-Verlag, Berlin Heidelberg (2002) 31–40Google Scholar
  11. 11.
    Seber G.A.F.: The Estimation of Animal Abundance. Charles Griffin, London (1982)Google Scholar
  12. 12.
    Mood, A. M., Graybill, F.A., Boes, D.C.: Introduction to the Theory of Statistics. 3rd edn. McGraw-Hill, New York (1973)Google Scholar
  13. 13.
    Johnson, N.L., Kotz, S.: Discrete Distributions. Wiley, New York (1969)zbMATHGoogle Scholar
  14. 14.
    Liu, C.L.: Introduction to Combinatorial Mathematics. McGraw-Hill, New York (1968)zbMATHGoogle Scholar
  15. 15.
    Kauffman, S.A.: Adaptation on Rugged Fitness Landscapes. In: Lectures in the Sciences of Complexity, Vol. I of SFI studies, Addison-Wesley (1989) 619–712Google Scholar
  16. 16.
    Golay, M.J. E.: Series for Low-Autocorrelation Binary Sequences. IEEE Trans. Inform. Theory. 23 (1977) 43–51zbMATHCrossRefGoogle Scholar
  17. 17.
    Johnson, D.S., Trick, M.A.: Introduction to the Second DIMACS Challenge: Cliques, Coloring, and Satisfiability. In: Johnson, D.S., Trick, M.A. (eds.): Cliques, Coloring, and Satisfiability. DIMACS Series in Discrete Mathematics and Theoretical Computer Science, Vol. 26. AMS (1996) 1–10Google Scholar
  18. 18.
    Garey, M.R., Johnson, D.S.: Computers and Intractability. A Guide to the Theory of N P-Completeness. W.H. Freeman and Company, San Francisco (1979)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2003

Authors and Affiliations

  • Anton V. Eremeev
    • 1
  • Colin R. Reeves
    • 2
  1. 1.Omsk Branch of Sobolev Institute of MathematicsOmskRussia
  2. 2.School of Mathematicalé amp; Information SciencesCoventry UniversityCoventryUK

Personalised recommendations