Skip to main content
SpringerLink
Log in
Book cover

Frontiers of Evolutionary Computation pp 73–102Cite as

Open Problems in the Spectral Analysis of Evolutionary Dynamics

  • Chapter

Part of the book series: Genetic Algorithms and Evolutionary Computation ((GENA,volume 11))

Abstract

The dynamics of evolution can be completely characterized by the spectra of the operators that define the dynamics, under broad classes of selection and genetic operators, in both infinite and finite populations. These classes include frequency-independent selection, uniparental inheritance, and generalized mutation. Several open questions exist regarding these spectra:

  1. 1

    For a given fitness function, what genetic operators and operator intensities are optimal for finding the fittest genotype? The concept of rapid first hitting time, and analog of Sinclair’s “rapidly mixing” Markov chains, is examined.

  2. 2

    What is the relationship between the spectra of deterministic infinite population models, and the spectra of the Markov processes derived from them in the case of finite populations?

  3. 3

    Karlin proved a fundamental relationship between selection, rates of transformation under genetic operators, and the consequent asymptotic mean fitness o the population. Developed to analyze the stability of polymorphisms in subdivided populations, the theorem has been applied to unify the reduction principle for self-adaptation, and has other applications as well. Many other problems could be solved if it were generalized to account for the interaction of different genetic operators. Can Karlin’s theorem on operator intensity be extended to account for mixed genetic operators?

This is a preview of subscription content, log in via an institution.

Chapter
USD   29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD   84.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD   109.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD   109.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Learn about institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  • Ackley, D. H. (1987). A Connectionist Machine for Genetic Hill climbing, volume SECS28 of The Kluwer International Series in Engineering and Computer Science. Kluwer Academic Publishers, Boston.

    Google Scholar 

  • Altenberg, L. (1984). A Generalization of Theory on the Evolution of Modifier Genes. PhD thesis, Stanford University. Available from University Microfilms, Ann Arbor, MI.

    Google Scholar 

  • Altenberg, L. (1995). The Schema Theorem and Price’s Theorem. In Whitley, D. and Vose, M. D., editors, Foundations of Genetic Algorithms 3, pages 23–49. Morgan Kaufmann, San Mateo, CA.

    Google Scholar 

  • Altenberg, L. and Feldman, M. W. (1987). Selection, generalized transmission and the evolution of modifier genes. I. The reduction principle. Genetics, 117:559–572.

    Google Scholar 

  • Arora, S., Rabani, Y., and Vazirani, U. (1994). Simulating quadratic dynamical systems is PSPACE-complete. In Proceedings of the 26th Annual ACM Symposium on Theory of Computing, pages 459–467.

    Google Scholar 

  • Bäck, T. (1996). Evolutionary Algorithms in Theory and Practice: Evolutionary Strategies, Evolutionary Programming and Genetic Programming. Oxford University Press, Oxford.

    Google Scholar 

  • Cannings, C. (1974). The latent roots of certain Markov chains arising in genetics: a new approach, I. haploid models. Advances in Applied Probability, 6:260–290.

    MATH  MathSciNet  Google Scholar 

  • Christiansen, F. B. (2000). Population Genetics of Multiple Loci. John Wiley and Sons, LTD, Chichester.

    Google Scholar 

  • Davis, T. E. and Principe, J. C. (1993). A Markov chain framework for the simple genetic algorithm. Evolutionary Computation, l(3):269–288.

    Google Scholar 

  • Donsker, M. D. and Varadhan, S. R. S. (1975). On a variational formula for the principal eigenvalue for operators with maximum principle. Proceedings of the National Academy of Science, USA, 72:780–783.

    MathSciNet  Google Scholar 

  • Ewens, W. J. (1979). Mathematical Population Genetics. Springer-Verlag, Berlin.

    Google Scholar 

  • Feller, W. (1951). Diffusion processes in genetics. In Neyman, J., editor, Proceedings of the Second Berkeley Symposium on Mathematical Statistics and Probability, pages 227–246. University of California Press, Berkeley.

    Google Scholar 

  • Fisher, R. A. (1930). The Genetical Theory of Natural Selection. Clarendon Press, Oxford.

    Google Scholar 

  • Gantmacher, F. R. (1959). The Theory of Matrices, volume 2. Chelsea Publishing Company, New York.

    Google Scholar 

  • Goldberg, David E. and Deb, Kalyanmoy (1991). A comparative analysis of selection schemes used in genetic algorithms. In Rawlins, G., editor, Foundations of Genetic Algorithms, pages 69–93. Morgan Kaufmann, San Mateo, CA.

    Google Scholar 

  • Goldberg, David E. and Segrest, P. (1987). Finite Markov chain analysis of genetic algorithms. In Proceedings of the Second International Conference on Genetic Algorithms, pages 1–8.

    Google Scholar 

  • Horn, J., Goldberg, David E., and Deb, Kalyanmoy (1994). Long path problems. In Schwefel, H. P. and R. Männer, editors, Parallel Problem Solving from Nature—PPSN III, volume 866, Berlin. Springer-Verlag.

    Google Scholar 

  • Karlin, S. (1982). Classification of selection-migration structures and conditions for a protected polymorphism. In Hecht, M. K., Wallace, B., and Prance, G. T., editors, Evolutionary Biology, volume 14, pages 61–204. Plenum Publishing Corporation.

    Google Scholar 

  • Kondrashov, A. S. (1988). Deleterious mutations and the evolution of sexual reproduction. Nature (London), 336:435–440.

    Article  Google Scholar 

  • Liepins, G. and Vose, M. D. (1990). Representational issues in genetic optimization. Journal of Experimental and Theoretical Artificial Intelligence, 2(2):101–115.

    Google Scholar 

  • Nix, A. E. and Vose, M. D. (1991). Modeling genetic algorithms with Markov chains Annals of Mathematics and Artificial Intelligence, 5:79–88.

    MathSciNet  Google Scholar 

  • Palmer, R. G. (1982). Broken ergodicity. Advances in Physics, 31:669–735.

    Article  Google Scholar 

  • Rabani, Y., Rabinovich, Y., and Sinclair, A. (1995). A computational view of population genetics. In Annual ACM Symposium on the Theory of Computing, pages 83–92.

    Google Scholar 

  • Rabinovich, Y., Sinclair, A., and Wigderson, A. (1992). Quadratic dynamical systems. In IEEE Symposium on Foundations of Computer Science, pages 304–313.

    Google Scholar 

  • Rabinovich, Y. and Wigderson, A. (1999). Techniques for bounding the convergence rate of genetic algorithms. Random Structures Algorithms, 14:111–138.

    Article  MathSciNet  Google Scholar 

  • Rudolph, G. (1997). Convergence properties of evolutionary algorithms. Verlag Kovač, Hamburg.

    Google Scholar 

  • Schmitt, F. and Rothlauf, F. (2001a). On the importance of the second largest eigenvalue on the convergence rate of genetic algorithms. In Spector, L., Goodman, E. D., Wu, A., Langdon, W. B., Voigt, H.-M., Gen, M., Sen, S., Dorigo, M., Pezeshk, S., Garzon, M. H., and Burke, E., editors, Proceedings of the Genetic and Evolutionary Computation Conference (GECCO-2001), pages 559–564, San Francisco, California, USA. Morgan Kaufmann.

    Google Scholar 

  • Schmitt, F. and Rothlauf, F. (2001b). On the mean of the second largest eigenvalue on the convergence rate of genetic algorithms. Technical Report Working Paper 1/2001, University of Bayreuth, Department of Information Systems, Universitaetsstrasse 30, D-95440 Bayreuth, Germany. Working Papers in Information Systems.

    Google Scholar 

  • Schwefel, H.-P. (1987). Collective phenomena in evolutionary systems. Preprints of the 31st Annual Meeting of the International Society for General Systems Research, Budapest, 2:1025–1033.

    Google Scholar 

  • Sinclair, A. (1992). Algorithms for random generation and counting: A Markov chain approach. Birkhäuser, Boston.

    Google Scholar 

  • Suzuki, J. (1995). A Markov chain analysis on simple genetic algorithms. IEE Transactions on Systems, Man and Cybernetics, 25(4):655–659.

    Google Scholar 

  • van Nimwegen, E. J. (1999). The Statistical Dynamics of Epochal Evolution. PhD thesis, Universiteit Utrecht, Amsterdam.

    Google Scholar 

  • van Nimwegen, E. J., Crutchfield, J. P., and Huynen, M. (1999). Metastable evolutionary dynamics: Crossing fitness barriers or escaping via neutral paths? Bulletin of Mathematical Biology, 62:799–848.

    Google Scholar 

  • Vitanyi, P. (2000). A discipline of evolutionary programming. Theoretical Computer Science, 241(1–2):3–23.

    MATH  MathSciNet  Google Scholar 

  • Wolpert, D. H. and Macready, W. G. (1995). No free lunch theorems for search. Technical Report SFI-TR-95-02-010, Santa Fe Institute, Santa Fe, NM.

    Google Scholar 

  • Wolpert, D. H. and Macready, W. G. (1997). No free lunch theorems for optimization. IEEE Transactions on Evolutionary Computation, I(l):67–82.

    Google Scholar 

  • Wright, S. (1931). Evolution in Mendelian populations. Genetics, 16:97–159.

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Information and Computer Sciences, University of Hawai’i at Manoa, Honolulu, HI, USA

    Lee Altenberg

Authors
  1. Lee Altenberg
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Editors and Affiliations

  1. Product Soft, Inc., Pittsburgh, Pennsylvania, USA

    Anil Menon

Rights and permissions

Reprints and permissions

Copyright information

© 2004 Kluwer Academic Publishers

About this chapter

Cite this chapter

Altenberg, L. (2004). Open Problems in the Spectral Analysis of Evolutionary Dynamics. In: Menon, A. (eds) Frontiers of Evolutionary Computation. Genetic Algorithms and Evolutionary Computation, vol 11. Springer, Boston, MA. https://doi.org/10.1007/1-4020-7782-3_4

Download citation

  • DOI: https://doi.org/10.1007/1-4020-7782-3_4

  • Publisher Name: Springer, Boston, MA

  • Print ISBN: 978-1-4020-7524-7

  • Online ISBN: 978-1-4020-7782-1

  • eBook Packages: Springer Book Archive

Publish with us

Policies and ethics

Search

Navigation