Diophantine Benchmarks for the B-Cell Algorithm

  • P. Bull
  • A. Knowles
  • G. Tedesco
  • A. Hone
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 4163)


The B-cell algorithm (BCA) due to Kelsey and Timmis is a function optimization algorithm inspired by the process of somatic mutation of B cell clones in the natural immune system. So far, the BCA has been shown to be perform well in comparison with genetic algorithms when applied to various benchmark optimisation problems (finding the optima of smooth real functions). More recently, the convergence of the BCA has been shown by Clark, Hone and Timmis, using the theory of Markov chains. However, at present the theory does not predict the average number of iterations that are needed for the algorithm to converge. In this paper we present some empirical convergence results for the BCA, using a very different non-smooth set of benchmark problems. We propose that certain Diophantine equations, which can be reformulated as an optimization problem in integer programming, constitute a much harder set of benchmarks for evolutionary algorithms. In the light of our empirical results, we also suggest some modifications that can be made to the BCA in order to improve its performance.


Markov Chain Model Diophantine Equation Natural Immune System Diophantine Problem Smooth Real Function 
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.


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  1. [Andrews 2006]
    Andrews, P.: Private communication (2006), opt-aiNet code Available at:
  2. [Burger 2000]
    Burger, E.: Exploring the Number Jungle: a Journey into Diophantine Analysis. American Mathematical Society, Providence (2000)zbMATHGoogle Scholar
  3. [de Castro and Timmis 2002a]
    de Castro, L., Timmis, J.: An Artificial Immune Network for Multimodal Function Optimisation. In: Proceedings of IEEE World Congress on Evolutionary Computation, pp. 669–674 (2002)Google Scholar
  4. [de Castro and Timmis 2002b]
    de Castro, L., Timmis, J.: Artificial Immune Systems: A New Computational Intelligence Approach. Springer, Heidelberg (2002)zbMATHGoogle Scholar
  5. [Clark et al. 2005]
    Clark, E., Hone, A., Timmis, J.: A Markov Chain Model of the B-Cell Algorithm. In: Jacob, C., Pilat, M.L., Bentley, P.J., Timmis, J.I. (eds.) ICARIS 2005. LNCS, vol. 3627, pp. 318–330. Springer, Heidelberg (2005)CrossRefGoogle Scholar
  6. [Dasgupta and McGregor 1992]
    Dasgupta, D., McGregor, D.R.: Nonstationary Function Optimization using the Structured Genetic Algorithm, Parallel Problem Solving from Nature 2, Proc. In: 2nd Int. Conf. on Parallel Problem Solving from Nature, Brussels, pp. 145–154. Elsevier, Amsterdam (1992)Google Scholar
  7. [Dasgupta 1994]
    Dasgupta, D.: Handling Deceptive Problems Using a different Genetic Search. In: Proceedings of the First IEEE Conference on Evolutionary Computation 1994, IEEE World Congress on Computational Intelligence, pp. 807–811 (1994)Google Scholar
  8. [De Jong 1992]
    De Jong, K.: Are genetic algorithms function optimizers? Parallel Problem Solving from Nature 2. In: Proceedings of the Second Conference on Parallel Problem Solving from Nature, pp. 3–13. North-Holland, Brussels (1992)Google Scholar
  9. [Dyer et al. 2006]
    Dyer, M., Goldberg, L.A., Jerrum, M., Martin, R.: Markov chain comparison. Probability Surveys 3, 89–111 (2006)CrossRefMathSciNetzbMATHGoogle Scholar
  10. [Gale 1991]
    Gale, D.: The strange and surprising saga of the Somos sequences. Mathematical Intelligence 13 (1), 40–42 (1991)CrossRefGoogle Scholar
  11. [Hone and Kelsey 2004]
    Hone, A., Kelsey, J.: Optima, extrema, and artificial immune systems. In: Nicosia, G., Cutello, V., Bentley, P.J., Timmis, J. (eds.) ICARIS 2004. LNCS, vol. 3239, pp. 80–90. Springer, Heidelberg (2004)CrossRefGoogle Scholar
  12. [Hone 2006]
    Hone, A.N.W.: Diophantine non-integrability of a third order recurrence with the Laurent property. J. Phys. A: Math. Gen. 39, L171–L177 (2006)CrossRefMathSciNetGoogle Scholar
  13. [Hunter 2003]
    Hunter, J.J.: Mixing Times with Applications to Perturbed Markov Chains. Institute of Information and Mathematical Sciences, Massey University (preprint, 2003)Google Scholar
  14. [Jerrum 2005]
    Jerrum, M.: Algorithmically feasible sampling: what are the limits? Talk at London Mathematical Society meeting. University College, London (October 7, 2005)Google Scholar
  15. [Kelsey and Timmis 2003]
    Kelsey, J., Timmis, J.: Immune Inspired Somatic Contiguous Hypermutation for Function Optimisation. In: Cantú-Paz, E., Foster, J.A., Deb, K., Davis, L., Roy, R., O’Reilly, U.-M., Beyer, H.-G., Kendall, G., Wilson, S.W., Harman, M., Wegener, J., Dasgupta, D., Potter, M.A., Schultz, A., Dowsland, K.A., Jonoska, N., Miller, J., Standish, R.K. (eds.) GECCO 2003. LNCS, vol. 2724, pp. 207–218. Springer, Heidelberg (2003)CrossRefGoogle Scholar
  16. [Kelsey et al. 2003]
    Kelsey, J., Timmis, J., Hone, A.: Chasing Chaos. In: Sarker, R., et al. (eds.) Proceedings of the Congress on Evolutionary Computation, Canberra, Australia, December 2003, pp. 413–419. IEEE, Los Alamitos (2003)Google Scholar
  17. [Kelsey 2006]
    Kelsey, J.: Private communication (2006)Google Scholar
  18. [Krishnakumar 1989]
    Krishnalumar, K.: Micro-genetic algorithms for stationary and non-stationary function optimization. In: SPIE Proceedings: Intelligent Control and Adaptive Systems, pp. 289–296 (1987)Google Scholar
  19. [Lamlum 1999]
    Lamlum, H., et al.: The type of somatic mutation at APC in familial adenomatous polyposis is determined by the site of the germline mutation: a new facet to Knudson’s ’two-hit’ hypothesis. Nature Medicine 5, 1071–1075 (1999)CrossRefGoogle Scholar
  20. [Lydyard et al. 2004]
    Lydyard, P., Whelan, A., Fanger, M.: Immunology, 2nd edn. Taylor & Francis, Abington (2004)Google Scholar
  21. [Manin and Panchishkin 2005]
    Manin, Y.I., Panchishkin, A.A.: Introduction to Modern Number Theory, 2nd edn. Springer, Heidelberg (2005)zbMATHGoogle Scholar
  22. [Mordell 1969]
    Mordell, L.J.: Diophantine Equations. Academic Press, London (1969)zbMATHGoogle Scholar
  23. [Timmis and Edmonds 2004]
    Timmis, J., Edmonds, C.: A Comment on Opt-AiNET: An Immune Network Algorithm for Optimisation. In: Deb, K., et al. (eds.) GECCO 2004. LNCS, vol. 3102, pp. 308–317. Springer, Heidelberg (2004)CrossRefGoogle Scholar
  24. [Villalobos et al. 2004]
    Villalobos-Arias, M., Coello Coello, C.A., Hernández-Lerma, O.: Convergence analysis of a multiobjective artificial immune system algorithm. In: Nicosia, G., Cutello, V., Bentley, P.J., Timmis, J. (eds.) ICARIS 2004. LNCS, vol. 3239, pp. 226–235. Springer, Heidelberg (2004)CrossRefGoogle Scholar
  25. [Vose 1995]
    Vose, M.D.: Modeling simple genetic algorithms. Evolutionary Computation 3(4), 453–472 (1996)CrossRefGoogle Scholar
  26. [Zagier 1982]
    Zagier, D.: On the Number of Markoff Numbers Below a Given Bound. Mathematics of Computation 39(160), 709–723 (1982)zbMATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2006

Authors and Affiliations

  • P. Bull
    • 1
  • A. Knowles
    • 2
  • G. Tedesco
    • 3
  • A. Hone
    • 4
  1. 1.Department of Computer ScienceUniversity of AberystwythAberystwythU.K.
  2. 2.Department of ElectronicsUniversity of YorkU.K.
  3. 3.School of Computer ScienceUniversity of NottinghamNottinghamU.K.
  4. 4.Institute of Mathematics, Statistics & Actuarial ScienceUniversity of KentCanterburyU.K.

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