Skip to main content

Advertisement

SpringerLink
Log in

Sporadic model building for efficiency enhancement of the hierarchical BOA

  • Original Paper
  • Published:
Genetic Programming and Evolvable Machines Aims and scope Submit manuscript

Abstract

Efficiency enhancement techniques—such as parallelization and hybridization—are among the most important ingredients of practical applications of genetic and evolutionary algorithms and that is why this research area represents an important niche of evolutionary computation. This paper describes and analyzes sporadic model building, which can be used to enhance the efficiency of the hierarchical Bayesian optimization algorithm (hBOA) and other estimation of distribution algorithms (EDAs) that use complex multivariate probabilistic models. With sporadic model building, the structure of the probabilistic model is updated once in every few iterations (generations), whereas in the remaining iterations, only model parameters (conditional and marginal probabilities) are updated. Since the time complexity of updating model parameters is much lower than the time complexity of learning the model structure, sporadic model building decreases the overall time complexity of model building. The paper shows that for boundedly difficult nearly decomposable and hierarchical optimization problems, sporadic model building leads to a significant model-building speedup, which decreases the asymptotic time complexity of model building in hBOA by a factor of \(\Uptheta(n^{0.26})\) to \(\Uptheta(n^{0.5}),\) where n is the problem size. On the other hand, sporadic model building also increases the number of evaluations until convergence; nonetheless, if model building is the bottleneck, the evaluation slowdown is insignificant compared to the gains in the asymptotic complexity of model building. The paper also presents a dimensional model to provide a heuristic for scaling the structure-building period, which is the only parameter of the proposed sporadic model-building approach. The paper then tests the proposed method and the rule for setting the structure-building period on the problem of finding ground states of 2D and 3D Ising spin glasses.

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

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10

Similar content being viewed by others

References

  1. M. Abramowitz, I.A. Stegun (eds.), Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables (Dover publications, Ninth printing, 1972)

  2. D.H. Ackley, An empirical study of bit vector function optimization. Genet. Algor. Simulat. Anneal. 170–204 (1987)

  3. L.A. Albert, Efficient Genetic Algorithms using Discretization Scheduling. Master’s thesis, University of Illinois at Urbana-Champaign, Department of General Engineering, Urbana, IL, 2001

  4. T.F.H. Allen, T. Starr (eds.), Hierarchy: Perspectives for Ecological Complexity (University of Chicago Press, Chicago, IL, 1982)

    Google Scholar 

  5. S. Baluja, Population-based Incremental Learning: A Method for Integrating Genetic Search Based Function Optimization and Competitive Learning (Tech. Rep. No. CMU-CS-94-163). (Carnegie Mellon University, Pittsburgh, PA, 1994)

  6. S. Baluja, Using a priori knowledge to create probabilistic models for optimization. Int. J. Approx. Reason. 31(3), 193–220 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  7. E. Bengoetxea, Inexact Graph Matching using Estimation of Distribution Algorithms. Doctoral dissertation (Department of Image and Signal Treatment, Ecole Nationale Superieure des Telecommunications (ENST), Paris, France, 2002)

  8. K. Binder, A. Young, Spin-glasses: experimental facts. theoretical concepts and open questions. Rev. Mod. Phys. 58, 801 (1986)

    Article  Google Scholar 

  9. P.A.N. Bosman, D. Thierens, Linkage information processing in distribution estimation algorithms. Proceedings of the Genetic and Evolutionary Computation Conference (GECCO-99), I, 60–67 (1999)

  10. E. Cantú-Paz, Efficient and Accurate Parallel Genetic Algorithms (Kluwer, Boston, MA, 2000)

    MATH  Google Scholar 

  11. D.M. Chickering, D. Heckerman, C. Meek, A Bayesian Approach to Learning Bayesian Networks with Local Structure (Technical Report MSR-TR-97-07). (Microsoft Research, Redmond, WA, 1997)

  12. P. Dayal, S. Trebst, S. Wessel, D. Würtz, M. Troyer, S. Sabhapandit, S. Coppersmith, Performance limitations of flat histogram methods and optimality of Wang-Langdau sampling. Phys. Rev. Lett. 92(9), 097201 (2004)

    Article  Google Scholar 

  13. K. Deb, D.E. Goldberg, Analyzing Deception in Trap Functions (IlliGAL Report No. 91009). (University of Illinois at Urbana-Champaign, Illinois Genetic Algorithms Laboratory, Urbana, IL, 1991)

  14. K. Deb, D.E. Goldberg, Sufficient conditions for deceptive and easy binary functions. Ann. Math. Artif. Intell. 10, 385–408 (1994)

    Article  MATH  MathSciNet  Google Scholar 

  15. R. Etxeberria, P. Larrañaga, Global optimization using Bayesian networks. Second Symposium on Artificial Intelligence (CIMAF-99), 332–339 (1999)

  16. Y. Faihe, Hierarchical Problem Solving Using Reinforcement Learning : Methodology and Methods. Doctoral dissertation, (University of Neuchâtel, Neuchâtel, Switzerland, 1999)

  17. W. Feller, An Introduction to Probability Theory and its Applications (Wiley, New York, NY, 1970)

    Google Scholar 

  18. K. Fischer, J. Hertz, Spin Glasses (Cambridge University Press, Cambridge, 1991)

    Google Scholar 

  19. S. Fischer, I. Wegener, The Ising model on the ring: Mutation versus recombination. Proceedings of the Genetic and Evolutionary Computation Conference (GECCO-2004), 1113–1124 (2004)

  20. N. Friedman, M. Goldszmidt, Learning Bayesian networks with local structure. in Graphical Models, ed. by M.I. Jordan (MIT Press, Cambridge, MA, 1999), pp. 421–459

    Google Scholar 

  21. N. Friedman, Z. Yakhini, On the sample complexity of learning Bayesian networks. in Proceedings of the Conference on Uncertainty in Artificial Intelligence (UAI-96), ed. by E. Horvitz, F. Jensen (Morgan Kaufmann Publishers, San Francisco, 1996), pp. 274–282

    Google Scholar 

  22. A. Galluccio, M. Loebl, A theory of Pfaffian orientations. I. Perfect matchings and permanents. Electron. J. Combinatorics 6(1). Research Paper 6 (1999a)

  23. A. Galluccio, M. Loebl, A theory of Pfaffian orientations. II. T-joins, k-cuts, and duality of enumeration. Electron. J. Combinatorics, 6(1). Research Paper 7 (1999b)

  24. D.E. Goldberg, Simple genetic algorithms and the minimal, deceptive problem. in: Genetic Algorithms and Simulated Annealing, Ch. 6, ed. by L. Davis (Morgan Kaufmann, Los Altos, CA, 1987), pp. 74–88

    Google Scholar 

  25. D.E. Goldberg, Genetic Algorithms in Search, Optimization, and Machine Learning (Addison-Wesley, Reading, MA, 1989)

    MATH  Google Scholar 

  26. D.E. Goldberg, Using time efficiently: Genetic-evolutionary algorithms and the continuation problem. Proceedings of the Genetic and Evolutionary Computation Conference (GECCO-99), 212–219 (1999)

  27. D.E. Goldberg, The Design of Innovation: Lessons from and for Competent Genetic Algorithms, Volume 7 of Genetic Algorithms and Evolutionary Computation. (Kluwer Academic Publishers, 2002)

  28. D.E. Goldberg, K. Deb, J.H. Clark, Genetic algorithms, noise, and the sizing of populations. Complex Syst.6, 333–362 (1992)

    MATH  Google Scholar 

  29. D.E. Goldberg, S. Voessner, Optimizing global-local search hybrids. Proceedings of the Genetic and Evolutionary Computation Conference (GECCO-99), 220–228 (1999)

  30. G. Harik, Linkage learning via probabilistic modeling in the ECGA (IlliGAL Report No. 99010). (University of Illinois at Urbana-Champaign, Illinois Genetic Algorithms Laboratory, Urbana, IL,1999)

  31. G. Harik, Cantú-Paz E., D.E. Goldberg, B.L. Miller, The gambler’s ruin problem, genetic algorithms, and the sizing of populations. Evol. Comput. 7(3), 231–253 (1999)

    Google Scholar 

  32. G.R. Harik, Finding multimodal solutions using restricted tournament selection. Proceedings of the International Conference on Genetic Algorithms (ICGA-95), 24–31 (1995)

  33. A.K. Hartmann, Cluster-exact approximation of spin glass ground states. Physica A 224, 480 (1996)

    Article  Google Scholar 

  34. A.K. Hartmann, Ground-state clusters of two, three and four-dimensional ± J Ising spin glasses. Phys. Rev. E 63, 016106 (2001)

    Article  Google Scholar 

  35. A.K. Hartmann, H. Rieger, Optimization Algorithms in Physics (Wiley-VCH, Weinheim, 2001)

    Google Scholar 

  36. A.K. Hartmann, H. Rieger (eds.), New Optimization Algorithms in Physics (Wiley-VCH, Weinheim, 2004)

    MATH  Google Scholar 

  37. A.K. Hartmann, M. Weigt, Phase Transitions in Combinatorial Optimization Problems (Wiley-VCH, Weinheim, 2005)

    MATH  Google Scholar 

  38. D. Heckerman, D. Geiger, D.M. Chickering, Learning Bayesian networks: The combination of knowledge and statistical data (Technical Report MSR-TR-94-09). (Microsoft Research, Redmond, WA, 1994)

  39. M. Henrion, Propagating uncertainty in Bayesian networks by probabilistic logic sampling. in Uncertainty in Artificial Intelligence, ed. by J.F. Lemmer, L.N. Kanal (Elsevier, Amsterdam, London, New York, 1988), pp. 149–163

    Google Scholar 

  40. J.H. Holland, Adaptation in Natural and Artificial Systems (University of Michigan Press, Ann Arbor, MI, 1975)

    Google Scholar 

  41. R. Höns, Estimation of distribution algorithms and minimum relative entropy. Doctoral dissertation (University of Bonn, Germany, 2005)

  42. R.A. Howard, J.E. Matheson, Influence diagrams. in Readings on the Principles and Applications of Decision Analysis, vol. II, ed. by R.A. Howard, J.E. Matheson (Strategic Decisions Group, Menlo Park, CA, 1981), pp. 721–762

    Google Scholar 

  43. J.R. Koza, Genetic Programming: On the Programming of Computers by Means of Natural Selection (The MIT Press, Cambridge, MA, 1992)

    MATH  Google Scholar 

  44. V.V. Kulish, Hierarchical Methods: Hierarchy and Hierarchical Asymptotic Methods in Electrodynamics (Kluwer, Dordrecht, 2002)

    MATH  Google Scholar 

  45. P. Larrañaga, J.A. Lozano (eds.), Estimation of Distribution Algorithms: A New Tool for Evolutionary Computation (Kluwer, Boston, MA, 2002)

    MATH  Google Scholar 

  46. C.F. Lima, M. Pelikan, K. Sastry, M.V. Butz, D.E. Goldberg, F.G. Lobo, Substructural neighborhoods for local search in the Bayesian optimization algorithm. in Parallel Problem Solving from Nature (PPSN IX), ed. by T.P. Runarsson, et al. (Springer Verlag, Berlin, 2006), pp. 232–241

    Chapter  Google Scholar 

  47. C.F. Lima, K. Sastry, D.E. Goldberg, F.G. Lobo, Combining competent crossover and mutation operators: A probabilistic model building approach. in Proceedings of the Genetic and Evolutionary Computation Conference (GECCO-2005), ed. by H.-G. Beyer et al. (ACM Press, New York, 2005), pp. 735–742

  48. A. Mendiburu, J.A. Lozano, J. Miguel-Alonso, Parallel implementation of edas based on probabilistic graphical models. IEEE Trans. Evol. Comput. 9(4), 406–423 (2005)

    Article  Google Scholar 

  49. M. Mezard, G. Parisi, M. Virasoro, Spin Glass Theory and Beyond (World Scientific, Singapore, 1987)

    MATH  Google Scholar 

  50. H. Mühlenbein, T. Mahnig, Convergence theory and applications of the factorized distribution algorithm. J. Comp. Inform. Technol. 7(1), 19–32 (1998)

    Google Scholar 

  51. H. Mühlenbein, T. Mahnig, FDA–A scalable evolutionary algorithm for the optimization of additively decomposed functions. Evol. Comput. 7(4), 353–376 (1999)

    Google Scholar 

  52. H. Mühlenbein, T. Mahnig, Evolutionary optimization and the estimation of search distributions with aplication to graph bipartitioning. Int. J. Approx. Reason.. 31(3), 157–192 (2002)

    Article  MATH  Google Scholar 

  53. H. Mühlenbein, G. Paaß, From recombination of genes to the estimation of distributions I. Binary parameters. in Parallel Problem Solving from Nature (PPSN IV), ed. by H.-M. Voigt, et al. (Springer Verlag, Berlin, 1996), pp. 178–187

  54. H. Mühlenbein, D. Schlierkamp-Voosen, Predictive models for the breeder genetic algorithm: I. Continuous parameter optimization. Evol. Comput. 1(1), 25–49 (1993)

    Google Scholar 

  55. B. Naudts, J. Naudts, The effect of spin-flip symmetry on the performance of the simple GA. in Parallel Problem Solving from Nature (PPSN V), ed. by A.E. Eiben, T. Bäck, M. Schoenauer, H.-P. Schwefel (Springer Verlag, Berlin, 1998), pp. 67–76

    Chapter  Google Scholar 

  56. J. Ocenasek, M. Pelikan, Parallel mixed Bayesian optimization algorithm: A scaleup analysis. in Workshop Proceedings of the Genetic and Evolutionary Computation Conference (GECCO-2004), ed. by S. Cagnoni (Electronic publication, 2004)

  57. J. Ocenasek, J. Schwarz, The parallel Bayesian optimization algorithm. in Proceedings of the European Symposium on Computational Inteligence (Physica Verlag, Heidelberg, 2000), pp. 61–67

  58. J. Ocenasek, J. Schwarz, M. Pelikan, Design of multithreaded estimation of distribution algorithms. in Proceedings of the Genetic and Evolutionary Computation Conference (GECCO-2003), ed. by E. Cantú-Paz, et al. (Springer, Berlin, 2003), pp. 1247–1258

  59. H.H. Pattee (eds.), Hierarchy Theory: The Challenge of Complex Systems (Braziller, New York, 1973)

    Google Scholar 

  60. J. Pearl, Probabilistic Reasoning in Intelligent Systems: Networks of Plausible Inference (Morgan Kaufmann, San Mateo, CA, 1988)

    Google Scholar 

  61. M. Pelikan, Bayesian optimization algorithm: From single level to hierarchy. Doctoral dissertation, (University of Illinois at Urbana-Champaign, Urbana, IL, 2002)

  62. M. Pelikan, Hierarchical Bayesian Optimization Algorithm: Toward a New Generation of Evolutionary Algorithms (Springer-Verlag, 2005)

  63. M. Pelikan, D.E. Goldberg, Hierarchical problem solving and the Bayesian optimization algorithm. in Proceedings of the Genetic and Evolutionary Computation Conference (GECCO-2000), ed. by D. Whitley, et al. (Morgan Kaufmann, San Francisco, CA, 2000a), pp. 275–282

  64. M. Pelikan, D.E. Goldberg, Research on the Bayesian optimization algorithm. in Workshop Proceedings of the Genetic and Evolutionary Computation Conference (GECCO-2000), ed. by A. Wu (Morgan Kaufmann, San Fransisco, CA, 2000b), pp. 216–219

    Google Scholar 

  65. M. Pelikan, D.E. Goldberg, Escaping hierarchical traps with competent genetic algorithms. in Proceedings of the Genetic and Evolutionary Computation Conference (GECCO-2001), ed. by L. Spector, et al. (Morgan Kaufmann, San Francisco, CA, 2001), pp. 511–518

  66. M. Pelikan, D.E. Goldberg, Hierarchical BOA solves Ising spin glasses and MAXSAT. in Proceedings of the Genetic and Evolutionary Computation Conference (GECCO-2003), vol. II, ed. by E. Cantú-Paz, et al. (Morgan Kaufmann, San Francisco, CA, 2003a), pp. 1275–1286

  67. M. Pelikan, D.E. Goldberg, A hierarchy machine: Learning to optimize from nature and humans. Complexity. 8(5), 36–45 (2003b)

    Article  Google Scholar 

  68. M. Pelikan, D.E. Goldberg, Hierarchical Bayesian optimization algorithm. in Scalable Optimization via Probabilistic Modeling: From Algorithms to applications, ed. by E. Cantú-Paz, M. Pelikan, K. Sastry (Springer, 2006)

  69. M. Pelikan, D.E. Goldberg, E. Cantú-Paz, BOA: The Bayesian optimization algorithm. in Proceedings of the Genetic and Evolutionary Computation Conference (GECCO-99), vol. I, ed. by W. Banzhaf, et al. (Morgan Kaufmann, San Fransisco, CA, 1999), pp. 525–532

  70. M. Pelikan, D.E. Goldberg, F. Lobo, A survey of optimization by building and using probabilistic models. Comput. Optim. Appl. 21(1), 5–20 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  71. M. Pelikan, A.K. Hartmann, Searching for ground states of Ising spin glasses with hierarchical BOA and cluster exact approximation. in Scalable Optimization via Probabilistic Modeling: From Algorithms to Applications, ed. by E. Cantú-Paz, M. Pelikan, K. Sastry (Springer, 2006), pp. 333–349

  72. M. Pelikan, K. Sastry, Fitness inheritance in the Bayesian optimization algorithm. in Proceedings of the Genetic and Evolutionary Computation Conference (GECCO-2004), vol. 2, ed. by K. Deb, et al. (Springer Verlag, Berlin, 2004), pp. 48–59

  73. M. Pelikan, K. Sastry, E. Cantú-Paz (eds.), Scalable Optimization via Probabilistic Modeling: From Algorithms to Applications (Springer, 2006)

  74. M. Pelikan, K. Sastry, D.E. Goldberg, Scalability of the Bayesian optimization algorithm. Int. J. Approx. Reason. 31(3), 221–258 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  75. I. Rechenberg, Evolutionsstrategie: Optimierung technischer Systeme nach Prinzipien der biologischen Evolution (Frommann-Holzboog, Stuttgart, 1973)

    Google Scholar 

  76. E.D. Sacerdoti, The nonlinear nature of plans. in Proceedings of the Fourth Annual International Joint Conference on Aritificial Intelligence (Tbilisi, Georgia, USSR, 1975), pp. 206–214

  77. R. Santana, Estimation of distribution algorithms with Kikuchi approximations. Evol. Comput. 13(1), 67–97 (2005)

    Article  Google Scholar 

  78. K. Sastry, Evaluation-relaxation schemes for genetic and evolutionary algorithms. Master’s thesis, University of Illinois at Urbana-Champaign. (Department of General Engineering, Urbana, IL, 2001)

  79. K. Sastry, D.E. Goldberg, On Extended Compact Genetic Algorithm (IlliGAL Report No. 2000026). (University of Illinois at Urbana-Champaign, Illinois Genetic Algorithms Laboratory, Urbana, IL, 2000)

  80. K. Sastry, D.E. Goldberg, M. Pelikan, Don’t evaluate, inherit. in Proceedings of the Genetic and Evolutionary Computation Conference (GECCO-2001), ed. by L. Spector, et al. (Morgan Kaufmann, San Francisco, CA, 2001), pp. 551–558

  81. K. Sastry, M. Pelikan, D.E. Goldberg, Efficiency enhancement of genetic algorithms via building-block-wise fitness estimation, in Proceedings of the IEEE Conference on Evolutionary Computation (IEEE Press, 2004), pp. 720–727

  82. K. Sastry, M. Pelikan, D.E. Goldberg, Efficiency enhancement of estimation of distribution algorithms. in Scalable Optimization via Probabilistic Modeling: From Algorithms to Applications, ed. by E. Cantú-Paz, M. Pelikan, K. Sastry (Springer, 2006), pp. 161–185

  83. J. Schwarz, J. Ocenasek, A problem-knowledge based evolutionary algorithm KBOA for hypergraph partitioning. in Proceedings of the Fourth Joint Conference on Knowledge-Based Software Engineering (IO Press, Brno, Czech Republic, 2000), pp. 51–58

  84. H.A. Simon, The Sciences of the Artificial (The MIT Press, Cambridge, MA, 1968)

    Google Scholar 

  85. A. Sinha, D.E. Goldberg, Verification and extension of the theory of global-local hybrids. in Proceedings of the Genetic and Evolutionary Computation Conference (GECCO-2001), ed. by L. Spector, et al. (Morgan Kaufmann, San Francisco, CA, 2001), pp. 591–597

  86. R. Srivastava, D.E. Goldberg, Verification of the theory of genetic and evolutionary continuation. in Proceedings of the Genetic and Evolutionary Computation Conference (GECCO-2001), ed. by L. Spector, et al (Morgan Kaufmann, San Francisco, CA, 2001), pp. 551–558

  87. M. Stefik, Planning and meta-planning (MOLGEN: Part 2). Artif. Intell.16(2), 141–170 (1981a)

    Article  Google Scholar 

  88. M. Stefik, Planning with constraints (MOLGEN: Part 1). Artif. Intell. 16(2), 111–140 (1981b)

    Article  Google Scholar 

  89. D. Thierens, Analysis and design of genetic algorithms. Doctoral dissertation, (Katholieke Universiteit Leuven, Leuven, Belgium, 1995)

  90. D. Thierens, D.E. Goldberg, A.G. Pereira, Domino convergence, drift, and the temporal-salience structure of problems. in Proceedings of the International Conference on Evolutionary Computation (ICEC-98) (IEEE Press, Picataway, NJ, 1998), pp. 535–540

  91. C. Van Hoyweghen, Detecting spin-flip symmetry in optimization problems. in Theoretical Aspects of Evolutionary Computing, ed. by L. Kallel, et al. (Springer, Berlin, 2001), pp. 423–437

    Google Scholar 

  92. R.A. Watson, G.S. Hornby, J.B. Pollack, Modeling building-block interdependency. in Parallel Problem Solving from Nature (PPSN V), ed. by A.E. Eiben, T. Bäck, M. Schoenauer, H.-P. Schwefel (Springer Verlag, Berlin, 1998), pp. 97–106

    Chapter  Google Scholar 

  93. L.D. Whitley, Fundamental principles of deception in genetic search. in Foundations of Genetic Algorithms ed. by G. Rawlins (Morgan Kaufmann, San Mateo, CA, 1991), pp. 221–241

    Google Scholar 

  94. A. Young (eds.), Spin Glasses and Random Fields (World Scientific, Singapore, 1998)

    Google Scholar 

Download references

Acknowledgments

This work was supported by the National Science Foundation under NSF CAREER grant ECS-0547013 (at UMSL) and ITR grant DMR-03-25939 (at Materials Computation Center, UIUC), by the Air Force Office of Scientific Research, Air Force Materiel Command, USAF, under grant FA9550-06-1-0096, and by the University of Missouri in St. Louis through the High Performance Computing Collaboratory sponsored by Information Technology Services, and the Research Award and Research Board programs. The experiments presented in this work were done using the hBOA software developed by Martin Pelikan and David E. Goldberg at the University of Illinois at Urbana-Champaign. Most experiments were completed at the Beowulf cluster at the University of Missouri at St. Louis. The U.S. Government is authorized to reproduce and distribute reprints for government purposes notwithstanding any copyright notation thereon.

Author information

Authors and Affiliations

  1. Missouri Estimation of Distribution Algorithms Laboratory, 321 CCB, Department of Mathematics and Computer Science, University of Missouri in St. Louis, One University Blvd., St. Louis, MO, 63121, USA

    Martin Pelikan

  2. Illinois Genetic Algorithms Laboratory, 117 TB, Department of Industrial and Enterprise Systems Engineering, University of Illinois at Urbana-Champaign, 104 S. Mathews Ave., Urbana, IL, 61801, USA

    Kumara Sastry & David E. Goldberg

Authors
  1. Martin Pelikan
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Kumara Sastry
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. David E. Goldberg
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Martin Pelikan.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Pelikan, M., Sastry, K. & Goldberg, D.E. Sporadic model building for efficiency enhancement of the hierarchical BOA. Genet Program Evolvable Mach 9, 53–84 (2008). https://doi.org/10.1007/s10710-007-9052-8

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10710-007-9052-8

Keywords

Search

Navigation