Abstract
We propose a method to build discrete Markov chain models of continuous stochastic optimisers that can approximate them on arbitrary continuous problems to any precision. We discretise the objective function using a finite element method grid which produces corresponding distinct states in the search algorithm. Iterating the transition matrix gives precise information about the behaviour of the optimiser at each generation, including the probability of it finding the global optima or being deceived. The approach is tested on a (1 + 1)-ES, a bare bones PSO and a real-valued GA . The predictions are remarkably accurate.
This is a preview of subscription content, log in via an institution.
Buying options
Tax calculation will be finalised at checkout
Purchases are for personal use only
Learn about institutional subscriptionsPreview
Unable to display preview. Download preview PDF.
References
Babuska, I.: The Finite Element Method for Infinite Domains. I. Mathematics of Computation 26(117), 1–11 (1972)
Bäck, T.: Evolutionary Algorithms in Theory and Practice: Evolution Strategies, Evolutionary Programming, Genetic Algorithms. Oxford University Press, New York (1996)
Bathe, K.J.: Finite Element Procedures in Engineering Analysis. Prentice-Hall, Englewood Cliffs, New Jersey (1982)
Clerc, M.: Stagnation analysis in particle swarm optimisation or what happens when nothing happens. Technical Report Technical Report CSM-460, Department of Computer Science, University of Essex, Edited by Riccardo Poli (August 2006)
Clerc, M., Kennedy, J.: The particle swarm - explosion, stability, and convergence in a multidimensional complex space. IEEE Transaction on Evolutionary Computation 6(1), 58–73 (2002)
Givoli, D.: Numerical methods for problems in infinite domains. Elsevier, New York (1992)
Goldstein, C.: The Finite Element Method with Nonuniform Mesh Sizes for Unbounded Domains. Mathematics of Computation 36(154), 387–404 (1981)
Kennedy, J.: The behavior of particles. In: Evolutionary Programming VII: Proceedings of the Seventh Annual Conference on evolutionary programming, pp. 581–589, San Diego, CA (1998)
Kennedy, J.: Bare bones particle swarms. In: Proceedings of the IEEE Swarm Intelligence Symposium 2003 (SIS 2003), pp. 80–87, Indianapolis, Indiana (2003)
Kennedy, J., Eberhart, R.C., Shi, Y.: Swarm Intelligence. Morgan Kaufmann, San Francisco (2001)
Krohling, R.A.: Gaussian particle swarm with jumps. In: Corne, D., Michalewicz, Z., Dorigo, M., Eiben, G., Fogel, D., Fonseca, C., Greenwood, G., Chen, T.K., Raidl, G., Zalzala, A., Lucas, S., Paechter, B., Willies, J., Guervos, J.J.M., Eberbach, E., McKay, B., Channon, A., Tiwari, A., Volkert, L.G., Ashlock, D., Schoenauer, M. (eds.) Proceedings of the 2005 IEEE Congress on Evolutionary Computation, Edinburgh, UK, September 2-5, 2005, vol. 2, pp. 1226–1231. IEEE Press, New York (2005)
Nomura, T., Shimohara, K.: An analysis of two-parent recombinations for real-valued chromosomes in an infinite population. Evolutionary Computation 9(3), 283–308 (2001)
Ozcan, E., Mohan, C.K.: Particle swarm optimization: surfing the waves. In: Proceedings of the IEEE Congress on evolutionary computation (CEC 1999), Washington, DC (1999)
Rudolph, G.: Convergence of evolutionary algorithms in general search spaces. In: International Conference on Evolutionary Computation, pp. 50–54 (1996)
Spears, W.M.: The Role of Mutation and Recombination in Evolutionary Algorithms. PhD thesis, George Mason University, Fairfax, Virginia, USA (1998)
Storn, R.: Designing digital filters with differential evolution. In: Corne, D., Dorigo, M., Glover, F. (eds.) New Ideas in Optimization, Advanced Topics in Computer Science, Ch. 7, pp. 109–125. McGraw-Hill, Maidenhead, Berkshire, England (1999)
Thompson, L., Pinsky, P.: Space-time finite element method for structural acoustics in infinite domains. Part 1: formulation, stability and convergence. Computer Methods in Applied Mechanics and Engineering 132(3), 195–227 (1996)
Trelea, I.C.: The particle swarm optimization algorithm: convergence analysis and parameter selection. Information Processing Letters 85(6), 317–325 (2003)
van den Bergh, F.: An Analysis of Particle Swarm Optimizers. PhD thesis, Department of Computer Science, University of Pretoria, Pretoria, South Africa (November 2001)
Author information
Authors and Affiliations
Editor information
Rights and permissions
Copyright information
© 2007 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Poli, R., Langdon, W.B., Clerc, M., Stephens, C.R. (2007). Continuous Optimisation Theory Made Easy? Finite-Element Models of Evolutionary Strategies, Genetic Algorithms and Particle Swarm Optimizers. In: Stephens, C.R., Toussaint, M., Whitley, D., Stadler, P.F. (eds) Foundations of Genetic Algorithms. FOGA 2007. Lecture Notes in Computer Science, vol 4436. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-73482-6_10
Download citation
DOI: https://doi.org/10.1007/978-3-540-73482-6_10
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-73479-6
Online ISBN: 978-3-540-73482-6
eBook Packages: Computer ScienceComputer Science (R0)