Theoretical analysis of evolutionary algorithms with an infinite population size in continuous space. Part I: Basic properties of selection and mutation

This paper aims at establishing fundamental theoretical properties for a class of "genetic algorithms" in continuous space (GACS). The algorithms employ operators such as selection, crossover, and mutation in the framework of a multidimensional Euclidean space. The paper is divided into two parts. The first part concentrates on the basic properties associated with the selection and mutation operators. Recursive formulae for the GACS in the general infinite population case are derived and their validity is rigorously proven. A convergence analysis is presented for the classical case of a quadratic cost function. It is shown how the increment of the population mean is driven by its own diversity and follows a modified Newton's search. Sufficient conditions for monotonic increase of the population mean fitness are derived for a more general class of fitness functions satisfying a Lipschitz condition. The diversification role of the crossover operator is analyzed in Part II. The treatment adds much light to the understanding of the underlying mechanism of evolution-like algorithms.

[1]  Thomas Bäck,et al.  A Survey of Evolution Strategies , 1991, ICGA.

[2]  M. Feldman,et al.  On the evolutionary effect of recombination. , 1970, Theoretical population biology.

[3]  David E. Goldberg,et al.  Genetic Algorithms and Walsh Functions: Part I, A Gentle Introduction , 1989, Complex Syst..

[4]  W. Ebeling,et al.  Stochastic Theory of Molecular Replication Processes with Selection Character , 1977 .

[5]  J. Kingman.  Uses of Exchangeability , 1978 .

[6]  David B. Fogel,et al.  Evolving artificial intelligence , 1992 .

[7]  Heinz Mühlenbein,et al.  Evolution algorithms in combinatorial optimization , 1988, Parallel Comput..

[8]  Gunar E. Liepins,et al.  Punctuated Equilibria in Genetic Search , 1991, Complex Syst..

[9]  S Karlin,et al.  Principles of polymorphism and epistasis for multilocus systems. , 1979, Proceedings of the National Academy of Sciences of the United States of America.

[10]  S Karlin,et al.  Models of multifactorial inheritance: II. The covariance structure for a scalar phenotype under selective assortative mating and sex-dependent symmetric parental-transmission. , 1979, Theoretical population biology.

[11]  Gunar E. Liepins,et al.  Polynomials, Basis Sets, and Deceptiveness in Genetic Algorithms , 1991, Complex Syst..

[12]  David E. Goldberg,et al.  Real-coded Genetic Algorithms, Virtual Alphabets, and Blocking , 1991, Complex Syst..

[13]  X. Qi,et al.  Analyses of the genetic algorithms in the continuous space , 1992, [Proceedings 1992] IJCNN International Joint Conference on Neural Networks.

[14]  A. E. Eiben,et al.  Global Convergence of Genetic Algorithms: A Markov Chain Analysis , 1990, PPSN.

[15]  Dean C. Karnopp,et al.  Random search techniques for optimization problems , 1963, at - Automatisierungstechnik.

[16]  S Karlin,et al.  The reduction property for central polymorphisms in nonepistatic systems. , 1982, Theoretical population biology.

[17]  J´nos Pintér,et al.  Convergence properties of stochastic optimization procedures , 1984 .

[18]  Feller William,et al.  An Introduction To Probability Theory And Its Applications , 1950 .

[19]  Kalyanmoy Deb,et al.  A Comparative Analysis of Selection Schemes Used in Genetic Algorithms , 1990, FOGA.

[20]  L. D. Whitley,et al.  The Traveling Salesman and Sequence Scheduling : , 1990 .

[21]  John Frank Charles Kingman,et al.  Coherent random walks arising in some genetical models , 1976, Proceedings of the Royal Society of London. A. Mathematical and Physical Sciences.

[22]  Marc Lipsitch,et al.  Adaptation on Rugged Landscapes Generated by Iterated Local Interactions of Neighboring Genes , 1991, ICGA.

[23]  S. M. Ermakov,et al.  On Random Search for a Global Extremum , 1984 .

[24]  H. Haario,et al.  Simulated annealing process in general state space , 1991, Advances in Applied Probability.

[25]  John H. Holland,et al.  Adaptation in Natural and Artificial Systems: An Introductory Analysis with Applications to Biology, Control, and Artificial Intelligence , 1992 .

[26]  L. Darrell Whitley,et al.  Genetic Reinforcement Learning with Multilayer Neural Networks , 1991, ICGA.

[27]  Larry J. Eshelman,et al.  On Crossover as an Evolutionarily Viable Strategy , 1991, ICGA.

[28]  L. Ginzburg,et al.  Multilocus population genetics: relative importance of selection and recombination. , 1980, Theoretical population biology.

[29]  Hans-Paul Schwefel,et al.  Numerical optimization of computer models , 1981 .

[30]  Günter Rudolph,et al.  Convergence analysis of canonical genetic algorithms , 1994, IEEE Trans. Neural Networks.

[31]  Roger J.-B. Wets,et al.  Minimization by Random Search Techniques , 1981, Math. Oper. Res..

[32]  T. Ohta,et al.  Theoretical aspects of population genetics. , 1972, Monographs in population biology.

[33]  S. Karlin.  Equilibrium behavior of population genetic models with non-random mating. , 1968 .

[34]  Samuel H. Brooks.  A Discussion of Random Methods for Seeking Maxima , 1958 .

[35]  José Carlos Príncipe,et al.  A Simulated Annealing Like Convergence Theory for the Simple Genetic Algorithm , 1991, ICGA.

[36]  Samuel Karlin,et al.  Analysis of central equilibria in multilocus systems: A generalized symmetric viability regime☆ , 1981 .