Computer Science

This paper is concerned with the application of genetic algorithms to optimization problems over several real parameters. It is shown that k-point crossover (for k small relative to the number of parameters) can be viewed as a crossover operation on the vector of parameters plus perturbations of some of the parameters. Mutation can also be considered as a perturbation of some of the parameters. This suggests a genetic algorithm that uses real parameter vectors as chromosomes, real parameters as genes, and real numbers as alleles. Such an algorithm is proposed with two possible crossover methods. Schemata are defined for this algorithm, and it is shown that Holland's Schema theorem holds for one of these crossover methods. Experimental results are given that indicate that this algorithm with a mixture of the two crossover methods outperformed the binary-coded genetic algorithm on 7 of 9 test problems.

A general form of stochastic search is described (random heuristic search) and some of its general properties are proved. This provides a framework in which the simple genetic algorithm (SGA) is a special case. The framework is used to illuminate relationships between seemingly di erent probabilistic perspectives of SGA behavior. Next, the SGA is formalized as an instance of random heuristic search. The formalization is then used to show expected population tness is a Lyapunov function in the in nite population model when mutation is zero and tness is linear. In particular, 1 This work was done while visiting the computer science department of the University of Tennessee.

Given a polynomial equation of degree d over the complex domain, the Fundamental Theorem of Algebra tells us that there are d solutions, assuming that the solutions are counted by multiplicity. These solutions can be approximated by deforming a standard «th degree equation into the given equation, and following the solutions through the deformation. This is called the homotopy method. The Fundamental Theorem of Algebra can be proved by the same technique.

This paper is the first part of a two part series. It proves a number of direct relationships between the Fourier transform and the simple genetic algorithm. (For a binary representation, the Walsh transform is the Fourier transform.) The results are of a theoretical nature and are based on the analysis of mutation and crossover. The Fourier transform of the mixing matrix is shown to be sparse. An explicit formula is given for the spectrum of the differential of the mixing transformation. By using the Fourier representation and the fast Fourier transform, one generation of the infinite population simple genetic algorithm can be computed in time O(c log 2 3 ), where c is arity of the alphabet and is the string length. This is in contrast to the time of O(c 3 ) for the algorithm as represented in the standard basis. There are two orthogonal decompositions of population space which are invariant under mixing. The sequel to this paper will apply the basic theoretical results obtained here to inverse problems and asymptotic behavior.

It is supposed that the finite search space Ω has certain symmetries which can be described in terms of a group of permutations acting upon it. If crossover and mutation respect these symmetries then these operators can be described in terms of a mixing matrix and a group of permutation matrices. Conditions under which certain subsets of Ω are invariant under crossover are investigated, leading to a generalization of the term schema. Finally, it is sometimes possible for the group acting on Ω to induce a group structure on Ω itself.

In the Infinite Population Simple Genetic Algorithm, stability of fixed points is considered when mutation is zero. The analysis is based on the spectrum of the differential of the mapping which defines the transition from one generation to the next. Based on a simple formula for this spectrum, fully nondeceptive functions having exponentially many non-optimal fixed points are constructed.

This paper addresses the problem of discovering the structure of a fitness function from binary strings to the reals under the assumption of bounded epistasis. Two loci (string positions) are epistatically linked if the effect of changing the allele (value) at one locus depends on the allele at the other locus. Similarly, a group of loci are epistatically linked if the effect of changing the allele at one locus depends on the alleles at all other loci of the group. Under the assumption that the size of such groups of loci are bounded, and assuming that the function is given only as a "black box function", this paper presents and analyzes a randomized algorithm that finds the complete epistatic structure of the function in the form of the Walsh coefficients of the function.

This paper continues the development, begun in part I, of the relationship between the simple genetic algorithm and the Walsh transform. The mixing scheme (comprised of crossover and mutation) is essentially "triangularized" when expressed in terms of the Walsh basis. This leads to a formulation of the inverse of the expected next generation operator. The fixed points of the mixing scheme are also determined, and a formula is obtained giving the fixed point corresponding to any starting population. Geiringer's theorem follows from these results in the special case corresponding to zero mutation.

Abstract We empirically searched for cycling and chaotic behavior in the infinite population Simple Genetic Algorithm. We found examples of period 2 cycling (which we expected) and long period cycling (which we didn't expect). These examples had mutation and crossover distributions which do not correspond to the way that mutation and crossover are normally used in practice. We also searched unsuccessfully for stable polymorphic fixed points in the sero mutation case.

Abstract Estimation of distribution algorithms construct an explicit model of the problem to be solved, and then use this model to guide the search for good solutions. For an important class of fitness functions, namely those with k-bounded epistasis, it is possible to construct a complete explicit representation of the fitness function by sampling the fitness function. A very natural model of the problem to be solved is the Boltzmann distribution of the fitness function, which is an exponential of the fitness normalized to a probability distribution.

In this paper we study, theoretically, the search biases produced by GP subtree crossover when applied to linear representations, such as those used in linear GP or in variable length GAs. The study naturally leads to generalisations of Geiringer's theorem and of the notion of linkage equilibrium, which, until now, were applicable only to fixed-length representations.

Estimation of distribution algorithms (EDA) are similar to genetic algorithms except that they replace crossover and mutation with sampling from an estimated probability distribution. We develop a framework for estimation of distribution algorithms based on the principle of maximum entropy and the conservation of schema frequencies. An algorithm of this type gives better performance than a standard genetic algorithm (GA) on a number of standard test problems involving deception and epistasis (ie Trap and NK).

We consider complex systems that are composed of many interacting elements, evolving under some dynamics. We are interested in characterizing the ways in which these elements may be grouped into higher-level, macroscopic states in a way that is compatible with those dynamics. Such groupings may then be thought of as naturally emergent properties of the system. We formalize this idea and, in the case that the dynamics are linear, prove necessary and sufficient conditions for this to happen.

A major goal of genetic algorithm theory is a tractible model of a GA that gives quantitative predictions over multiple generations. The Vose dynamical system model is exact in the infinite-population limit, but it is intractible for quantitative results due to the fact that the number of variables in the model is exponential in the string length. What is needed is a “coarse-grained” model that reduces the number of variables in the model.

The 8th Workshop on the Foundations of Genetic Algorithms, FOGA-8, was held at the University of Aizu in Aizu-Wakamatsu City, Japan, January 5–9, 2005. This series of workshops was initiated in 1990 to encourage further research on the theoretical aspects of genetic algorithms, and the workshops have been held biennially ever since. The papers presented at these workshops are revised, edited and published as volumes during the year following each workshop.

Abstract In recent work we have formulated a model of emergent coordinated behaviour for a population of interacting entities. The model is a modified spring mass model where masses can perceive the environment and generate external forces. As a result of the interactions the population behaves like a single organism moving under the effect the vector sum of the external forces generated by each entity.

Although much progress has been made in recent years in the theory of GAs and GP, there is still a conspicuous lack of tools with which to derive systematic, approximate solutions to their dynamics. In this article we propose and study perturbation theory as a potential tool to fill this gap. We concentrate mainly on selection-mutation systems, showing different implementations of the perturbative framework, developing, for example, perturbative expansions for the eigenvalues and eigenvectors of the transition matrix.

Abstract We compare the behavior of a GA with and without crossover. A simple GA with crossover can have two stable fixed points (bistability) on the single-peak landscapes for string lengths at least 8. There are catastrophic transitions from one to two and back to one stable fixed point as the mutation rate increases. Crossover also causes the fixed-point population distribution to change.