Niching methods extend genetic algorithms to domains that require the location and maintenance of multiple solutions. Such domains include classification and machine learning, multimodal function optimization, multiobjective function optimization, and simulation of complex and adaptive systems. This study presents a comprehensive treatment of niching methods and the related topic of population diversity. Its purpose is to analyze existing niching methods and to design improved niching methods. To achieve this purpose, it first develops a general framework for the modelling of niching methods, and then applies this framework to construct models of individual niching methods, specifically crowding and sharing methods. Using a constructed model of crowding, this study determines why crowding methods over the last two decades have not made effective niching methods. A series of tests and design modifications results in the development of a highly effective form of crowding, called determin...

We approach the difficult task of analyzing the complex behavior of even the simplest learning classifier system (LCS) by isolating one crucial subfunction in the LCS learning algorithm: covering through niching. The LCS must maintain a population of diverse rules that together solve a problem (e.g., classify examples). To maintain a diverse population while applying the GA's selection operator, the LCS must incorporate some kind of niching mechanism. The natural way to accomplish niching in an LCS is to force competing rules to share resources (i.e., rewards). This implicit LCS fitness sharing is similar to the explicit fitness sharing used in many niched GAs. Indeed, the LCS implicit sharing algorithm can be mapped onto explicit fitness sharing with a one-to-one correspondence between algorithm components. This mapping is important because several studies of explicit fitness sharing, and of niching in GAs generally, have produced key insights and analytical tools for understanding th...

. The theory of evolutionary computation has been enhanced rapidly during the last decade. This survey is the attempt to summarize the results regarding the limit and finite time behavior of evolutionary algorithms with finite search spaces and discrete time scale. Results on evolutionary algorithms beyond finite space and discrete time are also presented but with reduced elaboration. Keywords: evolutionary algorithms, limit behavior, finite time behavior 1. Introduction The field of evolutionary computation is mainly engaged in the development of optimization algorithms which design is inspired by principles of natural evolution. In most cases, the optimization task is of the following type: Find an element x 2 X such that f(x ) f(x) for all x 2 X , where f : X ! IR is the objective function to be maximized and X the search set. In the terminology of evolutionary computation, an individual is represented by an element of the Cartesian product X \Theta A, where A is a possibly...

In the binary evolutionary optimization framework, two mutation operators are theoretically investigated. For both the standard mutation, in which all bits are flipped independently with the same probability, and the 1-bit-flip mutation, which flips exactly one bit per bitstring, the statistical distribution of the first hitting times of the target are thoroughly computed (expectation and variance) up to terms of order l (the size of the bitstrings) in two distinct situations: without any selection, or with the deterministic (1+1)-ES selection on the OneMax problem. In both cases, the 1-bit-flip mutation convergence time is smaller by a constant (in terms of l) multiplicative factor. These results extend to the case of multiple independent optimizers. Keywords Evolutionary algorithms, stochastic analysis, binary mutations, Markov chains, hitting times. 1 Introduction One known drawback of Evolutionary Algorithms as function optimizers is the amount of computational efforts they re...

In this paper a new, general and exact schema theory for genetic programming is presented. The theory includes a microscopic schema theorem applicable to crossover operators which replace a subtree in one parent with a subtree from the other parent to produce the offspring. A more macroscopic schema theorem is also provided which is valid for crossover operators in which the probability of selecting any two crossover points in the parents depends only on their size and shape. The theory is based on the notions of Cartesian node reference systems and variable-arity hyperschemata both introduced here for the first time. In the paper we provide examples which show how the theory can be specialised to specific crossover operators and how it can be used to derive an exact definition of effective fitness and a size-evolution equation for GP. 1

A few schema theorems for Genetic Programming (GP) have been proposed in the literature in the last few years. Since they consider schema survival and disruption only, they can only provide a lower bound for the expected value of the number of instances of a given schema at the next generation rather than an exact value. This paper presents theoretical results for GP with one-point crossover which overcome this problem. Firstly, we give an exact formulation for the expected number of instances of a schema at the next generation in terms of microscopic quantities. Thanks to this formulation we are then able to provide an improved version of an earlier GP schema theorem in which some (but not all) schema creation events are accounted for. Then, we extend this result to obtain an exact formulation in terms of macroscopic quantities which makes all the mechanisms of schema creation explicit. This theorem allows the exact formulation of the notion of effective fitness in GP and opens the way to future work on GP convergence, population sizing, operator biases, and bloat, to mention only some of the possibilities.

Two main weaknesses of GA and GP schema theorems axe that they provide only information on the expected value of the number of instances of a given schema at the next generation E[m(H,t + 1)], and they can only give a lower bound for such a quantity. This paper presents new theoretical results on GP and GA schemata which laxgely overcome these weaknesses. Firsfly, unlike previous results which concentrated on schema survival and disruption, our results extend to GP recent work on GA theory by Stephens and Waelbroeck, and make the effects and the mechanisms of schema creation explicit. This allows us to give an exact formulation (rather than a lower bound) for the expected number of instances of a schema at the next generation. Thanks to this formulation we are then able to provide in improved version for an eaxlier GP schema theorem in which some schema creation events axe accounted for, thus obtaining a tighter bound for E[m(H, t + 1)]. This bound is a function of the selection probabilities of the schema itself and of a set of lower-order schemata which one-point crossover uses to build instances of the schema. This result supports the existence of building blocks in GP which, however, axe not necessaxily all short, low-order or highly fit. Building on eaxlier work, we show how Stephens and Waelbroeck 's GA results and the new GP results described in the paper can be used to evaluate schema vaxiance, signal-to-noise ratio and, in general, the probability distribution of re(H, t + 1). In addition, we show how the expectation operator can be removed from the schema theorem so as to predict with a known probability whether re(H, t + 1) (rather than Elm(H, t + 1)]) is going to be above a given threshold.

The common explanation of the manner in which genetic algorithms (GAs) process individuals in a population of contending solutions relies on the "building block hypothesis." This suggests that successively better solutions are generated by combining useful parts of extant solutions. An alternative explanation is presented which focuses on the collective phenomena taking place in populations that undergo recombination. The new explanation is derived from investigations in evolution strategies (ESs). The principles studied are general, and hold for all evolutionary algorithms (EAs), including genetic algorithms (GAs). Further, they appear to be somewhat analogous to some theories and observations on the benefits of sex in biota. Keywords building block hypothesis, evolutionary algorithms, multi-recombination 1 Introduction Although specific theoretical investigations into the properties of genetic algorithms have gained considerable recent attention, there still is no satisfa...

This article tries to connect two separate strands of literature concerning genetic algorithms. On the one hand, extensive research took place in mathematics and closely related sciences in order to find out more about the properties of genetic algorithms as stochastic processes. On the other hand, recent economic literature uses genetic algorithms as a metaphor for social learning. This paper will face the question what an economist can learn from the mathematical branch of research, especially concerning the convergence and stability properties of the genetic algorithm. It is shown that genetic algorithm learning is a compound of three different learning schemes. First, every particular scheme is analyzed. Then it will be pointed out that it is the combination of the three schemes that gives genetic algorithm learning its special flair: A kind of stability somewhere in between asymptotic convergence and explosion. 1 Introduction As a consequence of the discussion concer...

In this paper we present a Markov chain model for GP and variable-length GAs with homologous crossover: a set of GP operators where the offspring are created preserving the position of the genetic material taken from the parents. We obtain this result by using the core of Vose’s model for GAs in conjunction with a specialisation of recent GP schema theory for such operators. The model is then specialised for the case of GP operating on 0/1 trees: a tree-like generalisation of the concept of binary string. For these symmetries exist that can be exploited to obtain further simplifications. In the absence of mutation, the theory presented here generalises Vose’s GA model to GP and variable-length GAs. 1