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Evolution via natural selection of a randomly chosen population of individuals can be thought of as a search through the space of possible chromosome values In that sense, an evolutionary algorithm (EA) is a stochastic search for an optimal solution to a given problem The evolutionary search process is in uenced by the following main components of an EA: an encoding of solutions to the problem as a chromosome; a function to evaluate the fitness, or survival strength of individuals; initialization of the initial population; selection operators; and reproduction operators Algorithm 81 shows how these components are combined to form a generic EA Algorithm 81 Generic Evolutionary Algorithm Let t = 0 be the generation counter; Create and initialize an nx -dimensional population, C(0), to consist of ns individuals; while stopping condition(s) not true do Evaluate the tness, f (xi (t)), of each individual, xi (t); Perform reproduction to create o spring; Select the new population, C(t + 1); Advance to the new generation, ie t = t + 1; end
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82 Representation -- The Chromosome
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The steps of an EA are applied iteratively until some stopping condition is satis ed (refer to Section 87) Each iteration of an EA is referred to as a generation The di erent ways in which the EA components are implemented result in di ferent EC paradigms: Genetic algorithms (GAs), which model genetic evolution Genetic programming (GP), which is based on genetic algorithms, but individuals are programs (represented as trees) Evolutionary programming (EP), which is derived from the simulation of adaptive behavior in evolution (ie phenotypic evolution) Evolution strategies (ESs), which are geared toward modeling the strategic parameters that control variation in evolution, ie the evolution of evolution Differential evolution (DE), which is similar to genetic algorithms, di ering in the reproduction mechanism used Cultural evolution (CE), which models the evolution of culture of a population and how the culture in uences the genetic and phenotypic evolution of individuals Co-evolution (CoE), where initially dumb individuals evolve through cooperation, or in competition with one another, acquiring the necessary characteristics to survive These paradigms are discussed in detail in the chapters that follow in this part of the book With reference to Algorithm 81, both parts of Darwin s theory are encapsulated within this algorithm: Natural selection occurs within the reproduction operation where the best parents have a better chance of being selected to produce o spring, and to be selected for the new population Random changes are e ected through the mutation operator
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Representation The Chromosome
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In nature, organisms have certain characteristics that in uence their ability to survive and to reproduce These characteristics are represented by long strings of information contained in the chromosomes of the organism Chromosomes are structures of compact intertwined molecules of DNA, found in the nucleus of organic cells Each chromosome contains a large number of genes, where a gene is the unit of heredity Genes determine many aspects of anatomy and physiology through control of protein production Each individual has a unique sequence of genes An alternative form of a gene is referred to as an allele In the context of EC, each individual represents a candidate solution to an optimization problem The characteristics of an individual is represented by a chromosome, also
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8 Introduction to Evolutionary Computation
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referred to as a genome These characteristics refer to the variables of the optimization problem, for which an optimal assignment is sought Each variable that needs to be optimized is referred to as a gene, the smallest unit of information An assignment of a value from the allowed domain of the corresponding variable is referred to as an allele Characteristics of an individual can be divided into two classes of evolutionary information: genotypes and phenotypes A genotype describes the genetic composition of an individual, as inherited from its parents; it represents which allele the individual possesses A phenotype is the expressed behavioral traits of an individual in a speci c environment; it de nes what an individual looks like Complex relationships exist between the genotype and phenotype [570]: pleiotropy, where random modi cation of genes causes unexpected variations in the phenotypic traits, and polygeny, where several genes interact to produce a speci c phenotypic trait An important step in the design of an EA is to nd an appropriate representation of candidate solutions (ie chromosomes) The e ciency and complexity of the search algorithm greatly depends on the representation scheme Di erent EAs from the di erent paradigms use di erent representation schemes Most EAs represent solutions as vectors of a speci c data type An exception is genetic programming (GP) where individuals are represented in a tree format The classical representation scheme for GAs is binary vectors of xed length In the case of an nx -dimensional search space, each individual consists of nx variables with each variable encoded as a bit string If variables have binary values, the length of each chromosome is nx bits In the case of nominal-valued variables, each nominal value can be encoded as an nd -dimensional bit vector where 2nd is the total number of discrete nominal values for that variable To solve optimization problems with continuous-valued variables, the continuous search space problem can be mapped into a discrete programming problem For this purpose mapping functions are needed to convert the space {0, 1}nb to the space Rnx For such mapping, each continuous-valued variable is mapped to an nd -dimensional bit vector, ie : R (0, 1)nd (81)
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The domain of the continuous space needs to be restricted to a nite range, [xmin , xmax ] A standard binary encoding scheme can be used to transform the individual x = (x1 , , xj , , xnx ), with xj R to the binary-valued individual, b = (b1 , , bj , , bnx ), where bj = (b(j 1)nd +1 , , bjnd ), with bl {0, 1} and the total number of bits, nb = nx nd Decoding each bj back to a oating-point representation can be done using the function, j : {0, 1}nd [xmin,j , xmax,j ], where [39] nd 1 xmax,j xmin,j bj(nd l) 2l j (b) = xmin,j + (82) 2nd 1
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Holland [376] and De Jong [191] provided the rst applications of genetic algorithms to solve continuous-valued problems using such a mapping scheme It should be noted
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