Evolutionary Computation : A Unified Approach

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6.2. ANALYZING EA DYNAMICS 119 The alternative is to construct finite-population models that are closer to “real” EAs, but may be less mathematically tractable. Notationally, the finite-population models are nearly identical to their infinite-population counterparts, namely: P (t) = However, c i (t) now represents the number of copies of genotype x i in a population of size m at time t. Hence, all c i (t) mustsatisfy0≤ c i (t) ≤ m, and the elements of any particular population vector must sum to m. For example, the vector < 50, 50, 0, ..., 0 > describes a population of size 100 that consists of exactly two genotypes, x 1 and x 2 , in equal proportions. Of course, we can always normalize such vectors by dividing each element by m so that finite-population vectors represent points in the same (r − 1)-dimensional simplex as the infinite population case. Note, however, that these normalized finite-population vectors can represent only a discrete subset of points in the simplex, and hence finite-population trajectories are restricted to visiting only this subset of points. This normalization provides us with a useful mathematical view of the effect of increasing the population size m, namely, increasing in a uniform way the number of points a finitepopulation EA can visit in the (r − 1)-dimensional simplex. In particular, it puts us in the position of being able to characterize the extent to which the trajectories of finite-population EAs match up with the trajectories of infinite-population models as a function of m. A useful exercise for the reader at this point is to calculate how many unique populations of size m can be constructed from r genotypes. Just as we did in the infinite-population case, since P (t) =, we seek to develop a set of r dynamical equations of the form: c i (t +1)=evolve i (P (t)), i =1... r However, for finite population models, the c i can only take on a discrete set of values of the form k/m where m is the population size and k is an integer in the range 0...m. This creates difficulties mathematically since the most natural way to write down equations for evolve i involves the probabilities associated with the stochastic elements of an EA. Since these probabilities can take on arbitrary real values between 0.0 and 1.0, there is no easy way to guarantee c i values of the form k/m. The simplest approach to dealing with this issue is to shift to “expected value” models in which the population update rules are of the form: E[c i (t +1)]=evolve i (P (t)), i =1... r and make predictions about the makeup of future finite populations “on average”. In the simplex, trajectories of these models start at one of the finite-population points, but then are free to visit any point in the simplex. The concern, of course, is the extent to which the trajectories of “real” EAs resemble these “average” trajectories. As we will see, they can and do vary quite dramatically from their expected trajectories. This does not mean that expected value models provide little in the way of additional insights into the behavior of actual EAs. Historically they have proved to be quite helpful.
120 CHAPTER 6. EVOLUTIONARY COMPUTATION THEORY The alternative to adopting an expected value model is to appeal to a more powerful (and more complex) stochastic analysis framework. The one most frequently used is a Markov chain analysis that allows one to characterize how entire probability distributions evolve over time rather than just the means of the distributions. This is as much as can be said at this level of generality about modeling EA population dynamics. To go further, we must model specific aspects of EA components, namely how selection and reproduction are implemented. Since these EA components interact in complex, nonlinear ways, our strategy here will be to understand first the effects of selection and reproduction in isolation, and then analyze their interactions. 6.3 Selection-Only Models If we remove reproductive variation from our models, then we are left with models of the form: P (t +1)=evolve(P (t)) = survival selection(parent selection(P (t))) There are two major categories to be considered: 1) non-overlapping-generation models in which all of the parents die and only the offspring compete for survival, and 2) overlappinggeneration models in which both parents and offspring compete for survival. We explore both of these in the following sections. 6.3.1 Non-overlapping-Generation Models In the case of non-overlapping-generation models, we can initially simplify things further by assuming that all of the offspring survive, i.e., P (t +1)=evolve(P (t)) = parent selection(P (t)) This is the standard interpretation of infinite-population models and is also true for nonoverlapping finite-population models in which the sizes of parent and offspring populations are identical. As noted earlier, since P (t) =, we seek to develop a set of r dynamical equations of the form: c i (t +1)=evolve i (P (t)), i =1... r For parent-selection-only EAs this is quite straightforward: c i (t +1) = prob select i (t), i =1... r since the number of x i genotypes in the next generation is completely determined by the number of times x i is chosen to be a parent. The actual probabilities will be a function of two things: the frequency of the different genotypes in the current population, and the degree to which selection is biased by genotype fitness. The following subsections summarize these issues for the various forms of selection typically used in practice.
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Evolutionary Computation
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c○ 2006 Massachusetts Institute o
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vi CONTENTS 4 A Unified View of Sim
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viii CONTENTS 6.5.7 Selection, Repr
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Chapter 1 Introduction The field of
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1.2. EV: A SIMPLE EVOLUTIONARY SYST
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1.2. EV: A SIMPLE EVOLUTIONARY SYST
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1.3. EV ON A SIMPLE FITNESS LANDSCA
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1.4. EV ON A MORE COMPLEX FITNESS L
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1.4. EV ON A MORE COMPLEX FITNESS L
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1.5. EVOLUTIONARY SYSTEMS AS PROBLE
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1.6. EXERCISES 21 1.6 Exercises 1.
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24 CHAPTER 2. A HISTORICAL PERSPECT
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Chapter 3 Canonical Evolutionary Al
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3.3. EVOLUTIONARY PROGRAMMING 35 55
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3.4. EVOLUTION STRATEGIES 37 55 50
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3.4. EVOLUTION STRATEGIES 39 55 50
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3.5. GENETIC ALGORITHMS 41 Define s
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3.5. GENETIC ALGORITHMS 43 3.5.2 Un
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3.5. GENETIC ALGORITHMS 45 Populati
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3.6. SUMMARY 47 3.6 Summary The can
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50 CHAPTER 4. A UNIFIED VIEW OF SIM
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170 CHAPTER 6. EVOLUTIONARY COMPUTA
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Chapter 7 Advanced EC Topics The pr
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7.2. DYNAMIC LANDSCAPES 213 nested
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7.2. DYNAMIC LANDSCAPES 215 10 8 Fi
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7.2. DYNAMIC LANDSCAPES 217 10 8 Fi
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7.3. EXPLOITING PARALLELISM 219 7.2
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7.4. EVOLVING EXECUTABLE OBJECTS 22
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7.5. MULTI-OBJECTIVE EAS 223 be of
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7.7. BIOLOGICALLY INSPIRED EXTENSIO
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Chapter 8 The Road Ahead As a well-
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Appendix A Source Code Overview Rat
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A.1. EC1: A VERY SIMPLE EC SYSTEM 2
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A.3. EC3: A MORE FLEXIBLE EC SYSTEM
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A.3. EC3: A MORE FLEXIBLE EC SYSTEM
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Bibliography Altenberg, L. (1994).
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BIBLIOGRAPHY 243 Collins, R. and D.
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BIBLIOGRAPHY 245 Frank, S. A. (1995
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BIBLIOGRAPHY 247 Jansen, T., K. De
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BIBLIOGRAPHY 249 Potter, M. A., J.
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BIBLIOGRAPHY 251 Spears, W. (2000).
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Index 1-point crossover operator, 2
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INDEX 255 graph structures, 74-76,
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Evolutionary Computation : A Unified Approach

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