Power Magnetic Devices. Scott D. Sudhoff. Читать онлайн. Newlib. NEWLIB.NET

Автор: Scott D. Sudhoff
Издательство: John Wiley & Sons Limited
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Жанр произведения: Техническая литература
Год издания: 0
isbn: 9781119674634
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increasing the sheer number of genotypes of the gametes, crossover plays another critical role because it allows beneficial genes (traits) on a given chromosome to be decoupled from detrimental genes (traits). Crossover will play a very important role in the operation of GAs.

      A final concept from biology that will serve our needs for an optimization engine is the idea of natural selection and the survival of the fittest, an idea stemming from Charles Darwin’s voyages of the H.M.S Beagle, during a period of time roughly contemporary with the work of Mendel and the American Civil War. The idea that the most fit individuals of a population survive to reproduce is directly used in GAs. These algorithms are based on an explicit fitness function, which will be used to determine which individuals “survive” and will be placed into a mating pool.

      Clearly, the discussion in this section is at a high level and has been greatly simplified. The interested reader is referred to Crow [2] for a more thorough introduction to the topic.

      A century after the work of Mendel and Darwin, but a mere decade after the work of Watson and Crick, John Holland, a professor at the University of Michigan, proposed using the principles of biological genetics as a computation algorithm for optimization, a concept instantiated by a GA [3]. In this section, we will begin our consideration of GAs with a canonical GA similar to Holland’s original vision.

      GAs are quite different from traditional optimization algorithms. First of all, GAs operate not on the argument of the function being optimized, but rather on an encoding of the argument. Second, rather than iterating to improve an estimate for an optimizer, GAs iterate to improve a large number of different estimates of the optimizer. This collection of estimates will be referred to as a population. The use of a population of estimated solutions improves the chances of finding a global optimum. Third, GA operations are based only on the values of the objective function—gradients and Hessians are not used, nor even estimated. This property is useful in function with discontinuities or with a discrete or mixed search space. Finally, GA operations are based on probabilistic rather than deterministic computations.

      The first concept that must be set forth in a GA is that it, like evolution, operates on a population, not on an individual. We will denote the population within the GA as P[k], where k is the generation number. The kth generation consists of a number of individuals, that is,

      (1.5-1)equation

      where θi is the genetic code for the ith individual in the kth generation of the population and where Np denotes the number of individuals in the population, which should be an even number. The genetic code for the ith individual may be organized as

      The fact that the genes are encoded results in a limitation of the domain of the parameter vector. In other words, the domain of possible values of each element of the parameter vector is inherently limited. In some cases, this property is very convenient, but in other cases this limitation on the domain of the parameter vectors is disadvantageous.

      Associated with the genetic code for each population member, we will have a decoding function that translates the genetic code into a parameter vector. In particular,

      where xi is the parameter vector of the ith member of the population and is structured as

      (1.5-4)equation

      As can be seen, xi has one element (denoted with a subscript) for each gene. However, it is not partitioned into chromosomes.

      Based on the parameter vector of ith population member, the objective function can be evaluated. In particular,

      In the case of a GA, the objective function is referred to as a fitness function. It will be used in a “survival of the fittest” sense to determine which members of the population will mate to form the next generation. In the context of a GA, fitness is viewed in a positive sense, thus it is assumed that we wish to maximize the fitness function. Fortunately, it is a straightforward matter to convert between maximization of a function and the minimization of a function. At this point, we have enough background to discuss the computational aspects of a GA. However, before doing this, it is appropriate to briefly pause in our development and consider an example.

      Suppose the 13th member of the population has the genetic code

      The decoding algorithm is on a gene‐by‐gene basis and is of the form