Relación entre valores de dureza y densidad

Abstract

Real-valued vector representations are described and consideration is given to the relationship between the cardinality of the alphabet of symbols and implicit parallelism. Attention is also given to mechanisms of altering data structures under real-valued representation.

C1.3.1 Object variables

When posed with a real-valued function optimization problem of the form find the vector x such that F (x) : Rn _{→ R is minimized (or maximized), evolution strategies (B¨ack and Schwefel 1993) and B1.3} evolutionary programming (Fogel 1995, pp 75–84, 136–7) typically operate directly on the real-valued B1.4 vector x (with the components of x identified as object parameters). In contrast, traditional genetic B1.2 algorithms operate on a coding (often binary) of the vector x(Goldberg 1989, pp 80–4). The choice to use a separate coding rather than operating on the parameters themselves relies on the fundamental belief that it is useful to operate on subsections of a problem and try to optimize these subsections (i.e. building blocks) in isolation, and then subsequently recombine them so as to generate improved solutions. More specifically, Goldberg (1989, p 80) recommends

The user should select a coding so that short, low-order schemata are relevant to the underlying problem and relatively unrelated to schemata over other fixed positions.

The user should select the smallest alphabet that permits a natural expression of the problem.

Although the smallest alphabet generates the greatest implicit parallelism (see Section B2.5), there is B2.5 no empirical evidence to indicate that binary codings allow for greater effectiveness or efficiency in solving real-valued optimization problems (see the tutorial by Davis (1991, p 63) for a commentary on the ineffectiveness of binary codings).

Evolution strategies and evolutionary programming are not generally concerned with the recombination of building blocks in a solution and do not consider schema processing. Instead, solutions are viewed in their entirety, and no attempt is made to decompose whole solutions into subsections and assign credit to these subsections.

With the belief that maximizing the number of schemata being processed is not necessarily useful, or may even be harmful (Fogel and Stayton 1994), there is no compelling reason in a real-valued optimization problem to act on anything except the real values of the vectorxthemselves. Moreover, there has been a general trend away from binary codings within genetic algorithm research (see e.g. Davis 1991, Belew and Booker 1991, Forrest 1993, and others). Michalewicz (1992, p 82) indicated that for real-valued numerical optimization problems, floating-point representations outperform binary representations because they are more consistent and more precise and lead to faster execution. This trend may reflect a growing rejection of the building block hypothesis as an explanation for how genetic algorithms act as optimization

procedures (see Section B2.5). B2.5

With evolution strategies and evolutionary programming, the typical method for searching a real- valued solution space is to add a multivariate zero-mean Gaussian random variable to each parent involved in the creation of offspring (see Section C3.2.2). In consequence, this necessitates the setting of the C3.2.2

Real-valued vectors

covariance matrix for the Gaussian perturbation. If the covariances between parameters are ignored, only a vector of standard deviations in each dimension is required. There are a variety of methods for setting these standard deviations. Section C3.2.2 offers a variety of procedures for mutating real-valued vectors. C3.2.2

C1.3.2 Object variables and strategy parameters

It has been recognized since 1967 (Rechenberg 1994, Reed et al 1967) that it is possible for each solution to possess an auxiliary vector of parameters that determine how the solution will be changed. Two general procedures for adjusting the object parameters via Gaussian mutation have been proposed (Schwefel 1981, Fogel et al 1991) (see Section C3.2.2). In each case, a vector of strategy parameters for self-adaptation is included with each solution and is subject to random variation. The vector may include covariance or rotation information to indicate how mutations in each parameter may covary. Thus the representation consists of two or three vectors:

(x,σ,α)

wherexis the vector of object parameters (x1, . . . , xn),σ is the vector of standard deviations, andα is the vector of rotation angles corresponding to the covariances between mutations in each dimension, and may be omitted if these covariances are set to be zero. The vectorσmay havencomponents(σ1, . . . , σn) where each entry corresponds to the standard deviation in theith dimension,i=1, . . . , n. The vectorσ may also degenerate to a scalarσ in which case this single value is used as the standard deviation in all dimensions. Intermediate numbers of standard deviations are also possible, although such implementation is uncommon (this also applies to the rotation angles α). Very recent efforts by Ostermeier et al (1994) offer a variation on the methods of Schwefel (1981) and further study is required to determine the general effectiveness of this new technique (see Section C3.2.2).

Recent efforts in genetic algorithms have also included self-adaptive procedures (see e.g. Spears 1995) C7.1 and these may incorporate similar real-valued coding for variation operators including crossover and point mutations, on both real-valued or binary or otherwise coded object parameters.

References

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