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# Has been compared the direct and grammar codification methods

106 Manrique, Ríos & Rodríguez-Patón

1.

The morphological gradient operator, gb(fi): Dfi→ℜ, is applied on each vector fi, i = 0, 1, .., l-1, with a structuring element b:Db→ℜ defined as:

2.

Let ϕ: ℜ → ℜ be a given function. The maximum gene is defined as gimax = max(fi)-ϕ(gi). Likewise, the minimum gene is defined as gimin = min(fi)+ ϕ(gi). These values determine the crossover interval Ci=[gimin,gimax], from where the desired value oi is randomly taken. The ith gene for the other descendant o’iiscalculated from inside the crossover interval using the following formula: o’i = gimax + gimin – oi.

The objective is now to get a rule that can dynamically control the range of the crossover interval to make trapping in local minima less likely and to get a high convergence speed. When the individuals to be crossed are diverse (which implies a high value of the gradient), the crossover interval is made narrower according to the values max(fi) and min(fi), and the interval can be searched for the optimum much faster. On the other hand, if the individuals to be crossed are very similar (gradient close to zero), which means that the population is converging, then it is advisable to expand the interval [min(fi),max(fi)] to allow exploration of new points in the domain, thus avoiding the possible convergence to a local optimum. This possi-bility of expanding or narrowing the crossover interval depending on the value of the gradient gi is given by the function ϕ, which is shown in Figure 7. It is defined in domain [0,1], so the population has to be normalised in the same range. The application of this function in the described crossover operator is very efficient, as each pair of new individuals can be generated by only one multiplication.

The Evolutionary System for

Designing and Training

Figure 7. Function j used by the mathematical morphology crossover

0 .4

0 0 .2
1 ϕ (g ) i
(0.2·g ) i 0.01 if g i≤

(0.5·g ) i

0.1

-0 .0 1
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