Limit Properties of Evolutionary Algorithms
Witold Kosiski, Stefan Kotowski · InTech eBooks · 2008
One of the main results reported in this Chapter is the limiting algorithm and populations' distribution at the end of infinite steps. Theorem 5 does not tell about the form of the next population when actual population is known; it gives rather the limit distribution of all possible populations of the algorithm considered. The limiting algorithm describes globally the action of the genetic algorithm. It plays the role of the law of big numbers, known from the probability theory, however, for genetic algorithms. Knowledge the limiting algorithm could help in standard calculations: just in one step one could obtain the limit distribution. It could accelerate calculations and gives chance to omit the infinite numbers of calculation steps. If the limiting algorithm is known an extra classification tool is for our disposal, and new hierarchial classification method can be suggested. It will base not only on entropy, fractal and dimensions of trajectory, but on transition matrix T, its eigenvalues, eigenvectors and limiting matrix Q. This hierarchie could be as follows: ? ? ? Two genetic algorithms are equivalent if their transition matrices are the same. Two genetic algorithms are equivalent if they have the same limit distribution . Two genetic algorithms are equivalent if their limiting algorithm, described by the matrix Q is the same.