Modified genetic algorithm of global extremum search in combination with directional methods

In the paper, the stochastic method of global extremum search is discussed, modified and tested. The method is based on normal distribution modeling and provides covariance matrix adaptation. The method is iterative; a genetic algorithm has been developed on its basis. The coordinates of the trial points of each generation are determined using the "best" points of the previous generation and the values of standard normal random variables. Thus, at each stage of the search, a normal distribution is simulated, and its parameters (the mean and the covariance matrix) are estimated through the positions of the "best" points of the previous generation. In this case, there is no need to calculate, store and transform the covariance matrix, which is indisputable advantage of this method.
Practice has shown that the dispersion ellipsoid of normal distribution shrinks rapidly with generation number increasing, which can lead to an excessive narrowing the scanning area and obtaining a local extremum instead of a global one. The proposed modification of the method avoids this situation. The trial points are divided into two groups, which are simulated using normal random variables with different standard deviations, at least one of which is greater than 1. Thus, a kind of mutation of the population is carried out, which makes it possible to provide a sufficient number of sample points both near the “best” one and at a distance from it.
The modified genetic algorithm is applied to solving the problem of estimating the parameters of nonlinear parametric regression. A successful minimization of the multiextremal function is performed. The stochastic method is used in combination with directional. The numerical results presented confirm the effectiveness of the introduced modification of the genetic algorithm and make it possible to choose from two directed methods the more efficient one for the problem under consideration.