A Modified Hybrid Quasi-Newton/Genetic Algorithm For Nonlinear Equations

Recently, much attention has been paid to hybrid algorithms based on genetic algorithm and traditional iterative methods like steepest decent method, simplex method, quasi-Newton method and etc., which utilize both advantages of fast convergence of iterative methods and global convergence of genetic algorithms. And promising numerical results have been reported. It is needed to do further research on combining ways of the two kinds of methods. In this paper, a modified version of hybrid quasi-Newton/genetic algorithm is given, and numerical examples are given to show validity of the modified algorithm.The operator of crossover, mutation and selection in HGA have function of macroscopically search, that deals with a large scale search, while the quasi-Newton operator has function of local search, and deals with a small circle search and problem of accelerating of search.HGA have well combined their own advantages of GA and Quasi-Newton method, so HGA possess group searching and global convergence of GA, and have local strong searching ability of quasi-Newton method with upper convergent rate and precision. HGA overcome the shortcoming of GA's poor local searching ability, and when GA comes to its real solution, the convergence rate of GA is so slow as to result in "premature convergence" and low precision, and HGA effectively solve sensitivity to initial point of quasi-Newton method.In the fifth chapter, we select several typical examples to carry on the numerical experiment. We analyze and compare the new HGA with quasi-Newton method and SGA and the early HGA from indexes of convergent reliability, computational cost and applicability etc. The result shows: compared with the other three algorithms, the new HGA is more easily to converge to the overall solution, and the convergence rate is faster.