Solutions to systems of nonlinear equations via a genetic algorithm

Solving systems of nonlinear equations is perhaps the most difficult problem in all of numerical computation. It is also a problem that occurs frequently in a spectrum of engineering applications such as electric power generation and distribution, multi-objective optimization, and trajectory/path-planning applications. Although numerous methods have been developed to attack this class of numerical problems, one of the simplest and oldest methods, Newton's method, is arguably the most commonly used. Like most numerical methods for solving systems of nonlinear equations, the convergence and performance characteristics of Newton's method can be highly sensitive to the initial guess of the solution supplied to the method. In this paper, a hybrid scheme is presented, in which a genetic algorithm is used to locate efficient initial guesses, which are then supplied to a Newton method for solving a system of nonlinear equations. The hybrid scheme is tested on a specific example that is representative of this class of problems-one of determining the coefficients used in Gauss-Legendre numerical integration. Results show that the hybrid of a genetic algorithm and Newton's method is effective, and represents an efficient approach to solving systems of nonlinear equations. (C) 1998 Elsevier Science Ltd. All rights reserved.