求解非线性方程组的混合遗传算法

Hybrid Genetic Algorithm for solving systems of nonlinear equations

非线性方程组的求解是数值计算领域中最困难的问题,大多数的数值求解算法例如牛顿法的收敛性和性能特征在很大程度上依赖于初始点。但是对于很多非线性方程组,选择好的初始点是一件非常困难的事情。本文结合遗传算法和经典算法的优点,提出了一种用于求解非线性方程组的混合遗传算法。该混合算法充分发挥了遗传算法的群体搜索和全局收敛性,有效地克服了经典算法的初始点敏感问题;同时在遗传算法中引入经典算法(Powell法、拟牛顿迭代法)作局部搜索,克服了遗传算法收敛速度慢和精度差的缺点。选择了几个典型非线性方程组,从收敛可靠性、计算成本和适用性等指标分析对比了不同算法。计算结果表明所设计的混合遗传算法有着可靠的收敛性和较高的收敛速度和精度,是求解非线性方程组的一种成功算法。

Solving systems of nonlinear equations is perhaps the most difficult problem in all of numerical computation. For most numerical methods such as Newton's method for solving systems of nonlinear equations, their convergence and performance characteristics can be highly sensitive to the initial guess of the solution supplied to the methods. However, it is very difficult to select a good initial guess for most systems of nonlinear equations. Aiming at these problems, a hybrid genetic algorithm (HGA) was put forward, which combined the advantages of genetic algorithm (GA) and classical algorithms. The HGA sufficiently exerted the advantages of GA such as group search and global convergence can efficiently overcome the problem of high sensitivity to initial guess; and it also has a high convergence rate and solution precision just because it uses those high local-convergence classical algorithms (Powell, Quasi-Newton Method) for local search. Convergence reliability, computational cost and applicability of different algorithms were compared by testing several classical equations of nonlinear equations. The numerical computations show that hybrid approach for solving systems of nonlinear equations has reliable convergence probability, high convergence rate and solution precision, and is a successful approach in solving systems of nonlinear equations.