【作者】 叶海；

【导师】 马昌凤；

【作者基本信息】 福建师范大学 ， 基础数学， 2009， 硕士

【摘要】 本文针对非线性数值的问题,结合经典优化算法和遗传算法,构造新的混合遗传算法,数值模拟试验表明,该算法具有很高的精确性和较好的收敛性,是求解非线性数值问题的一种有效算法。在第一章中,概述了遗传算法的基本理论,论述了混合遗传算法的研究现状。在第二章中,分析和论述了非线性方程组的基本解法,主要介绍了牛顿型算法、信赖域方法和遗传算法在求解非线性方程组中应用。在第三章中,提出了一种求解非线性互补问题(NCP)的混合遗传算法,首先将NCP转化为等价的最优问题,然后利用浮点遗传算法快速求出NCP等价优化问题的接近精确解的较优解,将其作为牛顿型算法的初始迭代值,利用牛顿算法的局部寻优能力强的特点,快速迭代至满足精度要求的数值解。该算法保证了全局收敛性,克服了传统算法的缺点,数值试验结果验证了这种混合遗传算法的有效性。在第四章中,提出将非线性不等式组求解问题转化为等价的求解无约束极小化问题的全局最优解,基于浮点遗传算法全局群体搜索能力强和起始搜索速度快的特点,用浮点遗传算法快速求出无约束极小化问题的接近精确解的较优解,作为(拟)牛顿算法的初始迭代值,然后转入牛顿型迭代求得满足精度要求的近似解。这种算法充分利用了牛顿型方法收敛速度快的优点,又解决了牛顿型方法初始值选取的困难,数值试验结果验证了该混合遗传算法的有效性。更多还原

【Abstract】 For solve the nonlinear numerical problems, the traditional optimization and genetic algorithm will be combined for proposing a new hybrid genetic algorithm. Numerical experiments show that this algorithm has highly precision and reliable convergent resulting.In Chapter 1, we review the basic theory of genetic algorithm, introduce the research state and advances of hybrid genetic algorithm.In Chapter 2, we analyze some basic methods for solving the nonlinear equations. And introduce three methods for solving the nonlinear equations.In Chapter 3, we propose a hybrid genetic algorithm for solving nonlinear complement-tarity problems (denoted by NCP). At first, we transform NCP into the equivalent optimization problems.Then taking advantage of excellence of the floating genetic algorithms, we gain the superior results which close to precise solutions quickly, and then taking the results as the initial values of Newton or quasi-Newton iterations, which has strong ability in locally converging to precise solution, we obtain satisfactory approximation solution. The hybrid genetic algorithm absorbs fully the merits of the floating genetic algorithm and the Newton-type’methods.Some numerical results show that this method is effective.In Chapter 4, we discuss a method that transform the problems of solving nonlinear equations into an unconstrained optimization problem.Then taking advantage of excellence of the floating genetice algorithms,we obtain the superior results which close to precise solution quickly,and then taking the results as the initial values of Newton or quasi-Newton interations.This algorithm absorbs fully the merits of the Newton methods and overcome the limitations of initial values. The numerical results show the effectiveness of the algorithm.更多还原