非线性最优化计算方法与算法

毕业论文

题目非线性最优化计算方法与算法学院数学科学学院

专业信息与计算科学

班级计算1201

学生陶红

学号***********

指导教师邢顺来

二〇一六年五月二十五日

摘要

非线性规划问题是一般形式的非线性最优化问题。本文针对非线性规划的最优化问题进行方法和算法分析。传统的求解非线性规划的方法有最速下降法、牛顿法、可行方向法、函数逼近法、信赖域法，近来研究发现了更多的求解非线性规划问题的方法如遗传算法、粒子群算法。本文对非线性规划分别从约束规划和无约束规划两个方面进行理论分析。

利用最速下降法和牛顿法两种典型算法求解无约束条件非线性规划问题，通过MATLAB程序求解最优值，探讨其收敛性和稳定性。另外给出了阻尼牛顿法，探讨其算法的收敛性和稳定性，求解无约束非线性规划比牛顿法的精确度更高，收敛速度更快。惩罚函数是经典的求解约束非线性的方法，本文采用以惩罚函数法为核心的遗传算法求解有约束条件非线性规划问题，通过MATLAB程序求解最优值，探讨其收敛性和稳定性。并改进遗传算法，给出适应度函数，通过变换适应度函数，提高算法的收敛性和稳定性。

关键词：非线性规划；最速下降法；牛顿法；遗传算法

ABSTRACT

Nonlinear programming problem is the general form of the nonlinear optimization problem. In this paper, we carry on the analysis of the method and algorithm aiming at the optimization problem of nonlinear programming. The traditional methods of solving nonlinear programming problems include steepest descent method, Newton method, the feasible direction method, function approximation method and trust region method. Recent studies found more method of solving nonlinear programming problems, such as genetic algorithm, particle swarm optimization (pso) algorithm. In this paper, the nonlinear programming is analyzed from two aspects: the constraint programming and the unconstrained programming.

We solve unconstrained condition nonlinear programming problem by steepest descent method and Newton's method, and get the optimal value through MATLAB. Then the convergence and stability are discussed. Besides, the damped Newton method is furnished. By discussing the convergence and stability of the algorithm, the damped Newton method has higher accuracy and faster convergent speed than Newton's method in solving unconstrained nonlinear programming problems.Punishment function is a classical method for solving constrained nonlinear. This paper solves nonlinear programming problem with constraints by using genetic algorithm method, the core of which is SUMT. Get the optimal value through MATLAB, then the convergence and stability are discussed. Improve genetic algorithm, give the fitness function, and improve the convergence and stability of the algorithm through transforming the fitness function.

Key words：Nonlinear Programming; Pteepest Descent Method; Newton Method; GeneticAlgorithm

目录

摘要 ........................................................................................................................................ I ABSTRACT .......................................................................................................................... I I 1 前言 .. (1)

1.1 引言 (1)

1.2 非线性规划的发展背景 (2)

1.3 国内外研究现状 (2)

1.4 研究主要内容及研究方案 (3)

1.4.1 研究的主要内容 (3)

1.4.2 研究方案 (3)

1.5 研究难点 (4)

2 预备知识 (5)

2.1 向量和矩阵范数 (5)

2.1.1 常见的向量范数 (5)

2.1.2 谱范数 (6)

2.2符号和定义 (6)

2.3 数值误差 (7)

2.4 算法的稳定性 (7)

2.5 收敛性 (9)

3 非线性规划模型 (10)

3.1 非线性规划模型 (10)

3.2 无约束非线性规划 (11)

3.2.1 最速下降法 (13)

3.2.2 牛顿法 (15)

3.2.2 阻尼牛顿法 (16)

3.3 约束非线性规划 (17)

3.3.1 惩罚函数法 (18)

3.3.2 遗传算法 (18)

3.3.3 自适应遗传算法 (20)

结论 (23)

参考文献 (24)

致谢 (25)

附录 (26)