【作者】 王薇；

【导师】 张连生；

【作者基本信息】 上海大学 ， 运筹学与控制论， 2005， 博士

【摘要】 最优化是一门应用相当广泛的学科,它讨论决策问题的最佳选择,构造寻求最佳解的计算方法并研究这些方法的理论性质及实际计算表现。由于社会的进步和科学技术的发展,最优化问题广泛见于经济计划,工程设计,生产管理,交通运输,国防军事等重要领域,因此受到高度重视。伴随着计算机的高速发展和最优化工作者的努力,最优化的理论分析和计算方法得到了极大提高。本论文主要工作就是讨论,研究了非线性最优化问题的几个算法及理论分析。 本文包含五章内容。第一章简述了目前国内外几种主要的全局最优化问题和算法及本论文所要用到的非线性规划的一些基本概念及性质。后面四章由四篇基本独立的文章组成。 第二章和第三章主要讨论求解无约束全局最优化问题的变换函数法。求解一般函数的全局最优解问题是热点课题之一。对全局问题有两个困难需要解决。一是如何从一个局部极小解出发找到更好的局部解,另一个是全局最优解的判定问题。打洞函数法和填充函数法是解决第一个困难的实用方法。它们的共同点是如果已经找到了一个局部极小x₁^*,但它不是全局最小,我们可以在x₁^*处构造一个辅助函数-打洞函数或填充函数使迭代点列离开x₁^*所在的谷域,找到更好的点x′（即x′处的函数值比x^*处的函数值更小）。然后以x′为起点找出更优的局部极小点。第二章定义了两类变换函数,在适当的条件下证明了它们兼具打洞函数和填充函数的特点和性质,即填充函数法和打洞函数法两种方法存某种意义下是可以统一的,因此可称其为T-F函数。第三章给出了几个简单,易于计算且函数性态较好的变换函数,同样它们兼具打洞函数和填充函数的特点和性质。文章证明了第二,三章定义的变换函数的主要性质:在f（x）的值比当前局部极小值f（x₁^*）大的水平集上变换函数没有极小点或稳定点;在比当前局部极小值小的水平集上变换函数一定有极小值点。当然这两章也给出了数值试验结果。 第四章将用于无约束全局最优问题的思想方法拓广到求解带有约束的非线性规划问题的全局最优问题。首先,对于求解带有线性约束的非线性规划问题的更多还原

【Abstract】 The optimization is a widely used discipline, which discusses the characters of optimal choice on decision problems and constructs computing approaches to find the optimal solution. Due to the advancement of society and the development of science and technology, the optimization problems are often discovered in the field of economic planning administration, engineering design, production management, traffic transportation, national defence and so on. They are so important that meet with much recognition. With the speedy development of computer and the hard work of scientists, the theoretic analysis and computational methods on optimization have been highly improved .This paper mainly consists of five chapters.In the first chapter, some mainly methods for global optimization problems are briefly presented. And several basis concepts and characters on generally nonlinear programming are introduced.In the second and third chapter the transformation functions for unconstrained global optimal problems are mainly discussed. To find the effective methods for finding the global optimal solutions of a general multi-mininiizers function is one of the hot topics. There two difficulties in global optimization. One is how to leave from a local minimizer to a smaller one and the other is how to judge that the current minimizer is global. The tunnelling function proposed by Levy and Montalvo (1985) and filled function algorithms introduced by Ge and Qin (1987) are two well-known and practical methods for settling the first difficulty. They have common character. If a local minimizer x*1 has been found, we can make a auxiliary function, such as tunnelling function or filled function, such that iterative sequential points leave the valley in which x*1 lies to find a better point x’ in the lower valley (i.e. f(x’) < f(x*1). Then let x’ be a new initial point to search for a better minimizer. In second chapter two classes of transformation functions for global optimization are defined and it is proved theoretically and computationally that they possess the both characters of tunnelling functions and filled functions under some general assumptions. In third chapter some easy and computable transformation functions are presented. They have the both characters of tunnelling functions and filled functions as well. We proved the main characters of transformation functions, that is, the transformation functions have no any minimizer or stationary point on the region {x : f(x) ≥ f(x*1)} and have at least one minimizer on the region {x : f(x) < f(x*1)} if {x : f(x) ≥ f(x*1)} ≠ 0. Certainly, the numerical results are listed in these two chapters.In chapter four, the idea for unconstrained global optimization is extended to nonlinear global problems with constraints. First, there exist many effective methods for local minimizers of nonlinear programming. Because these methods更多还原