两类广义多乘积规划问题的优化算法

发布时间：2018-10-22 19:08

【摘要】：非凸规划问题作为一类重要的优化问题,能广泛应用于经济金融、信息技术、工业制造等多个重要领域.通常情况下,该类问题往往存在多个非全局最优的局部最优解,因此寻找其全局最优解极其困难.由于非凸优化问题在现实生活中的广泛应用,近年来引起越来越多研究者的关注,一些优化方法相继被提出,本文针对非凸优化问题中的两类广义多乘积规划问题分别提出了相应的优化算法.和已有的方法相比,本文提出的分支定界算法和迭代算法在保证最优解的质量的同时还很大程度上提高了其执行效率.主要内容如下:第一章,首先给出本文所研究的优化问题模型.其次对该优化问题的应用背景、理论意义及当前的研究工作做简要介绍,最后呈现本文的主要工作.第二章,根据广义线性多乘积优化问题的特点,提出了一个新的分支定界算法.首先通过引入变量获得原问题的等价问题,接着采用凸松弛技巧将等价问题转化为凸规划问题,然后基于一个新的分支规则来求解一系列的凸规划问题从而获得原问题的全局最优解,最后从理论上证明该算法的全局收敛性.数值实验结果说明本章算法对于求解广义线性多乘积规划问题具有一定的优势.第三章.针对广义多项式乘积优化问题,给出了一个迭代算法.首先引入变量得到与原问题等价的广义几何规划问题,其次运用算术-几何平均不等式及罚函数思想将广义几何规划问题转化成标准几何规划形式,然后通过求解一系列的标准几何规划问题得到原问题的最优解,最后给出迭代算法的收敛性.数值实验结果表明该算法是有效可行的.
[Abstract]:As an important optimization problem, non-convex programming problem can be widely used in many important fields, such as economy and finance, information technology, industrial manufacturing and so on. In general, there are many non-global optimal local optimal solutions for this kind of problems, so it is very difficult to find their global optimal solutions. Because of the wide application of non-convex optimization problem in real life, more and more researchers pay attention to it in recent years, and some optimization methods have been proposed one after another. In this paper, two kinds of generalized multi-product programming problems in non-convex optimization problems are proposed. Compared with the existing methods, the proposed branch-and-bound algorithm and iterative algorithm not only guarantee the quality of the optimal solution, but also greatly improve its execution efficiency. The main contents are as follows: in the first chapter, the optimization model studied in this paper is given. Secondly, the application background, theoretical significance and current research work of the optimization problem are briefly introduced. Finally, the main work of this paper is presented. In chapter 2, according to the characteristics of generalized linear multiple product optimization problem, a new branch and bound algorithm is proposed. First, the equivalent problem of the original problem is obtained by introducing variables, and then the equivalent problem is transformed into a convex programming problem by using convex relaxation technique. Then a series of convex programming problems are solved based on a new branching rule to obtain the global optimal solution of the original problem. Finally, the global convergence of the algorithm is proved theoretically. Numerical results show that this algorithm has some advantages in solving generalized linear multiple product programming problems. Chapter 3 An iterative algorithm is proposed for generalized polynomial product optimization. The generalized geometric programming problem equivalent to the original problem is obtained by introducing variables, and then the generalized geometric programming problem is transformed into a standard geometric programming form by using the arithmetic-geometric mean inequality and penalty function. Then the optimal solution of the original problem is obtained by solving a series of standard geometric programming problems. Finally, the convergence of the iterative algorithm is given. Numerical results show that the algorithm is effective and feasible.
【学位授予单位】：河南师范大学
【学位级别】：硕士
【学位授予年份】：2017
【分类号】：O221

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