等式约束优化与极大极小化问题的罚函数研究

发布时间：2018-04-26 01:40

本文选题：非线性优化 + 等式约束优化　；参考：《曲阜师范大学》2017年硕士论文

【摘要】：在现实生活中会遇到在众多方案中选择一类方案使得资源使用效益最大或者目标成本最低的问题,这样的一类问题称为最优化问题.最优化问题根据有无约束条件划分为约束优化问题和无约束优化问题.在理论推理和算法设计方面,约束优化问题和无约束优化问题有很大的不同,但此两类问题在某种情况下是可以相互转化的.一般情况下,无约束优化问题比约束优化问题的求解相对容易.本文选择非线性规划中的罚函数方法将约束优化问题转化为无约束优化问题,通过求解无约束的罚问题来求解带有等式或不等式的约束优化问题.对于传统的罚函数,若是简单光滑的,则一定不精确;若是简单精确的,则不光滑.因此本文的主要工作是改造传统罚函数,使简单罚函数既是精确的,又是光滑的.本文结构安排如下:第一章主要介绍约束优化问题和罚优化问题的基本概念、基础知识以及本文的主要工作.第二章针对等式约束优化问题,通过对约束函数增加变量,提出一类简单罚函数并结合K-K-T条件和Lagrange函数证明这一类简单罚函数在有界闭集上同时具有光滑性和精确性.本章提出一种新的算法解决此类等式约束优化问题并给出数值例子说明算法的可行性.第三章针对等式约束优化问题,提出一类新的简单罚函数并证明它是光滑精确的.最后给出数值例子说明本章所给算法的可行性.第四章针对不等式约束优化问题,引入目标罚因子和约束罚因子,提出一类新的简单精确罚函数.此罚函数同时惩罚目标函数和约束函数,使得约束函数的违反度减小的同时目标函数趋近于最优值.基于此类新的罚函数分别给出全局最优求解算法和局部最优求解算法,并且分别证明了算法的收敛性.最后给出数值算例,说明所给算法是可行的.
[Abstract]:In real life, there will be a problem of selecting a class of schemes in many schemes to make the use of the resources maximum or the lowest cost of the target. Such a problem is called the optimization problem. The optimization problem is divided into constrained optimization and unconstrained optimization problems based on unconstrained conditions. In theoretical reasoning and algorithm design, There are great differences between the constrained optimization problem and the unconstrained optimization problem, but the two kinds of problems can be converted to each other in some circumstances. In general, the unconstrained optimization problem is relatively easier than the constrained optimization problem. In this paper, the penalty function method in nonlinear programming is chosen to transform the constrained optimization problem into unconstrained optimization question. The problem of solving the constrained optimization problem with equality or inequality is solved by solving unconstrained penalty problems. For the traditional penalty function, if it is simple and smooth, it is not accurate; if it is simple and accurate, it is not smooth. Therefore, the main work of this paper is to transform the traditional penalty function and make the simple penalty function both accurate and smooth. The structure is arranged as follows: the first chapter mainly introduces the basic concepts, basic knowledge and the main work of the penalty optimization problem and penalty optimization problem. In the second chapter, a simple penalty function is proposed by adding variables to the constraint function, and the simple penalty is proved by combining the K-K-T condition and the Lagrange function. In this chapter, a new algorithm is proposed to solve this kind of equality constrained optimization problem and a numerical example is given to illustrate the feasibility of the algorithm. In the third chapter, a new simple penalty function is proposed for equality constrained optimization problem and it is proved to be smooth and accurate. Finally, a numerical example is given. In this chapter, the feasibility of the algorithm is explained. In the fourth chapter, a new simple exact penalty function is proposed by introducing the target penalty factor and constraint penalty factor for the inequality constrained optimization problem. This penalty function punishes both the target function and the constraint function, which makes the target function close to the optimal value at the same time, and the objective function is close to the optimal value. The new penalty function gives the global optimal solution algorithm and the local optimal solution algorithm respectively, and proves the convergence of the algorithm respectively. Finally, a numerical example is given to illustrate the feasibility of the proposed algorithm.

【学位授予单位】：曲阜师范大学
【学位级别】：硕士
【学位授予年份】：2017
【分类号】：O224

【参考文献】