线性互补问题数值算法研究

【作者】 刘志敏；

【导师】 杜守强；

【作者基本信息】 青岛大学 ， 计算数学， 2017， 硕士

【摘要】 互补问题是数学规划中一类重要的问题,在经济均衡问题和工程技术问题等研究领域有很多重要的应用。经过几十年的研究,互补问题的理论和算法都得到了很大的发展。由于现实问题中多含有不确定数据,近年来人们开始关注含有随机变量的随机互补问题。随机线性互补问题是随机互补问题中的基本问题,其理论和算法的研究对随机互补问题的求解有重要的参考意义。因此,在本文中我们对基本的线性互补问题和随机线性互补问题的求解算法进行了研究。本论文的结构和主要研究内容概括如下:第一章考虑了基本的线性互补问题,给出了Levenberg-Marquardt型算法,在一般条件下,得到了算法相应的全局收敛结果并给出了相应的数值实验表明算法的有效性。第二章考虑了一类离散型随机线性互补问题,给出了非光滑Levenberg-Marquardt型算法,并且给出了算法相应的全局收敛结果与相应的数值实验。第三章考虑了离散型广义随机线性互补问题,给出了一个新的共轭梯度投影算法,并在一般条件下,给出了算法相应的全局收敛结果与相应的数值实验。更多还原

【Abstract】 The complementarity problem is one of the most important research subjects in mathematical programming.It has a wide range of applications in engineering and economics.After decades of research,the subject has developed into a very fruitful discipline.The developments include a rich mathematical theory and a host of effective solution algorithms.Since some elements may involve uncertain data in many practical problems,the stochastic versions of complementarity problems has drawn much attention in the recent literature.The stochastic linear complementarity problem is the basic problem of stochastic complementarity problem.The study of theories and algorithms of stochastic linear complementarity problems has important reference value to stochastic complementarity problems.So we focus on the basic linear complementarity problems and the stochastic linear complementarity problems.The structure and main contents of this thesis are summarized as follows:In the first chapter,we consider the basic linear complementarity problems and the Levenberg-Marquardt-type methods is given.And in the general conditions,the global convergence result is also proved.The numerical experiments are presented to show the effectiveness of this method.In the second chapter,we consider a class of stochastic linear complementarity problems with finitely many elements.We propose a feasible nonsmooth Levenberg-Marquardt-type method.And the corresponding global convergence result and the numerical experiments are also given.In the third chapter,we consider the stochastic generalized linear complementarity problems and propose a new conjugate gradient projection method.In the general conditions,we give the global convergence result and the related numerical experiments of the given method.更多还原