几类变分不等式和互补问题的光滑化算法研究

发布时间：2018-04-28 23:06

本文选题：变分不等式和互补问题 + 非精确光滑牛顿算法　；参考：《淮北师范大学》2017年硕士论文

【摘要】：变分不等式和互补问题由于其重要的应用价值而备受关注.经过几十年的发展,变分不等式和互补问题在理论研究和算法求解方面都取得了丰硕成果.本文在阅读了大量文献的基础上,主要致力于几类变分不等式和互补问题的光滑化算法研究,其主要思想是通过光滑函数将变分不等式或互补问题转化为与之等价的光滑方程组,进而设计求解算法.本文主要内容概括如下:(1)设计一个互补函数的新光滑函数,利用该函数将箱约束变分不等式问题的KKT条件转化为光滑方程组,给出了一种光滑牛顿算法,证明了该算法具有适定性和全局收敛性,并给出了数值实验.(2)给出一个新的光滑互补函数,在此基础上将非线性互补问题转化为与之等价的光滑方程组,设计一种求解非线性互补问题的非精确光滑牛顿算法,在适当的假设下,证明了算法具有全局收敛性,给出了数值实验.(3)将交通均衡问题转化为随机线性互补问题,设计了一种求解该随机线性互补问题(1模型的光滑牛顿算法,证明了该算法的适定性和全局收敛性,数值实验说明算法是有效的.
[Abstract]:Variational inequalities and complementary problems have attracted much attention because of their important application value. After decades of development, variational inequalities and complementary problems have made great achievements in theoretical research and algorithm solving. On the basis of reading a large number of literatures, this paper focuses on the smoothing algorithms for several classes of variational inequalities and complementary problems. The main idea is to transform variational inequalities or complementary problems into smooth equations equivalent to them through smooth functions, and then design a solution algorithm. The main contents of this paper are summarized as follows: (1) A new smooth function of complementary function is designed. By using this function, the KKT condition of box constrained variational inequality problem is transformed into a smooth system of equations, and a smooth Newton algorithm is given. It is proved that the algorithm has the properties of fitness and global convergence, and a new smooth complementary function is given in the numerical experiment. On this basis, the nonlinear complementarity problem is transformed into a set of smooth equations equivalent to it. An inexact smooth Newton algorithm for solving nonlinear complementarity problems is designed. Under appropriate assumptions, the global convergence of the algorithm is proved, and numerical experiments are given to transform the traffic equilibrium problem into a stochastic linear complementarity problem. In this paper, a smooth Newton algorithm is designed to solve the model of the stochastic linear complementarity problem. The fitness and global convergence of the algorithm are proved. The numerical experiments show that the algorithm is effective.
【学位授予单位】：淮北师范大学
【学位级别】：硕士
【学位授予年份】：2017
【分类号】：O221

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