【作者】 林婷；

【导师】 柯艺芬；

【作者基本信息】 福建师范大学 ， 计算数学， 2023， 硕士

【摘要】 互补问题在金融、交通、力学、控制等许多领域有着重要的实际应用.因此,研究互补问题的高效数值方法具有十分重要的意义.本文旨在研究二阶锥水平线性互补问题、非负锥水平线性互补问题和非线性互补问题的高效数值算法.本文的主要内容如下:第1章,主要介绍了互补问题的应用背景和研究现状.第2章,考虑二阶锥水平线性互补问题(SOC-HLCP).首先将SOC-HLCP重新表述为基于二阶锥中定义的模的不动点方程.通过将不动点方程重构为2×2块非线性方程组,提出了一类基于2×2块系数矩阵分裂的类AOR迭代求解方法.其中,当类AOR迭代法中两个参数相等时,类AOR迭代法退化为类SOR迭代法.本文证明了所提出的迭代方法在适当的参数选择下可以收敛到SOC-HLCP的解.此外,还通过数值算例证明了类AOR模系矩阵分裂迭代算法在计算上的可行性和有效性.第3章,考虑非负锥水平线性互补问题(HLCP).首先将HLCP重新表述为基于非负锥中定义的模的不动点方程.通过将不动点方程重新表述为单调系统,提出了一类修正多元谱梯度投影法来求解该方程.本文证明了在给定的假设条件下,所提出的迭代方法可以收敛到HLCP的解.此外,还通过数值算例说明了模系修正多元谱梯度投影方法在计算上的可行性和有效性.第4章,提出了一类求解非线性互补问题的新修正谱梯度投影方法.首先将非线性互补问题等价地表述为一个非线性方程组.进而提出一类新的修正谱梯度投影方法求解所得的非线性方程组.新方法具有如下特点:谱梯度主要由修正的长BarzilliBorwein步长和修正的短Barzilli-Borwein步长的凸组合决定,并采用了一种新的线搜索技术.数值实验证明所提出的方法能够有效地求解非线性互补问题.第5章,对本文的研究内容进行总结,并提出未来可继续进行研究的方向.更多还原

【Abstract】 Complementarity problems have important practical applications in many fields such as mechanics,traffic,finance and control.Therefore,it is very important to study the efficient numerical method of complementarity problem.This aims to study the efficient numerical algorithms for the second-order cone horizontal linear complementarity problem,the nonnegative cone horizontal linear complementarity problem and the nonlinear complementarity problem.The main content of this paper is organized as follows:In Chapter 1,the application background and research status of complementarity problem are introduced.In Chapter 2,we consider the second-order cone horizontal linear complementarity problem(SOC-HLCP).We first reformulate the SOC-HLCP to a fixed-point equation based on modulus defined in the second-order cone.By reformulating the fixed-point equation as a 2 × 2 block nonlinear equation,we propose a class of AOR-like iteration method for solving it,which is based on a splitting of the 2 × 2 block coefficient matrix.In particular,when two parameters in AOR-like iteration method are equal,the AOR-like iteration method reduces to the SOR-like iteration method.We prove that the proposed iteration method will converge to the solution of the SOC-HLCP under suitable choices of the involved parameters.In addition,we also use numerical examples to show that the AOR-like modulus-based matrix splitting iteration algorithm is feasible and effective in computing.In Chapter 3,we consider the nonnegative cone horizontal linear complementarity problem(HLCP).We first reformulate the HLCP to a fixed-point equation based on modulus defined in the nonnegative cone.By reformulating the fixed-point equation as a monotone system,we propose a class of modified multivariate spectral gradient projection method for solving it.We prove that the proposed iteration method will converge to the solution of the HLCP under the given assumptions.In addition,we also use numerical examples to show that the modulus-based modified multivariate spectral gradient projection method is feasible and effective in computing.In Chapter 4,we propose a new modified spectral gradient projection method for solving nonlinear complementary problems.Firstly,the nonlinear complementarity problem is equitably expressed as a nonlinear system of equations.A new modified spectral gradient projection method is proposed to solve the nonlinear equations.The new method has the following characteristics: the spectral gradient is mainly determined by the convex combination of the modified long Barzilli-Borwein step size and the modified short BarzilliBorwein step size,and a new line search technique is adopted.Numerical experiments show that the proposed method can effectively solve nonlinear complementarity problems.In Chapter 5,we sum up the research content of this paper and put forward the future research direction.更多还原