【作者】 陈建芮；

【导师】 乌力吉；

【作者基本信息】 内蒙古工业大学 ， 计算数学， 2007， 硕士

【摘要】 变分不等式问题(VIP)是运筹学领域中的一个重要分支，广泛应用于自然科学、工程计算和经济均衡等诸多学科。特别是VIP的数值方法，近年来引起了众多学者的广泛兴趣。其中利用VIP的KKT条件来构造算法是一类重要方法，一般文献都采用Fischer-Burmeister将相应的KKT条件等价地转化为非光滑方程组形式。本文另辟蹊径，依据文献[6]～[9]中提出的互补问题的一种Lagrange乘子法思想，重新构造给出KKT条件两类新的等价形式。本文的具体工作如下：1．首先利用Levenberg-Marquardt方法给出求解带约束非线性方程组的一类新方法，该方法扩大了文献[4]中参数的取值范围，进而弥补了参数取唯一值在具体问题中的缺陷。在不要求梯度矩阵非奇异的条件下得到了算法的全局收敛性。并且在适当的假设条件下，得到了算法的局部超线性收敛及局部二次收敛性。2．根据变分不等式问题本身的特点，建立了变分不等式问题KKT条件与光滑带约束方程组的等价关系。该等价形式中的每个函数都是连续可微的，虽然引入了参数λ，但在每一个迭代步修正参数λ的方法非常简单。利用上述Levenberg-Marquardt的方法得到求解变分不等式问题的一种新算法。3．在上述等价形式的基础上，减少了等价形式中方程的个数。与已有利用Fis-cher函数求解变分不等式问题KKT条件的数值方法相比，该等价形式更简单，算法更易实现。基于该等价形式，本文提出了一种阻尼牛顿类算法，有效地弥补了Levenberg-Marquardt方法中必须计算Jacobi矩阵乘法的缺陷。在适当条件下，证明了算法的全局收敛性、局部超线性收敛或二次收敛性。数值实验结果表明该算法效率较高。更多还原

【Abstract】 The variational inequality problem (VIP) is an important branch in operationalresearch, which has widely applications in science, engineering and economics. Es-pecially, the numerical methods for VIP attracts many scholars’ more attention inrecent years. An important numerical method for VIP is to construct algorithms forthe corresponding KKT conditions.In general, the KKT conditions for VIP is equiva-lently transformed as a system of nonsmooth equations by using Fischer-Burmeisterfunction. Different from the methods in literature, we proposed two new numericalmethods for VIP in the paper as follows:1. At first, a revised Levenberg-Marquardt method was presented for constraintsmooth equations system, which overcome the drawbacks of the parameter in [4]must be a constant number. Same convergent results in [4] were obtained undermild assumptions.2. A new equivalent reformulation of constraint smooth equations system to theKKT conditions for VIP was proposed, and a new Levenberg-Marquardt methodfor solving VIP was constructed by using above mentioned method. The convergentresults were shown under some suitable conditions.3. A new equivalent reformulation of nonlinear equations system without con-straints condition to the KKT conditions for VIP was proposed. It has fewer vari-ables and simpler form, and the algorithm is easily implemented. A damped Newtontype algorithm was presented based on it, which needs not to compute the product ofJacobi matrix as in Levenberg-Marquardt method. Under some certain conditions,the global convergence, local superlinear or quadratical convergence were obtained.Numerical results suggested the method is efficient.更多还原