非线性互补问题若干方法的研究

发布时间：2018-05-03 10:46

本文选题：非线性互补问题 + 光滑牛顿算法　；参考：《安徽理工大学》2017年硕士论文

【摘要】：非线性互补问题的数值算法和理论研究是最优化理论研究领域中一个重要的研究课题,它们在航空,化工,数学规划,机械以及经济均衡等方面有着十分广泛的应用。光滑牛顿法是解决非线性互补问题最常用的方法之一。对于非线性互补问题可以通过利用光滑非线性互补函数,把其转化为等价的光滑方程组加以求解,从而可以建立求解非线性互补问题的光滑逼近算法。我们基于这种思想的考虑,借助原始的非线性互补函数,构造了一个新的连续可微的P0-函数,利用P0-函数把非线性互补问题转化为非线性方程组问题来加以研究;本文分析了 P0-函数一些好的性质,并利用这一新的连续可微的P0-函数,建立了对应的非线性互补问题的一个光滑牛顿算法;并在适当的条件下证明了该算法的全局收敛性以及局部收敛性;数值实验也表示该算法的可行性。接着我们又对原有的Fischer-Burmeister函数进行扰动,从而得到一个扰动的非线性互补函数;在此扰动函数的基础上,通过构造光滑互补函数将原有问题进行等价变形,建立了对应的求解非线性互补问题的光滑逼近算法;在水平集有界的条件下,证明了该方法的全局收敛性。对于非线性互补问题,我们讨论了原始的非线性互补问题在经过目标函数极小化变形之后的求解方法,我们利用增广的FB函数,构造了一个新的merit函数,在此函数的基础上把非线性互补问题转化为了约束极小化问题,并建立对应的无导数下降算法;而后进一步分析了此算法的全局收敛性,具体的数值实验例子也说明了本文所提出方法的有效性。
[Abstract]:Numerical algorithms and theoretical studies of nonlinear complementarity problems are an important research topic in the field of optimization theory. They are widely used in aviation, chemical engineering, mathematical programming, machinery and economic equilibrium. Smooth Newton method is one of the most common methods to solve nonlinear complementarity problems. For nonlinear complementarity problems, the smooth approximation algorithm for nonlinear complementarity problems can be established by transforming them into equivalent smooth equations by using smooth nonlinear complementary functions. Based on this idea, we construct a new continuous differentiable P0- function with the help of the original nonlinear complementary function, and transform the nonlinear complementarity problem into a nonlinear system of equations by using P0- function. In this paper, some good properties of P0-function are analyzed, and a smooth Newton algorithm for nonlinear complementarity problem is established by using this new continuous differentiable P0-function. The global convergence and local convergence of the algorithm are proved under appropriate conditions, and the feasibility of the algorithm is also demonstrated by numerical experiments. Then we perturb the original Fischer-Burmeister function and obtain a disturbed nonlinear complementary function. On the basis of the perturbation function, we construct a smooth complementary function to deform the original problem. The corresponding smooth approximation algorithm for nonlinear complementarity problems is established and the global convergence of the method is proved under the condition that the level set is bounded. For the nonlinear complementarity problem, we discuss the solution of the original nonlinear complementarity problem after minimization of the objective function. We construct a new merit function by using the augmented FB function. On the basis of this function, the nonlinear complementarity problem is transformed into a constrained minimization problem, and the corresponding derivative free descent algorithm is established, and then the global convergence of the algorithm is analyzed. Numerical examples also show the effectiveness of the proposed method.
【学位授予单位】：安徽理工大学
【学位级别】：硕士
【学位授予年份】：2017
【分类号】：O221.2

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