两类问题的互补求解方法及二阶锥互补问题解的性质

发布时间：2018-05-23 09:42

本文选题：互补问题 + NCP函数　；参考：《天津大学》2012年博士论文

【摘要】：本文的主要内容包括医学中张量成像问题的求解、一类自由边界问题的求解以及二阶锥互补问题解的性质的理论研究. 核磁共振成像已经被广泛的应用在各种临床治疗中.扩散张量成像(Difusion Tensor Imaging)是通过求解一个扩散张量的特征值来成像的技术.但这一理论总假设水分子在人体中的扩散是高斯型的.而在有纤维交叉的人体组织中,扩散张量成像技术无法正确显示各个纤维方向.本文研究扩散陡度成像(Difusion Kurtosis Imaging)模型,该技术利用一个4阶3维张量W的特征值来成像.以一类广义互补函数的性质为基础,利用该类互补函数将求解扩散陡度张量W的特征值问题重构为一类广义非光滑方程组,并设计光滑牛顿型算法求解该方程组,在适当的假设下,该算法的全局收敛性被证明.最后的数值结果表明该方法是有效的. 本文也利用互补问题求解自由边界问题.在将一类自由边界问题重构为一类非李普希兹连续的非线性互补问题的基础上,引入plus函数的一类广义光滑函数,分析其性质,然后利用该光滑函数将所得的互补问题重构为一系列光滑方程组,并设计一类带非单调线搜索的光滑牛顿型算法来求解该方程组以得到原问题的解.在很弱的条件下,该算法的全局收敛性和局部二次收敛性被证明.最后的数值结果显示该算法是有效的. 互补问题的应用非常广泛,而现实中的许多问题均可归结为二阶锥互补(SOCCP)问题.虽然二阶锥互补问题的理论与算法已经有许多的研究,但所研究的互补问题以单调互补问题为主.本文在二阶锥上引入一类新的非单调映射,称之为笛卡尔P_*(κ)映射,它是单调映射的推广,并讨论涉及这类映射的二阶锥互补问题的解的存在性和解集的有界性.在严格可行的假设下证明该类问题的解集是非空有界的.
[Abstract]:The main contents of this paper include the solution of the Zhang Liang imaging problem in medicine, the solution of a class of free boundary problems and the theoretical study of the solution of the second order cone complementarity problem. Magnetic resonance imaging has been widely used in various clinical treatments. Diffusion Zhang Liang imaging (Zhang Liang) is an imaging technique by solving the eigenvalues of a diffused Zhang Liang. However, this theory always assumes that the diffusion of water molecules in the human body is Gao Si type. In human tissues with intersected fibers, diffusive Zhang Liang imaging can not correctly show the direction of each fiber. In this paper, the diffusion steepness imaging fusion Kurtosis imaging model is studied. The technique uses an eigenvalue of 4-order 3D Zhang Liang W to image. Based on the properties of a class of generalized complementary functions, the eigenvalue problem for solving diffusion steepness Zhang Liang W is reconstructed into a class of generalized nonsmooth equations, and a smooth Newtonian algorithm is designed to solve the equations. Under appropriate assumptions, the global convergence of the algorithm is proved. The numerical results show that the method is effective. In this paper, the complementarity problem is also used to solve the free boundary problem. On the basis of reconstructing a class of free boundary problems into a class of non-Lipschitz continuous nonlinear complementarity problems, a class of generalized smooth functions of plus function is introduced and its properties are analyzed. Then, the complementarity problem is reconstructed into a series of smooth equations by using the smooth function, and a class of smooth Newtonian algorithm with non-monotone linear search is designed to solve the system to obtain the solution of the original problem. Under very weak conditions, the global convergence and local quadratic convergence of the algorithm are proved. The numerical results show that the algorithm is effective. The application of complementarity problem is very extensive, but many problems in reality can be attributed to the second order cone complementarity (SOCCP) problem. Although there have been many studies on the theory and algorithm of the second order cone complementarity problem, the monotone complementarity problem is the main one. In this paper, we introduce a new class of nonmonotone mappings on the second order cones, called Cartesian PSP (魏) mappings, which is a generalization of monotone mappings, and discuss the existence of solutions and the boundedness of solutions for the second order cone complementarity problems involving such mappings. It is proved that the solution set of this kind of problem is nonempty bounded under the strictly feasible assumption.
【学位授予单位】：天津大学
【学位级别】：博士
【学位授予年份】：2012
【分类号】：O241.6;R310

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