两类矩阵优化问题的扰动分析

发布时间：2018-01-13 06:34

本文关键词：两类矩阵优化问题的扰动分析　出处：《大连理工大学》2016年博士论文　论文类型：学位论文

【摘要】：矩阵优化问题(Matrix Optimization Problems)是指目标函数或约束函数中含有矩阵变量或者带有矩阵约束的优化问题.这类问题被广泛地应用在经济金融、工程计算等领域.在设计算法求解这些问题,尤其是在终止准则和收敛性分析中,扰动分析理论起着重要作用.因此,对矩阵优化问题进行扰动分析理论的研究是非常必要的.本论文主要研究两类矩阵优化问题,分别是由谱范数上图诱导的矩阵优化问题和半定矩阵优化问题.本论文所阐述的主要研究结果可概括如下：1.第三章研究的是由谱范数上图诱导的矩阵优化问题(MOSN)的最优性条件.我们首先给出由谱范数上图定义的锥的变分几何性质以及临界锥的刻画.由于MOSN的约束条件可以转化为半定矩阵约束,这样使得MOSN可以表述为一个半定规划(SDP)问题.所以针对约束非退化条件和强二阶充分条件,我们研究了两个问题之间的关系.证明了它们的强二阶充分条件是等价的,但是对于约束非退化条件,MOSN的比其SDP转化问题的弱,并举例加以说明.2.第四章研究的是由谱范数上图诱导的矩阵优化问题的扰动分析.首先,将原问题的一阶必要条件由一个非光滑方程来表示,通过对该非光滑方程中的投影算子进行光滑化,我们得到一个光滑方程.然后,我们研究光滑化投影算子的微分性质,并建立了最优解处的约束非退化条件和强二阶充分条件、该光滑方程在其解处的Clarke广义微分的非奇异性等一系列等价条件.最后利用此结果给出了采用光滑牛顿法求解此类问题的收敛性结果.3.第五章研究的是与半定矩阵有关的优化问题,包括半定矩阵广义方程和欧式距离矩阵优化问题.首先,在部分约束非退化和严格互补等条件下,给出半定矩阵广义方程解映射伴同导数的精等式刻画.由此,建立了解映射Aubin性质成立的等价条件和非线性凸半定规划问题的Karush-Kuhn-Tucker (KKT),点强正则性成立的充分条件.其次,证明了在严格Robinson约束规范和二阶充分条件下,欧式距离矩阵优化问题KKT映射的孤立平稳性成立.
[Abstract]:Matrix Optimization problems). Objective function or constraint function is an optimization problem with matrix variables or matrix constraints. This kind of problem is widely used in economy and finance. The perturbation analysis theory plays an important role in the design algorithm to solve these problems, especially in the termination criterion and convergence analysis. It is necessary to study the perturbation analysis theory of matrix optimization problem. In this paper, two kinds of matrix optimization problems are studied. These are matrix optimization problems induced by spectral norm and semidefinite matrix optimization problems, respectively. The main results of this paper can be summarized as follows:. 1. In Chapter 3, we study the matrix optimization problem induced by spectral norm. We first give the geometric properties of the cone defined by the spectral norm graph and the characterization of the critical cone. Since the constraint conditions of MOSN can be transformed into semi-definite matrix constraints. In this way, MOSN can be expressed as a semi-definite programming problem. Therefore, for constrained non-degenerate conditions and strong second-order sufficient conditions. We study the relationship between the two problems and prove that their strong second-order sufficient conditions are equivalent, but for constrained non-degenerate conditions, the SDP transformation of MOSN is weaker than that of its SDP transformation problem. An example is given to illustrate the perturbation analysis of matrix optimization problem induced by spectral norm. Firstly, the first order necessary condition of the original problem is represented by a nonsmooth equation. By smoothing the projection operator in the nonsmooth equation, we obtain a smooth equation. Then, we study the differential properties of the smooth projection operator. The constrained nondegenerate condition and the strong second order sufficient condition of the optimal solution are established. A series of equivalent conditions, such as the nonsingularity of the Clarke generalized differential at the solution of the smooth equation, are given. Finally, the convergence result of the smooth Newton method for solving this kind of problems is given. 3. 5th chapter. The optimization problem related to semidefinite matrix is studied. It includes the generalized equation of semidefinite matrix and the Euclidean distance matrix optimization problem. Firstly, under the condition of partial constraint nondegeneracy and strict complementarity, the exact equality characterizations of the mapping and derivative of the solution of the semidefinite matrix generalized equation are given. In this paper, we establish the equivalent conditions for the existence of Aubin properties of mappings and the Karush-Kuhn-Tucker KKT for nonlinear convex semidefinite programming problems. Secondly, it is proved that the isolated stationarity of KKT mapping for Euclidean distance matrix optimization problem is established under the strict Robinson constraint criterion and the second order sufficient condition.
【学位授予单位】：大连理工大学
【学位级别】：博士
【学位授予年份】：2016
【分类号】：O224

【相似文献】