半张量积下矩阵方程组求解及Stiefel流形约束矩阵优化问题的若干有效算法

发布时间：2018-04-06 22:24

本文选题：半张量积　切入点：矩阵方程组　出处：《桂林电子科技大学》2017年硕士论文

【摘要】：半张量积概念最初由中科院程代展教授提出[An introduction to Semi-Tensor product of matrices and its applications,World Scientific,2012],在高维数据排列、多线性函数矩阵表示、电力系统稳定性控制等领域应用广泛,且为布尔网络、密码学、图染色、模糊控制等研究提供一种新的研究工具.本文在前人研究基础上,考虑半张量积下矩阵方程组AX=B XC=D的可解性,相容解的具体解析表达式及其不相容情况下的最小二乘解.Stiefel流形约束矩阵优化问题是指自变量矩阵满足列正交约束下极小化目标函数,其广泛应用于稀疏主成分分析、线性与非线性特征值、二次分配、信息检索、低秩相关矩阵、原子化学等领域.本文从数值角度研究来源于原子化学中的Stiefel流形一类矩阵最小二乘问题.本文具体内容组织如下:第二章研究半张量积下AX=B,XC=D的可解理论.分两种情况即未知X为向量和矩阵展开讨论,并分别给出半张量积定义下维数相容条件,相容解存在的充要条件及其具体解析表达式.第三章继续讨论半张量积下AX = B,XC = D的最小二乘解.通过半张量积的定义,将该问题等价转化为普通矩阵乘积下的相关问题,并结合奇异值分解分别给出当未知X为向量和矩阵情形下最小二乘解的解析表达式.第四章从数值角度研究来源于原子化学中非线性矩阵方程XTAX=B的Stiefel流形约束最小二乘解.从可行和不可行方法两方面设计若干迭代算法,包括梯度下降法、采用Barziiai-Browein步长法则的曲线搜索法、交替方向法、分裂正交约束法和临近交替增广拉格朗日法.通过大量数值实验验证各算法的有效性并比较迭代效率.
[Abstract]:The concept of semi-tensor product was first put forward by Professor Cheng Daizhan of the Chinese Academy of Sciences [an introduction to Semi-Tensor product of matrices and its applications / World Science 2012]. It is widely used in the fields of high-dimensional data arrangement, multi-linear function matrix representation, power system stability control, and so on, and it is Boolean network and cryptography.The research of graph coloring and fuzzy control provides a new research tool.In this paper, on the basis of previous studies, we consider the solvability of matrix equations AX=B XC=D under semi-tensor product.The concrete analytical expression of the compatible solution and the least square solution. Stiefel manifold constrained matrix optimization problem in the case of incompatibility is that the independent variable matrix satisfies the column orthogonal constraints and minimizes the objective function, which is widely used in sparse principal component analysis.Linear and nonlinear eigenvalues, quadratic assignment, information retrieval, low rank correlation matrix, atomic chemistry and so on.In this paper, the least square problem of a class of matrices of Stiefel manifolds derived from atomic chemistry is studied numerically.The content of this paper is organized as follows: in Chapter 2, we study the solvable theory of ax BX XCU D under semi-tensor product.In this paper, we discuss the unknown X as vector and matrix in two cases, and give the necessary and sufficient conditions for the existence of the compatible solution and the necessary and sufficient conditions for the existence of the compatible solution under the definition of semi-tensor product.In chapter 3, we continue to discuss the least square solution of ax = BX XC = D under semi-tensor product.Through the definition of semi-tensor product, the problem is equivalent to the related problem under the product of ordinary matrix, and the analytic expression of the least square solution under the condition of unknown X as vector and matrix is given by combining singular value decomposition.In chapter 4, the Stiefel manifold constrained least squares solutions derived from the nonlinear matrix equation XTAX=B in atomic chemistry are studied numerically.Several iterative algorithms are designed from both feasible and infeasible methods, including gradient descent method, curve search method using Barziiai-Browein step size rule, alternating direction method, split orthogonal constraint method and adjacent alternating augmented Lagrangian method.A large number of numerical experiments are carried out to verify the effectiveness of the algorithms and to compare the iterative efficiency.
【学位授予单位】：桂林电子科技大学
【学位级别】：硕士
【学位授予年份】：2017
【分类号】：O241.6

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