线性约束矩阵最小二乘问题的解及稳定性研究

项目名称： 线性约束矩阵最小二乘问题的解及稳定性研究

项目编号： No.11201422

项目类型： 青年科学基金项目

立项/批准年度： 2013

项目学科： 数理科学和化学

项目作者： 裘渔洋

作者单位： 浙江工商大学

项目金额： 22万元

中文摘要： 在数学和工程计算中，许多问题可以转化成带各种线性约束的矩阵最小二乘问题。借助于特殊的矩阵分解和迭代法，一些带指定约束的特定问题已经解决。但是这种对特殊问题采取的特定方法，依赖于方程本身和约束条件，很难直接应用于一系列相关而又不同的问题，可移植性不是很好。本项目试图在理论和算法上给出框架式的研究方法。我们首先通过构造最小二乘右端矩阵的合适映射，将它转化成一个新的最小二乘问题，该问题和原问题等价，且解可以从与之相关的带更大范围约束的最小二乘问题的解集中重构出来。其次，借助于该方法，我们也考虑了解的稳定性。最后，我们通过约束空间的基将各种迭代纳入统一的构造格式。本项目所体现的算法思想和理论对处理线性约束矩阵最小二乘及相关问题具有一定的启发性和指导意义，为一般的约束矩阵最小二乘问题的求解和稳定性分析提供了一种新的解决模式。

中文关键词： 线性约束；最小二乘问题；通解构造；数值求解；稳定性分析

英文摘要： In the mathematics and engineering technology, many problems can be reduced to least squares problem with linear constraints on solutions.In terms of some special matrix-factorizations and iterative methods, some given questions have been solved. However, the produce factorization applied on matrix least squares problem requires proficient skill, depending on the system of matrix equations and the constraints on solutions themselves. It is difficult to directly apply a consistent factorization produce on those related but different problems. In this project, we will present a frame work in both theory and algorithms. Firstly, by suitable mapping on the right hand side matrix, the constrained least squares problem is equivalent to a new one, whose solutions can be constructed by solutions set of the least squares problem with more broad constraints. Secondly, we consider the stability of the solutions. Finally, we develope a unified iteration methods by the basis of the constrianed space. The ideas and theories in this project are adaptive for general constained least squares problem, which give a heuristic model to least squares problem.

英文关键词： linear constraints；least square problem；general solution；numerical solution；stability analysis