矩阵不等式约束矩阵最小二乘问题的投影算法研究

项目名称： 矩阵不等式约束矩阵最小二乘问题的投影算法研究

项目编号： No.11301107

项目类型： 青年科学基金项目

立项/批准年度： 2014

项目学科： 数理科学和化学

项目作者： 李姣芬

作者单位： 桂林电子科技大学

项目金额： 23万元

中文摘要： 本项目将主要研究源于金融理论、统计分析等领域的矩阵不等式约束矩阵最小二乘问题及其矩阵方程求解问题。结合矩阵理论和最优化理论及方法，从减少计算工作量角度出发，以降低问题复杂度为目的，构造高效、稳定的投影类算法求解。具体研究内容为：1）基于最佳逼近理论研究几类指定约束矩阵集合内投影矩阵的计算；2）研究矩阵不等式约束矩阵最小二乘问题，结合矩阵理论将问题等价转化为只含简单约束条件的可解矩阵凸优化问题或建立新的可解子问题，并设计高效、稳定且适用于不同问题规模的梯度投影类求解算法；3）研究矩阵不等式约束矩阵方程求解问题，从凸可行问题的角度考虑问题，并采取无限逼近迭代思想，设计保结构特征、快速、稳定的交替投影算法和次梯度投影算法，并提出不定步长选取原则以提高算法的收敛效果。 本项目的研究将有益于促进最优化理论在数值代数领域的应用，使线性矩阵方程问题的研究工作在研究范围和研究手段上有新的突破。

中文关键词： 矩阵最小二乘问题；交替投影法；矩阵不等式；交替方向法；图像恢复

英文摘要： This project will studies the matrix least squares problems and the linear matrix equation problems under matrix inequalities constraints, which come originally from financial theory and statistical analysis and so on. Combing the theories and the methods in the matrix theory and the optimization theory, and starting from reduce the computation quantity of algorithms and the complexity of the problems, we will design efficient and stable projection algorithms to solve the considered problems. The detailed research contents are listed as follows. 1) Establishing the calculation of the projection matrices onto several specified classes of constrained matrix sets basing on the optimal approximation theory. 2) Studying the matrix least squares problems under matrix inequalities constraints. By combing the matrix theory, we convert the original problems into the equivalent solvable matrix convex optimization problems which contains only some linear constraints and simple convex constraints, or some new solvable subproblems, then construct serval gradient projection algorithms with fast convergence, stable and suitable for different scale problems to solve the transformed problems. 3) Studying the problems of solving linear matrix equation under matrix inequalities constraints. From the perspective of convex feasible

英文关键词： Matrix least-squares problem；Alternating projiection method；Matrix inequality；Alternating direction method；Image restoration