可拓展概率逼近中一类矩阵优化问题的有效算法

An Eficient Algorithm for Solving a class of Matrix Optimization Problem in ScalableProbabilistic Approximation

研究来源于复杂系统离散逼近中的一类可拓展概率逼近模型,欧氏空间中该问题模型可重塑为一类由线性流形和斜流形组成的乘积流形约束矩阵优化问题.结合乘积流形的几何性质,基于Zhang-Hager技术拓展,本文设计一类适用于问题模型的黎曼非线性共轭梯度法,并给出算法全局收敛性分析.数值实验验证所提算法对于问题模型求解是高效可行的,且与其它黎曼梯度类算法及黎曼优化工具箱中已有的黎曼梯度类算法和二阶算法相比在迭代效率上有一定优势.

In this paper,we consider the model of scalable probabilistic approximation problem arising in discrete approximation of complex systems,which can be reformulated as a matrix optimization problem over product manifold consisting of a linear manifold and an oblique manifold.Combining the geometric properties of product manifold in question and extending the Zhang-Hager technique,a class of Riemannian nonlinear conjugate gradient method is designed for solving the underlying problem model,the global convergence analysis is also given.Numerical experiments are provided to illustrate the efficiency of the proposed method.Comparisons with some classical Riemannian gradient-type methods,the existing Riemannian version of limited-memory BFGS algorithm and trust-region algorithm in the MATLAB toolbox Manopt are also provided to show the merits of the proposed approach.