余稀疏优化的一类非凸松弛方法的理论与算法研究

Cosparse optimization methods for dealing with non-sparse data have received a lot of research due to their wide practical applications, but there are few studies on related nonconvex relaxation methods. How to develop the theory and algorithms of nonconvex relaxation methods for cosparse optimization is one of the topics that has received much attention. This project will study the theory and algorithm of the nonconvex Lp relaxation method for cosparse optimization, and specifically study the following problems: (1) For noiseless cosparse optimization problems, we will establish the exact recovery theory of the nonconvex Lp relaxation method, especially, study the equivalence of the original problem with nonconvex Lp relaxation models. And then, we will design an efficient alternating direction method of multipliers for solving large-scale noiseless nonconvex Lp relaxation models of cosparse optimization problems, establish the convergence theory of the algorithm and perform effective numerical verification. (2) For noisy cosparse optimization problems, we will establish the error estimation theory of the nonconvex Lp relaxation method, and also explore the accurate recovery of the nonconvex Lp relaxation models of the noise signal. And then, we will design an efficient alternating direction method of multipliers for solving large-scale noisy nonconvex Lp relaxation models of cosparse optimization problems, establish the convergence theory of the algorithm and perform effective numerical verification. The smooth implementation of this project may not only enrich the theory and method of cosparse optimization, but also provide algorithm support for the problems related to dealing with large-scale non-sparse data in practice.

处理非稀疏数据的余稀疏优化方法由于实际应用广泛而得到了大量的研究，但相关的非凸松弛方法少有研究，如何发展余稀疏优化的非凸松弛方法的理论与算法是目前备受关注的课题之一。本项目研究余稀疏优化的非凸Lp松弛方法的理论与算法，具体研究以下问题：（1）对于无噪声余稀疏优化问题，建立非凸Lp松弛方法的精确恢复理论，特别地，研究原问题与非凸Lp松弛模型的等价性；设计求解大规模无噪声余稀疏优化非凸Lp松弛模型的高效的交替方向乘子法，建立算法的收敛性理论并进行有效的数值验证。（2）对于噪声余稀疏优化问题，建立非凸Lp松弛方法的误差估计理论，同时探讨噪声信号的非凸Lp松弛模型的精确恢复性；设计求解大规模噪声余稀疏优化非凸Lp松弛模型的高效交替方向乘子法，建立算法的收敛性理论并进行有效的数值验证。本项目的顺利实施，不但可以丰富余稀疏优化的理论与方法，而且可以为实际中处理大规模非稀疏型数据的相关问题提供算法支撑。

余稀疏优化方法有很多的实际应用，因而得到了大量的研究，但是求解这类问题的非凸松弛方法少有研究。本项目旨在研究余稀疏优化问题的非凸Lp松弛方法的理论与算法，主要获得了以下成果：（1）对于噪声余稀疏优化问题，建立了非凸Lp松弛方法的误差估计理论，给出了噪声信号的非凸Lp松弛模型的精确恢复条件，并通过数值实验验证了所获得的理论结果；（2）设计了求解大规模噪声余稀疏优化模型的高效交替方向乘子法，并在混合变分不等式的框架下，建立了算法的收敛性理论。本项目的实施，丰富了处理非稀疏数据的余稀疏优化的理论与方法，为求解相关的大规模实际问题提供理论和算法支撑。

内容获取失败，请点击重试