结构稀疏优化模型的理论与算法  

Theory and algorithms of structured sparse optimization problems

作　　者：胡耀华 李昱帆 刘艳艳 覃静 Yaohua Hu;Yufan Li;Yanyan Liu;Jing Qin

机构地区：[1]深圳大学数学科学学院,深圳518060 [2]中山大学理学院,深圳518107 [3]中山大学药学院(深圳),深圳518107

出　　处：《中国科学：数学》2024年第7期1045-1070,共26页Scientia Sinica：Mathematica

基　　金：国家自然科学基金(批准号:12222112,12071306和32170655);广东省教育厅重点领域专项(批准号:2023ZDZX1017);深圳市科技计划(批准号:RCJC20221008092753082,202206193000001和20220817122906001);深圳大学科研团队培育项目(批准号:2023QNT011)资助项目。

摘　　要：结构稀疏优化在压缩感知、信号与图像处理、机器学习和生物医学等领域应用广泛,是优化领域非常热门的研究课题.本文首先介绍稀疏优化与组稀疏优化的相关模型,并概述混合稀疏优化与联合稀疏优化的相关模型及近年来的进展;其次从理论与算法两个方面阐述关于稀疏优化与组稀疏优化模型的研究进展,包括相合性理论与求解算法;最后提出一些结构稀疏优化领域值得深入研究的方向.总体来说,结构稀疏优化的研究虽已取得一些成果,但仍存在理论问题亟待解决.Structured sparse optimization is one of the most popular research topics in optimization,which has been widely used in a wide range of fields,such as compressed sensing,signal and image processing,machine learning,and biological sciences.In this paper,we first introduce the development of sparse optimization,group sparse optimization,mixed sparse optimization,and joint sparse optimization in recent years.We summarize the theoretical studies and optimization algorithms for structured sparse optimization,including the consistency theory and convergence theory.Finally,we also propose some interesting and important problems in structured sparse optimization that are worthy of further investigation.Generally speaking,a great deal of effort has been achieved in the development of structured sparse optimization;however,there are still several important problems to be resolved.

关 键 词：稀疏优化 组稀疏优化 相合性理论 一阶算法 二阶算法 收敛性理论 

分 类 号：TP18[自动化与计算机技术—控制理论与控制工程]