几类典型稀疏优化问题的算法、理论及应用

项目名称： 几类典型稀疏优化问题的算法、理论及应用

项目编号： No.11471101

项目类型： 面上项目

立项/批准年度： 2015

项目学科： 数理科学和化学

项目作者： 肖运海

作者单位： 河南大学

项目金额： 56万元

中文摘要： 稀疏优化问题在图像处理、机器学习、基因网络等领域有着广泛的应用。本项目研究几类典型稀疏优化问题的理论、快速算法及应用。主要包括：(1)基于求解光滑优化问题的牛顿法思想，研究求解L1正则化问题非单调谱梯度算法；利用精确罚函数和变量分裂技术，研究求解Lp(0

中文关键词： 稀疏优化；矩阵优化；压缩感知；Lp正则化问题；交替方向法

英文摘要： The sparse optimization problem has wide range of applications in image processing, machine learning, gene networks etc. This project aims to study the theory and fast algorihtms for sparse optimization problems and their applications. The main research content includes: (1) Inspired by Newton method in smooth optimization, we will study the nonmonotone spectral gradient method for L1-regularized minimization problems. Using the exact penalty function and variable splitting techniques, we will study the alternating directions method for Lp(0<p<1)-regularized minimization problems. The uncosntrained Lp-regularized model is reformulated as a linear constrained and separable convex minimization problem. Then, an alternating directions method is developed to solve the resulting problem, and it will be proved that the generated iterations converge to the KKT point of the resulting problem. (2) Based on the superiority of the two-step shringkage/thresholding method for recovering a large and sparse signal in compressive sensing, we will study the multi-step shringkage/thresholding algorithms for low-rank matrix optimization. Using the properties of dual norm, we will study the alternating directions method for matrix mixed-norm optimization problems. We will show that each subproblem admits closed-form solutions, and the dual version of the algorithm is superior to the primal one in theory and numerical performance. (3) Based on the linearized technique and proximal points method, we will study the alternating directions method for log-determinant sparse minimization problems. Finally, we will test the practical performance of each proposed algorithm and develop highly efficient software packages.

英文关键词： Sparse Optimization;Matrix Optimization;Compressive Sensing;Lp-regularized Problems;Alternating Directions Methods