稀疏优化的非凸松弛模型与加速算法研究

In the age of big data, a large number of large-scale sparse optimization problems have emerged in the areas of signal and image processing, data mining and complex networks. The aim of this project is to study the fundamental theory and acceleration algorithms for non-convex relaxations of sparse optimization. First, we will carry out the equivalence analysis and superiority comparison for different forms of non-convex relaxation models. Based on the data characteristics of the sparse optimization problems, the selection strategies of the non-convex relaxation models are explored. Furthermore, we will design the acceleration algorithm for the non-convex relaxations. Exploiting the special structure of the models, we will prove that the acceleration algorithm can achieve a better convergence rate than the non-acceleration algorithm. A series of numerical experiments will be carried out to assess the acceleration algorithm. Then, we will extend this result to general non-convex optimization problems. Finally, some improved strategies will be proposed to improve the convergence rate of the acceleration algorithm, avoiding the oscillation phenomenon near the optimal solution. Our goal is to recommend a suitable non-convex relaxation for sparse optimization problems, provide theoretical basis for the applications of non-convex acceleration algorithms and provide effective algorithm tools for the large-scale non-convex optimization problems.

随着大数据时代的来临，在信号与图像处理、数据挖掘、复杂网络等前沿科技领域涌现出大量的大规模稀疏优化问题。本项目主要研究稀疏优化非凸松弛模型的理论性质及其加速算法。首先，针对不同形式的非凸松弛模型，进行统一的等价性分析和优越性对比。从稀疏优化问题的观测数据和未知变量的先验信息出发，探讨非凸松弛模型的选择策略。其次，设计非凸松弛模型的加速算法框架，结合模型的结构特征，获得其加速算法较未加速算法优越性的理论支撑，并通过数值实验验证加速算法的优越性，同时将加速算法及其理论结果推广到一般的非凸优化问题。最后，针对非凸加速算法在最优解附近的振荡现象，探索振荡周期与目标函数的内在关系，提出改进策略，提高算法收敛率。通过本项目的研究，能够为实际应用中的稀疏优化问题推荐合适的非凸松弛模型，为非凸加速算法的广泛应用提供理论依据，为大规模非凸优化问题提供高效的算法工具。

随着大数据时代的来临，在信号与图像处理、数据挖掘、复杂网络等前沿科技领域涌现出大量的大规模稀疏优化问题。本项目主要研究稀疏优化非凸松弛模型的理论性质及其加速算法。首先，针对不同形式的非凸松弛模型，进行统一的等价性分析和优越性对比。从稀疏投资组合优化问题的观测数据和未知变量的先验信息出发，探讨非凸松弛模型的选择策略。其次，设计非凸松弛模型的加速算法框架，结合模型的结构特征，获得其加速算法较未加速算法优越性的理论支撑，并通过数值实验验证加速算法的优越性，同时将加速算法及其理论结果推广到一般的非凸优化问题。最后，针对非凸加速算法在最优解附近的振荡现象，探索振荡周期与目标函数的内在关系，提出改进策略，提高算法收敛率。通过本项目的研究，为投资组合中的稀疏优化问题推荐合适的非凸松弛模型，为非凸加速算法的广泛应用提供理论依据，为大规模非凸优化问题提供高效的算法工具。