非Lipschitz优化的高效光滑化信赖域方法及应用

Recently, non-Lipschitz optimization has attracted significant attention in many areas, such as statistical analysis, image processing, machine learning and data mining. However, due to the non-Lipschitz continuity, most of the algorithms for nonsmooth problems cannot be applied. In fact, the lack of effective solving methods has severely restricted the further use of non-Lipschitz optimization in practice. Therefore, we are going to investigate how to solve a class of widely used non-Lipschitz optimization efficiently in this project. Smoothing approximations have been studied for decades and are suitable for non-Lipschitz problems. Trust region is one of the two major strategies for smooth optimizations. So we plan to combining these two methods together and design new algorithms in the framework of Smoothing Trust Region (STR). The main work will include: (1) Optimality Conditions: Derive the easy-to-compute optimality conditions for local solutions. (2)Smoothing Aproximations：Construct the effective smoothing approximations based on the specific structure information of the problems. (3) Algorithms Design: Design the efficient STR algorithms for large scale problems. (4) Theoretical Analysis: Analyze the theoretical properties of the algorithms, including the global convergence and the worst case computational complexity. (5) Applications: Construct the new clustering models with the non-Lipschitz penalizations, and use our STR algorithms to solve them. Provide a new method for big data cluster analysis. For most part of our research contents in this project, some are totally new topics, some are still in their infancy. Besides enriching the theory and techniques in trust region and smoothing methods, the study of this project can also help to promote the further applications of Non-Lipschitz optimizations.

近年来，非Lipschitz优化在统计分析、图像处理、机器学习和数据挖掘等众多领域中备受关注，但缺乏有效解法严重制约了其实际应用。本项目针对一类具有广泛应用价值的非Lipschitz优化问题，设计基于光滑化信赖域框架的新算法，主要研究内容包括：（1）建立非Lipschitz优化问题的最优性条件；（2）根据问题结构特点，为非Lipschitz函数构造有效的光滑逼近；（3）设计光滑化信赖域算法，侧重研究如何利用子空间技术构造大规模问题的迭代格式；（4）建立能够更好体现中心点成对差异的非Lipschitz聚类模型并应用新算法求解，为聚类分析提供全新的求解方法。其研究成果既能丰富和发展最优化理论和技术，又可推动非Lipschitz优化在实际中的进一步应用，具有重要的理论意义和应用价值。

近年来，非Lipschitz优化在统计分析、图像处理、机器学习和数据挖掘等众多领域中备受关注，但缺乏有效解法严重制约了其实际应用。本项目针对一类具有广泛应用价值的非Lipschitz优化问题，设计基于光滑化信赖域框架的新算法，主要研究内容包括：（1）建立非Lipschitz优化问题的最优性条件；（2）根据问题结构特点，为非Lipschitz函数构造有效的光滑逼近；（3）设计光滑化信赖域算法，侧重研究如何利用子空间技术构造大规模问题的迭代格式；（4） 基于非Lipschitz优化，为聚类、特征选择、深度神经网络压缩等建立新模型，并应用新算法求解，为机器学习中的若干基本问题提供全新的求解方法。其研究成果既能丰富和发展最优化理论和技术，又可推动非Lipschitz优化在实际中的进一步应用，具有重要的理论意义和应用价值。