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Collaborative Research: CIF: Small: New Theory and Applications of Non-smooth and Non-Lipschitz Riemannian Optimization

合作研究：CIF：小：非光滑和非Lipschitz黎曼优化的新理论和应用

基本信息

批准号：
2007797
负责人：
Shiqian Ma
金额：
$ 31.68万
依托单位：
University of California-Davis
依托单位国家：
美国
项目类别：
Standard Grant
财政年份：
2020
资助国家：
美国
起止时间：
2020-10-01 至 2023-02-28
项目状态：
已结题

来源：
https://www.nsf.gov/awardsearch/showAward?AWD_ID=2007797&HistoricalAwards=false
关键词：
Collaborative Research CIF Small New

项目摘要

Non-convex optimization problems are ubiquitous in fields as diverse as data science, machine learning, and information science and engineering, thereby creating a need for algorithms to efficiently solve such problems. This project will study an important but less developed area in non-convex optimization, namely non-smooth and non-Lipschitz Riemannian optimization. The outcomes of this project will provide insights into important classes of non-convex optimization problems, and will lead to the development of new tools for solving them. New teaching material on non-convex optimization problems will be produced for educating the next generation students in this important class of applications. The societal impact of this exploration will be to benefit new applications in areas such as gene expression, autonomous driving and cancer studies. While existing theory and algorithms for Riemannian optimization usually require the objective function to be differentiable, in contrast this project focuses on non-smooth and non-Lipschitz Riemannian optimization. In particular, the project will study several algorithms for non-smooth optimization that are less developed in the Riemannian setting, including the manifold alternating direction method of multipliers, the inertial manifold proximal gradient method, the stochastic manifold proximal point algorithm, and the manifold prox-linear algorithm. For Riemannian optimization with a non-Lipschitz objective, the investigators will derive the corresponding optimality conditions and then design two algorithms that are based on a smoothing technique, namely the Riemannian smoothing gradient descent method and the Riemannian smoothing trust region method. The proposed algorithms will be implemented to solve real-world applications such as the clustering of single-cell RNA sequencing data, and 3D object detection and 3D tracking in autonomous driving.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.

非凸优化问题在数据科学、机器学习和信息科学与工程等领域普遍存在，因此需要有效解决此类问题的算法。本计画将研究非凸最佳化中一个重要但发展较少的领域，即非光滑与非Lipschitz黎曼最佳化。该项目的成果将为重要的非凸优化问题提供见解，并将导致开发新的工具来解决这些问题。新的教材非凸优化问题将产生教育下一代学生在这一重要类的应用。这一探索的社会影响将有利于基因表达、自动驾驶和癌症研究等领域的新应用。虽然现有的黎曼优化理论和算法通常要求目标函数是可微的，但相比之下，该项目专注于非光滑和非Lipschitz黎曼优化。特别是，该项目将研究几种在黎曼环境中开发较少的非光滑优化算法，包括乘子的流形交替方向方法，惯性流形近端梯度方法，随机流形近点算法和流形非线性算法。对于非Lipschitz目标的黎曼优化，研究人员将推导出相应的最优性条件，然后设计两种基于平滑技术的算法，即黎曼平滑梯度下降法和黎曼平滑信赖域法。该奖项反映了NSF的法定使命，并通过使用基金会的智力价值和更广泛的影响力审查标准进行评估，被认为值得支持。