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

非凸优化问题在数据科学、机器学习、信息科学和工程等领域无处不在，因此需要有效解决此类问题的算法。本项目将研究非凸优化中一个重要但欠发达的领域，即非光滑非lipschitz黎曼优化。这个项目的结果将为非凸优化问题的重要类别提供见解，并将导致解决这些问题的新工具的开发。关于非凸优化问题的新教材将被制作出来，以教育下一代学生学习这类重要的应用。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。现有的黎曼优化理论和算法通常要求目标函数是可微的，而本项目侧重于非光滑和非lipschitz黎曼优化。特别是，该项目将研究几种在黎曼设置中较少开发的非光滑优化算法，包括乘法器的流形交替方向法、惯性流形近端梯度法、随机流形近端点算法和流形近线性算法。对于非lipschitz目标的黎曼优化，研究者将推导相应的最优性条件，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。所提出的算法将用于解决现实世界的应用，如单细胞RNA测序数据的聚类，以及自动驾驶中的3D物体检测和3D跟踪。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

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.