Nonsmooth Analysis and Numerical Optimization Techniques beyond Convexity

凸分析和优化通过为解决各种领域的问题提供数学基础和方法而发挥着至关重要的作用。 同时，最近在这些领域的应用需要超越凸性的优化技术。 虽然凸优化技术和数值算法已经被广泛研究了50多年，但在不存在凸性的情况下解决大规模优化问题仍然是一个挑战。 在这个项目中，主要研究者的目的是开发新的理论成果，在凸和非光滑分析，和新的数值算法，为非凸函数的优化，不一定是可微的，特别是功能是凸函数的差异。 这类优化问题出现在多设施定位，聚类，机器学习，压缩传感和成像应用中。 研究人员和他的同事开发，实施和测试数值算法来解决这些问题。 这些数值算法不要求可微性和凸性，为解决不同应用领域的复杂优化问题带来了新的方法。本项目旨在发展新的非光滑分析理论和优化方法，以解决优化问题，而无需施加可微性或凸性条件。 基于变分几何方法，这个项目的第一个目标是在非光滑分析中发展新的结果，以处理目标函数不可微和非凸的优化问题。 这种方法提供了一个系统的发展，非光滑分析，使它可以从不同领域的研究人员。 该项目的第二个目标是开发用于解决非凸优化问题的数值算法，特别是那些目标函数可表示为凸函数差异的问题，并将其应用于多设施定位，聚类和分层聚类，机器学习，压缩传感和成像等问题。 研究者和他的同事们特别关注涉及不同规范或约束的问题，需要平滑和初始化技术的进步。 他们解决的重要问题的存在性和唯一性的最优解的模型，初始化技术的基础上的全局优化方法，实施的算法进行比较和测试的人工和真实的数据集，和算法的收敛速度。 结果有助于非光滑分析的发展，并使用其在建设和分析非光滑优化问题的数值算法是不凸的。

Convex analysis and optimization play a crucial role by providing the mathematical foundation and methods for solving problems in a variety of fields. At the same time, recent applications in these fields require optimization techniques beyond convexity. Although convex optimization techniques and numerical algorithms have been the topics of extensive research for more than 50 years, solving large-scale optimization problems without the presence of convexity remains a challenge. In this project, the principal investigator aims to develop new theoretical results in convex and nonsmooth analysis, and new numerical algorithms, for the optimization of nonconvex functions that are not necessarily differentiable, especially functions that are the difference of convex functions. Optimization problems of this sort arise in multi-facility location, clustering, machine learning, compressed sensing, and imaging applications. The investigator and his colleagues develop, implement, and test numerical algorithms for solving such problems. With no requirement on differentiability and convexity, these numerical algorithms bring new methods for solving complex optimization problems in different fields of application.This project aims to develop new theory of nonsmooth analysis and optimization methods for solving optimization problems without imposing conditions of differentiability or convexity. Based on a variational geometric approach, the first goal of this project is to develop new results in nonsmooth analysis to deal with optimization problems in which the objective functions are nondifferentiable and nonconvex. This approach provides a systematic development of nonsmooth analysis, making it accessible to researchers from different fields. The second goal of the project is to develop numerical algorithms for solving nonconvex optimization problems, especially those whose objective functions are representable as differences of convex functions, and to apply them to problems in multi-facility location, clustering and hierarchical clustering, machine learning, compressed sensing, and imaging. The investigator and his colleagues particularly focus on problems that involve different norms or constraints, requiring advances in smoothing and initialization techniques. They address the important issues of existence and uniqueness of optimal solutions of the models, initialization techniques based on global optimization methods, implementation of the algorithms for comparison and testing on artificial and real data sets, and the convergence rate of the algorithms. The results contribute to the development of nonsmooth analysis and its use in building and analyzing numerical algorithms for nonsmooth optimization problems that are not convex.