Nonsmooth Analysis and Numerical Optimization Techniques beyond Convexity

凸分析和优化为解决各种领域的问题提供了数学基础和方法，在其中起着至关重要的作用。与此同时，这些领域的最新应用需要超越凸性的优化技术。尽管凸优化技术和数值算法已经被广泛研究了50多年，但在没有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，主要研究者的目标是在凸和非光滑分析中发展新的理论结果，以及新的数值算法，用于优化不一定可微的非凸函数，特别是凸函数的差函数。这类优化问题出现在多设施定位、聚类、机器学习、压缩感知和成像应用中。研究者和他的同事开发、实现和测试数值算法来解决这些问题。这些数值算法在不要求可微性和凸性的情况下，为求解不同应用领域的复杂优化问题提供了新的方法。本项目旨在发展新的非光滑分析理论和优化方法，以解决不附加可微性或凸性条件的优化问题。基于变分几何方法，本项目的第一个目标是发展非光滑分析的新结果，以处理目标函数不可微和非凸的优化问题。这种方法为非光滑分析提供了系统的发展，使不同领域的研究人员可以使用它。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、压缩感知和成像等问题。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，这些问题需要平滑和初始化技术的进步。他们讨论了模型最优解的存在性和唯一性、基于全局优化方法的初始化技术、在人工和真实数据集上进行比较和测试的算法的实现以及算法的收敛速度等重要问题。这些结果有助于非光滑分析的发展及其在非凸非光滑优化问题的数值算法的建立和分析中的应用。

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.