Nonsmooth Analysis and Numerical Optimization Techniques beyond Convexity

凸性分析和最优化为解决各种领域的问题提供了数学基础和方法，因此起着至关重要的作用。同时，最近在这些领域的应用需要超越凸性的优化技术。尽管50多年来，凸优化技术和数值算法一直是人们广泛研究的主题，但解决不存在凸性的大规模优化问题仍然是一个挑战。在这个项目中，主要的研究人员致力于发展新的凸函数和非光滑分析的理论结果，以及新的数值算法，用于优化不一定可微的非凸函数，特别是作为凸函数的差的函数。这类优化问题出现在多设施选址、集群、机器学习、压缩传感和成像应用中。这位研究人员和他的同事开发、实施和测试了解决这类问题的数值算法。这些数值算法在不要求可微性和凸性的情况下，为解决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，这个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。这种方法提供了非光滑分析的系统发展，使来自不同领域的研究人员能够接触到它。该项目的第二个目标是开发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的问题，并将其应用于多设施选址、聚类和层次聚类、机器学习、压缩传感和成像等问题。这位研究人员和他的同事特别关注涉及不同规范或约束的问题，这些问题需要在平滑和初始化技术方面取得进展。它们解决了模型最优解的存在唯一性、基于全局优化方法的初始化技术、在人工数据集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。这些结果有助于发展非光滑分析，并将其用于建立和分析非光滑优化问题的数值算法。

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.