Nonsmooth Analysis and Numerical Optimization Techniques beyond Convexity

超越凸性的非光滑分析和数值优化技术

基本信息

批准号：
1716057
负责人：
Mau Nguyen
金额：
$ 12万
依托单位：
Portland State University
依托单位国家：
美国
项目类别：
Standard Grant
财政年份：
2017
资助国家：
美国
起止时间：
2017-08-15 至 2020-07-31
项目状态：
已结题

来源：
https://www.nsf.gov/awardsearch/showAward?AWD_ID=1716057&HistoricalAwards=false
关键词：
Nonsmooth Analysis Numerical Optimization Techniques

项目摘要

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.

凸分析和优化为解决各个领域的问题提供了数学基础和方法，发挥着至关重要的作用。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 50 多年来，凸优化技术和数值算法一直是广泛研究的主题，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，主要研究者的目标是开发凸和非光滑分析的新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。此类优化问题出现在多设施定位、集群、机器学习、压缩感知和成像应用中。研究人员和他的同事开发、实施和测试用于解决此类问题的数值算法。这些数值算法不需要可微性和凸性，为解决不同应用领域的复杂优化问题带来了新方法。该项目旨在开发新的非光滑分析理论和优化方法，用于在不施加可微性或凸性条件的情况下解决优化问题。基于变分几何方法，该项目的首要目标是开发非光滑分析的新结果，以处理目标函数不可微且非凸的优化问题。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、聚类和层次聚类、机器学习、压缩感知和成像。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，需要平滑和初始化技术的进步。他们解决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构建和分析非凸非光滑优化问题的数值算法中的应用。