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.

凸分析和优化通过提供数学基础和解决各个领域问题的方法来发挥关键作用。同时，这些领域的最新应用需要超出凸度的优化技术。尽管凸优化技术和数值算法已成为50多年来的广泛研究的主题，但在不存在凸度的情况下解决了大规模优化问题仍然是一个挑战。在该项目中，主要研究者旨在在凸面和非平滑分析中开发新的理论结果，以及新的数值算法，以优化不一定是可区分的非convex函数，尤其是凸函数差异的函数。此类优化问题在多物种位置，聚类，机器学习，压缩传感和成像应用中出现。研究人员及其同事开发，实施和测试用于解决此类问题的数值算法。对于不可不同的性能和凸度，这些数值算法带来了解决不同应用领域中复杂优化问题的新方法。该项目旨在开发非平滑分析的新理论和解决优化问题的新理论，而无需施加不同的条件或可不同的条件或凸度。基于一种变异几何方法，该项目的第一个目标是在非平滑分析中开发新的结果，以应对目标函数不可分割且不convex的优化问题。这种方法提供了非平滑分析的系统开发，使得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convex优化问题的数值算法，尤其是那些目标函数代表凸功能的差异，并将其应用于多易于性位置的问题，聚类和层次集群，机器学习，机器学习，压缩感应和成像。研究人员及其同事特别关注涉及不同规范或约束的问题，需要在平滑和初始化技术方面取得进步。他们解决了模型的最佳解决方案的存在和独特性，基于全球优化方法的初始化技术，实施算法，用于比较和对人工和真实数据集的比较和测试以及算法的收敛速率。结果有助于开发非平滑分析及其在建立和分析非凸的非平滑优化问题的数值算法中的使用。