Variational Analysis of Optimal Value Functions and Applications to Nonsmooth Optimization

最优值函数的变分分析及其在非光滑优化中的应用

• 批准号: 1411817
• 负责人: Mau Nguyen
• 金额: $ 11.23万
• 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: 2014
• 资助国家: 美国
• 起止时间: 2014-06-15 至 2017-07-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1411817&HistoricalAwards=false
• 关键词: Variational; Analysis; Optimal; Value; Functions

Variational analysis serves as the mathematical foundation for non-smooth optimization problems in which the cost functions to be minimized are not necessarily differentiable. Because of the non-differentiability, traditional calculus-based methods are not applicable. Through this research project, the principal investigator and his colleagues will develop new applications of variational analysis designed to solve a number of important non-smooth optimization problems in the areas of facility location, computational geometry, and machine learning. They will develop and implement numerical algorithms for large-scale location problems, some involving different types of distance metrics, etc. The methods being built will be used to study other non-smooth optimization models in computational geometry and machine learning. The new knowledge in variational analysis this project anticipates will advance the solution of practical models in non-smooth optimization.This project aims at developing new applications of variational analysis to non-smooth optimization. The principal investigator and his colleagues study generalized differentiation properties of a class of optimal value functions in both convex and non-convex settings. Functions of this type, are intrinsically non-differentiable, and play an important role in the theory of variational analysis and its applications. In particular, the PI and his colleagues focus on two classes of optimal value functions: the minimal time function, which is a natural extension of the closest distance function, and the maximal time function, which is an extension of the farthest distance function. Generalized differentiation properties of the optimal value function are used to study necessary and sufficient conditions on initial data that guarantee different properties of the optimal value function such as continuity, Lipschitz continuity, and differentiability. Results obtained here contribute to development of numerical algorithms for the solution of non-smooth optimization problems in facility location, computational geometry, and machine learning. Generalized differentiation properties of the optimal value function as well as advanced smoothing techniques and fast gradient methods are investigated in order to develop effective numerical algorithms for solving these problems.

变分分析作为非光滑优化问题的数学基础，其中要最小化的成本函数不一定是可微的。由于不可微性，传统的基于微积分的方法不适用。通过这个研究项目，首席研究员和他的同事将开发变分分析的新应用，旨在解决设施位置，计算几何和机器学习领域的一些重要的非光滑优化问题。 他们将开发和实现大规模定位问题的数值算法，其中一些涉及不同类型的距离度量等，所构建的方法将用于研究计算几何和机器学习中的其他非光滑优化模型。本计画所期待的变分分析新知识，将有助于非光滑最佳化的实际模型的求解。本计画的目的是发展变分分析在非光滑最佳化上的新应用。主要研究者和他的同事们研究了一类最优值函数在凸和非凸环境下的广义微分性质。这类函数本质上是不可微的，在变分分析理论及其应用中起着重要的作用。特别是，PI和他的同事专注于两类最优值函数：最小时间函数，这是最近距离函数的自然扩展，以及最大时间函数，这是最远距离函数的扩展。利用最优值函数的广义微分性质来研究初始数据上保证最优值函数连续性、Lipschitz连续性和可微性等不同性质的充分必要条件。这里得到的结果有助于发展的数值算法的解决方案的非光滑优化问题的设施选址，计算几何和机器学习。研究了最优值函数的广义微分性质以及先进的光滑技术和快速梯度法，以发展有效的数值算法来解决这些问题。