Variational Analysis of Optimal Value Functions and Applications to Nonsmooth Optimization

变分分析是非光滑优化问题的数学基础，在非光滑优化问题中，要最小化的代价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再适用。通过这个研究项目，首席研究员和他的同事们将开发变分分析的新应用，旨在解决设施选址、计算几何和机器学习领域中的一些重要的非光滑优化问题。他们将开发和实现大规模选址问题的数值算法，其中一些涉及不同类型的距离度量等。正在建立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。本项目所期望的变分分析新知识将促进非光滑优化中实用模型的求解。本项目旨在开发变分分析在非光滑优化中的新应用。主要研究人员和他的同事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和他的同事们专注于两类最优值函数：最小时间函数和最大时间函数，最小时间函数是最近距离函数的自然扩展，最大时间函数是最远距离函数的扩展。利用最优值函数的广义微分性质，研究了保证最优值函数具有不同性质的充要条件，如连续性、Lipschitz连续性和可微性。本文的结果有助于发展数值算法来解决设施选址、计算几何和机器学习中的非光滑优化问题。研究了最优值函数的广义微分性质，以及先进的光顺技术和快速梯度法，以开发有效的数值算法来求解这些问题。

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.