Applications of variational analysis in optimization and data science

大量的统计方法和学习算法归结为一个优化问题。随着计算机和信息时代的到来，这些问题在规模和复杂性上都出现了爆炸性增长。当今最重要的问题之一是如何从超大数据集中提取趋势和子结构。在这个数据科学领域，包括数据挖掘、机器学习、支持向量机或信号处理，稀疏和低阶优化或压缩感知等技术经常被使用。所得到的优化问题的一个共同特征是发生函数的非光滑性。因此，对非光滑优化方法的需求越来越大。这在很大程度上依赖于非光滑和集值分析中坚实的数学基础。因此，作为一个长期目标，本研究计划旨在开发新的变分工具，并将其应用于数据科学和学习等各种领域中的非光滑优化问题的求解方法。*优化的一个分水岭，特别是在非光滑环境下，是凸性。这是因为凸函数表现出许多理想的性质：它们允许强大的次微分和对偶演算。此外，驻点、局部极小点和全局极小点重合，并且它们具有理想的分析特征，如仿射小型化、(局部)Lipschitz性质或Prox正则性。*尽管实践中的许多问题都不是完全凸的，但经常存在可以(也应该)利用的凸子结构。我们以此作为这个项目的指导方针，在这个项目中，我们专注于非光滑优化中的两个密切相关的主题，这两个主题与统计和机器学习以及优化中的其他当前领域有很强的联系。*主要研究DC优化，即目标函数为两个凸函数之差的极小化问题。这一成熟的非凸问题类涵盖了大量的应用和许多我们感兴趣的问题。*我们还关注于各种应用中出现的一些具体的非光滑凸函数的变分分析，例如矩阵分式函数，它似乎在数据科学领域中无处不在，并连接了不同的主题，如二次优化、多任务学习和核范数平滑。*

An abundance of statistical methods and learning algorithms reduce to an optimization problem. With the advent of computers and the information age these problems have exploded both in size and complexity. One of the most important questions nowadays is how to extract trends and substructures in extremely large data sets. In this area of data science, which comprises data mining, machine learning, support vector machines or signal processing, techniques like sparse and low-rank optimization or compressed sensing are frequently used. ***A common feature of the resulting optimization problems is nonsmoothness of the occurring functions. Hence, there is an increased demand for nonsmooth optimization methods. These rely heavily on solid mathematical foundations in nonsmooth and set-valued analysis. Therefore, as a long-term goal, this research program aims at developing novel variational tools and bring them to bear on solution methods for nonsmooth optimization problems occurring in a variety of fields such as data science and learning.***A watershed in optimization, in particular in the nonsmooth setting, is convexity. This is due to the fact that convex functions exhibit many desirable properties: They allow for a powerful subdifferential and duality calculus. Moreover, stationary points, local and global minima coincide and they have desirable analytical features such as affine minorization, (local) Lipschitz properties or prox-regularity. ***Although many problems in practice are not fully convex, there are often convex substructures which can (and should) be exploited. We take this as a guideline for this program in which we focus on two intimately related topics in nonsmooth optimization which have strong connections to statistical and machine learning as well as to other current areas in optimization. ***Primarily, we would like to study DC optimization, i.e. minimization problems where the objective function is the difference of two convex functions. This well-established nonconvex problem class covers an abundance of applications and many of the problems of our interest.***We also lay a focus on the variational analysis of some of the concrete nonsmooth, convex functions occurring in various applications such as the matrix-fractional function, which seems to be ubiquitous in the area of data science and connects different topics such as quadratic optimization, multitask learning and nuclear norm smoothing.****** *****