Variational Techniques in Nonsmooth Optimization

在自然科学、商业和经济学、管理科学、信息科学和工程学中，有限维和无限维优化问题自然会出现在大量的应用和理论问题中。在现代方法、理论和应用中，一个共同的主题是不平滑。非光滑性可以直接通过建模结构产生，也可以通过最优值函数和解映射的变分特性间接产生。非光滑性在稀疏性优化、鲁棒统计的最大似然方法、机器学习、鲁棒优化、具有平衡约束的数学程序（mpec）、具有消失约束的数学程序（mpvc）、（广义）纳什均衡问题(G) nep)或特征值优化等领域发挥着重要作用。处理非光滑性的理论工具被归入“变分分析”的范畴，其中包括凸分析、非光滑分析和集值分析等。本研究项目的重点在于平滑方法，使用变分技术来构造和分析非光滑函数的光滑近似。平滑方法是通过求解一系列相关的无约束光滑逼近来解决非光滑和约束优化问题的标准方法。构造近似，使近似光滑问题解的聚类点或平稳点成为极限非光滑或约束优化问题的解或平稳点。在凸规划的背景下，现在人们对这些解决大规模问题的方法非常感兴趣，其中一阶凸非光滑优化方法已经非常成功。然而，在中小尺度环境中，二阶方法，特别是半光滑牛顿方法受到了广泛的关注，目前正被用于解决无限维环境下的pde约束优化问题。该项目强调所谓的epi-平滑函数，它依赖于泛函序列的epi-收敛的概念，并包括两个广泛的研究领域：第一个是对某类epi-平滑函数的二阶方法，即基于无穷卷积的函数，第二个旨在将epi-平滑的概念应用于无限维空间，因为epi-收敛在函数空间设置中的重要性。为了推广和加强这类重要的平滑函数的现有结果，除了外延平滑外，还将分析积分卷积平滑函数。

Finite and infinite dimensional optimization problems naturally arise in a vast array of applied and theoretical problems in the natural sciences, business and economics, management sciences, information sciences, and engineering. A common theme in modern methods, theory, and applications is nonsmoothness. Nonsmoothness arises either directly through the modeling structure or indirectly through the variational properties of the optimal value function and solution mapping. Prominent examples where nonsmoothness plays a central role is in sparsity optimization, maximum likelihood methods for robust statistics, machine learning, robust optimization, mathematical programs with equilibrium constraints (MPECs), mathematical programs with vanishing constraints (MPVCs), (generalized) Nash equilibrium problems ((G)NEPs) or eigenvalue optimization, just to name a few areas. The theoretical tools for dealing with nonsmoothness are subsumed under the label "variational analysis", which comprises convex, nonsmooth and set-valued analysis among other things. The focus of this research project lies on smoothing methods, using variational techniques for both constructing and analyzing smooth approximations of nonsmooth functions.Smoothing methods constitute a standard approach to solving nonsmooth and constrained optimization problems by solving a related sequence of unconstrained smooth approximations. The approximations are constructed so that cluster points of the solutions or stationary points of the approximating smooth problems are solutions or stationary points for the limiting nonsmooth or constrained optimization problem. In the setting of convex programming, there is now great interest in these methods for solving very large-scale problems, where first-order methods for convex nonsmooth optimization have been very successful. In a small- to medium-scale setting, however, second-order methods, in particular, semismooth Newton methods received much attention and are now being used in the infinite-dimensional setting to solve PDE-constrained optimization problems. This project emphasizes so-called epi-smoothing functions, which rely on the notionof epi-convergence of sequences of functionals, and includes two broad areas of study: The first is on second-order methods for a certain class of epi-smoothing functions, namely for those based on infimal convolution, and the second aims at adapting the concept of epi-smoothing to infinite-dimensional spaces due to the importance of epi-convergence in the function space setting. In addition to the epi-smoothing, integral convolution smoothing functions are to be analyzed in order to generalize and strengthen existing results for this important class of smoothing functions.