Variational Techniques in Nonsmooth Optimization

有限和无限维优化问题自然出现在自然科学、商业和经济学、管理科学、信息科学和工程中的大量应用和理论问题中。现代方法、理论和应用中的一个共同主题是非光滑性。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性质间接产生。在稀疏优化、稳健统计的最大似然方法、机器学习、稳健优化、带平衡约束的数学规划(MPEC)、带消失约束的数学规划(MPVC)、(广义)纳什均衡问题((G)N结)或特征值优化等领域中，非光滑性起着重要作用。处理非光滑性的理论工具被归入“变分分析”的标签下，它包括凸性、非光滑性和集值分析等。本研究的重点在于光滑化方法，它使用变分技术来构造和分析非光滑函数的光滑逼近，通过求解相关的无约束光滑逼近序列，构成了解决非光滑和约束优化问题的标准方法。所构造的逼近使得逼近光滑问题的解的聚集点或稳定点是极限非光滑或约束优化问题的解或稳定点。在凸规划的背景下，这些方法在解决超大规模问题方面引起了极大的兴趣，其中一阶凸非光滑优化方法已经非常成功。然而，在中小规模的环境中，二阶方法，特别是半光滑牛顿方法受到了极大的关注，现在正被用于无限维环境中求解偏微分方程组约束优化问题。这个项目强调所谓的表光滑函数，它依赖于泛函序列的表收敛的概念，并包括两个广泛的研究领域：第一个是关于某一类表光滑函数的二阶方法，即基于下界卷积的方法；第二个目的是将表光滑的概念适应于无限维空间，因为表收敛在函数空间中的重要性。除了表光滑外，还需要分析积分卷积光滑函数，以推广和加强这类重要光滑函数的现有结果。

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.