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Variational Techniques in Nonsmooth Optimization

非光滑优化中的变分技术

基本信息

批准号：
254179842
负责人：
Professor Dr. Tim Hoheisel
金额：
--
依托单位：
Department of Mathematics and Statistics
依托单位国家：
德国
项目类别：
Research Fellowships
财政年份：
2013
资助国家：
德国
起止时间：
2012-12-31 至 2013-12-31
项目状态：
已结题

来源：
https://gepris.dfg.de/gepris/projekt/254179842?language=en
关键词：
Variational Techniques Nonsmooth Optimization

项目摘要

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.

有限维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、信息科学和工程学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。非光滑性直接通过建模结构或间接通过最优值函数和解映射的变分性质产生。非光滑性发挥核心作用的突出例子是稀疏优化，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规划（MPEC），具有消失约束的数学规划（MPVC），（广义）纳什均衡问题（G）NEP）或特征值优化，仅举几个领域。处理非光滑性的理论工具被归入“变分分析”的标签下，其中包括凸，非光滑和集值分析等。该研究项目的重点在于光滑方法，使用变分技术来构造和分析非光滑函数的光滑逼近。光滑方法通过求解相关的无约束光滑逼近序列来构成解决非光滑和约束优化问题的标准方法。构造近似，使得近似光滑问题的解的聚类点或稳定点是极限非光滑或约束优化问题的解或稳定点。在凸规划的设置，现在有很大的兴趣在这些方法来解决非常大规模的问题，其中凸非光滑优化的一阶方法已经非常成功。然而，在中小规模的设置，二阶方法，特别是半光滑牛顿法受到了很大的关注，现在正在使用的无限维设置，以解决偏微分方程约束的优化问题。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：第一个是关于一类epi光滑函数的二阶方法，即基于下底卷积的epi光滑函数，第二个目标是适应epi平滑的概念，无限维空间，由于epi收敛的重要性，在功能空间设置。除了epi平滑，积分卷积平滑函数进行分析，以推广和加强现有的结果，这类重要的平滑函数。