论文标题
非平滑分析和优化简介
Introduction to Nonsmooth Analysis and Optimization
论文作者
论文摘要
本书旨在介绍通用的衍生概念，可用于得出必要的最佳条件和数值算法，用于在反问题，成像和PDE约束的优化中出现的无限差异非不同的优化问题。它们涵盖了凸的细分，二元性，单调算子和分解，莫罗 - Yosida正则化以及Clarke以及（短暂地）限制了细分。一阶（近端点和分裂）方法和二阶（半齿牛顿）方法均已处理。此外，讨论了设定值映射的分化，并用于得出最小化器的二阶最佳条件以及Lipschitz稳定性。对反问题的应用和部分微分方程的最佳控制说明了派生的结果和算法。还简要概述了功能分析和变化的计算所需的背景。
This book aims to give an introduction to generalized derivative concepts useful in deriving necessary optimality conditions and numerical algorithms for infinite-dimensional nondifferentiable optimization problems that arise in inverse problems, imaging, and PDE-constrained optimization. They cover convex subdifferentials, Fenchel duality, monotone operators and resolvents, Moreau--Yosida regularization as well as Clarke and (briefly) limiting subdifferentials. Both first-order (proximal point and splitting) methods and second-order (semismooth Newton) methods are treated. In addition, differentiation of set-valued mapping is discussed and used for deriving second-order optimality conditions for as well as Lipschitz stability properties of minimizers. Applications to inverse problems and optimal control of partial differential equations illustrate the derived results and algorithms. The required background from functional analysis and calculus of variations is also briefly summarized.
