Nonsmooth Analysis and Optimization

TL;DR

This paper provides a comprehensive overview of nonsmooth analysis techniques and optimization methods, focusing on their application to inverse problems, imaging, and PDE-constrained optimization, including subdifferentials, duality, and numerical algorithms.

Contribution

It offers a detailed synthesis of generalized derivative concepts and their use in deriving optimality conditions and algorithms for nonsmooth optimization problems.

Findings

Explains subdifferentials and Fenchel duality in nonsmooth optimization.

Describes proximal point and splitting methods for nonsmooth problems.

Summarizes semismooth Newton methods and their applications.

Abstract

These lecture notes for a graduate course cover generalized derivative concepts useful in deriving necessary optimality conditions and numerical algorithms for nondifferentiable optimization problems in inverse problems, imaging, and PDE-constrained optimization. Treated are convex functions and subdifferentials, Fenchel duality, monotone operators and resolvents, Moreau--Yosida regularization, proximal point and (some) first-order splitting methods, Clarke subdifferentials, and semismooth Newton methods. The required background from functional analysis and calculus of variations is also briefly summarized.