Novel Concepts for Nonsmooth Optimization and their Impact on Science and Technology

Abstract

A multitude of important problems can be cast as nonsmooth variational problems in function spaces, and hence in an infinite-dimensional, setting. Traditionally numerical approaches to such problems are based on first order methods. Only more recently Newton-type methods are systematically investigated and their numerical efficiency is explored. The notion of Newton differentiability combined with path following is of central importance. It will be demonstrated how these techniques are applicable to problems in mathematical imaging, and variational inequalities. Special attention is paid to optimal control with partial differential equations as constraints.