Lipschitz Lectures

Semi-smooth Newton methods for non-differentiable optimization problems

Karl Kunisch, Karl-Franzens-Universität, Graz

 

Date: November 26 - December 11, 2007

  • Mondays: 13:00-15:00
  • Tuesdays: 13:00-15:00

Location: kleiner Hörsaal, Wegelerstr. 10

Abstract:

Obstacle problems, the elasto-plastic torsion problems, Bingham fluids, optimal control of partial differential equations with control or state constraints, problems in image reconstruction or in portfolio optimization share an important common feature: They are optimization, or variational problems, in a function space setting, containing non-differentiable terms. There- fore Newton-type methods with faster than linear convergence properties appear to be ruled out, at first sight. In these lectures we show that nevertheless they can be advantageously applied in the general context of so-called semismooth Newton methods. They lead to super-linearly convergent schemes and frequently have a mesh-independent convergence behavior.

In these lectures we:

  1. Provide the theoretical background from convex analysis.
  2. Review semi-smooth Newton methods in the finite-dimensional context.
  3. Develop the concept of Newton differentiability in function spaces.
  4. Provide the necessary prerequisites for pde-constraint optimal control.
  5. Analyze semi-smooth Newton methods for selected pde-constraint optimization problems.
  6. Provide a duality framework for image reconstruction based on bounded variation type regularization.
  7. Explain path-following techniques for problems which are not sufficiently regular such that Newton-differentiability holds.