Sufficient optimality conditions and semi-smooth newton methods for optimal control of stationary variational inequalities

¹ Institute for Mathematics and Scientific Computing, Heinrichstraße 36, 8010 Graz, Austria
karl.kunisch@uni-graz.at
² Johann Radon Institute for Computational and Applied Mathematics (RICAM), Austrian Academy of Sciences, Altenbergerstraße 69, 4040 Linz, Austria
daniel.wachsmuth@oeaw.ac.at

Received: 14 July 2010
Revised: 2 February 2011

Abstract

In this paper sufficient second order optimality conditions for optimal control problems subject to stationary variational inequalities of obstacle type are derived. Since optimality conditions for such problems always involve measures as Lagrange multipliers, which impede the use of efficient Newton type methods, a family of regularized problems is introduced. Second order sufficient optimality conditions are derived for the regularized problems as well. It is further shown that these conditions are also sufficient for superlinear convergence of the semi-smooth Newton algorithm to be well-defined and superlinearly convergent when applied to the first order optimality system associated with the regularized problems.