This paper addresses complementarity problems motivated by constrained optimal control problems. It is shown that the primal-dual active set strategy, which is known to be extremely efficient for this class of problems, and a specific semismooth Newton method lead to identical algorithms. The notion of slant differentiability is recalled and it is argued that the max-function is slantly differentiable in L p-spaces when appropriately combined with a two-norm concept. This leads to new local convergence results of the primal-dual active set strategy. Global unconditional convergence results are obtained by means of appropriate merit functions.

In this paper, a primal-dual algorithm for total bounded variation (TV)-type image restoration is analyzed and tested. Analytically it turns out that employing a global L sregularization, with 1 < s ≤ 2, in the dual problem results in a local smoothing of the TVregularization term in the primal problem. The local smoothing can alternatively be obtained as the infimal convolution of the r-norm, with r −1 + s −1 = 1, and a smooth function. In the case r = s = 2, this results in Gauss-TV-type image restoration. The globalized primal-dual algorithm introduced in this paper works with generalized derivatives, converges locally at a superlinear rate, and is stable with respect to noise in the data. In addition, it utilizes a projection technique which reduces the size of the linear system that has to be solved per iteration. A comprehensive numerical study ends the paper.

A multi-scale total variation model for image restoration is introduced. The model utilizes a spatially dependent regularization parameter in order to enhance image regions containing details while still sufficiently smoothing homogeneous features. The fully automated adjustment strategy of the regularization parameter is based on local variance estimators. For robustness reasons, the decision on the acceptance or rejection of a local parameter value relies on a confidence interval technique based on the expected maximal local variance estimate. In order to speed-up the performance of the update scheme a generalized hierarchical decomposition of the restored image is used. The corresponding subproblems are solved by a superlinearly convergent algorithm based on Fenchel-duality and inexact semismooth Newton techniques. The paper ends by a report on numerical tests, a qualitative study of the proposed adjustment scheme and a comparison with popular total variation based restoration methods.

An optimal control problem governed by an elliptic variational inequality is studied. The feasible set of the problem is relaxed and a pathfollowing type method is used to regularize the constraint on the state variable. First order optimality conditions for the relaxed-regularized subproblems are derived and convergence of stationary points with respect to the relaxation and regularization parameters is shown. In particular, C-and strong stationarity as well as variants thereof are studied. The subproblems are solved by using semismooth Newton methods. The overall algorithmic concept is provided and its performance is discussed by means of examples, including problems with bilateral constraints and a nonsymmetric operator.

The problem of segmentation of a given image using the active contour technique is considered. An abstract calculus to find appropriate speed functions for active contour models in image segmentation or related problems based on variational principles is presented. The speed method from shape sensitivity analysis is used to derive speed functions which correspond to gradient or Newton-type directions for the underlying optimization problem. The Newton-type speed function is found by solving an elliptic problem on the current active contour in every time step. Numerical experiments comparing the classical gradient method with Newton's method are presented.

A nonconvex variational model is introduced which contains the q-"norm", q ∈ (0, 1), of the gradient of the image to be reconstructed as the regularization term together with a leastsquares type data fidelity term which may depend on a possibly spatially dependent weighting parameter. Hence, the regularization term in this functional is a nonconvex compromise between the minimization of the support of the reconstruction and the classical convex total variation model. In the discrete setting, existence of a minimizer is proven, a Newton-type solution algorithm is introduced and its global as well as locally superlinear convergence is established. The potential indefiniteness (or negative definiteness) of the Hessian of the objective during the iteration is handled by a trust-region based regularization scheme. The performance of the new algorithm is studied by means of a series of numerical tests. For the associated infinite dimensional model an existence result based on the weakly lower semicontinuous envelope is established and its relation to the original problem is discussed. * This research was supported by the Austrian Science Fund (FWF) through START project Y305 "Interfaces and Free Boundaries" and through SFB project F3204 "Mathematical Optimization and Applications in Biomedical Sciences".

An adaptive finite element semi-smooth Newton solver for the Cahn-Hilliard model with double obstacle free energy is proposed. For this purpose, the governing system is discretised in time using a semi-implicit scheme, and the resulting time-discrete system is formulated as an optimal control problem with pointwise constraints on the control. For the numerical solution of the optimal control problem, we propose a function space based algorithm which combines a Moreau-Yosida regularization technique for handling the control constraints with a semi-smooth-Newton method for solving the optimality systems of the resulting sub-problems. Further, for the discretization in space and in connection with the proposed algorithm, an adaptive finite element method is considered. The performance of the overall algorithm is illustrated by numerical experiments.

Abstract. Path-following methods for primal-dual active set strategies requiring a regularization parameter are introduced. Existence of a path and its differentiability properties are analyzed. Monotonicity and convexity of the primal-dual path value function are investigated. Both feasible and infeasible approximations are considered. Numerical path following strategies are developed and their efficiency is demonstrated by means of examples.