Nonsmooth Newton methods for set-valued saddle point problems

We present a new class of iterative schemes for large scale set-valued saddle point problems as arising, e.g., from optimization problems in the presence of linear and inequality constraints. Our algorithms can be regarded either as nonsmooth Newton-type methods for the nonlinear Schur complement or as Uzawa-type iterations with active set preconditioners. Numerical experiments with a control constrained optimal control problem and a discretized Cahn–Hilliard equation with obstacle potential illustrate the reliability and efficiency of the new approach

Efficient and Globally Convergent Minimization Algorithms for Small- and Finite-Strain Plasticity Problems

We present efficient and globally convergent solvers for several classes of plasticity models. The models in this work are formulated in the primal form as energetic rate-independent systems with an elastic energy potential and a plastic dissipation component. Different hardening rules are considered, as well as different flow rules. The time discretization leads to a sequence of nonsmooth minimization problems. For small strains, the unknowns live in vector spaces while for finite strains we have to deal with manifold-valued quantities. For the latter, a reformulation in tangent space is performed to end up with the same dissipation functional as in the small-strain case. We present the Newton-type TNNMG solver for convex and nonsmooth minimization problems and a newly developed Proximal Newton (PN) method that can also handle nonconvex problems. The PN method generates a sequence of penalized convex, coercive but nonsmooth subproblems. These subproblems are in the form of block-separable small-strain plasticity problems, to which TNNMG can be applied. Global convergence theorems are available for both methods. In several numerical experiments, both the efficiency and the flexibility of the methods for small-strain and finite-strain models are tested