Nonsmooth Newton Methods for Set-Valued Saddle Point Problems

Abstract

Abstract. We present a new class of iterative schemes for large scale setvalued saddle point problems as arising, e.g., from optimization problems in the presence of linear and inequality constraints. Our algorithms can be either regarded 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.

Solving the Cahn-Hilliard variational inequality with a semi-smooth Newton method

“…Iterative solution techniques for the resulting system are simple recursive methods using a semi-smooth reformulation of the complementary formulation of the variational inequality, see [7], splitting methods using an approach of Lions and Mercier [3,14,35] and nonlinear Gauss-Seidel and SOR methods [4,20]. Only recently multigrid methods for the discrete variational inequality have been investigated, see [2,[27][28][29] and the discussion below.…”

Fast Solvers for Nonsmooth Optimization Problems in Phase Separation

“…In [6], a nonsmooth Newton Schur method is proposed which is also interpreted as a preconditioned Uzawa iteration. For a given time step k, the Uzawa iteration reads: The first step (21) corresponds to a quadratic obstacle problem:…”

“…By nonlinear Gaussian elimination of the u * variables, the system above could be reduced to a nonlinear Schur complement system in w * variables [6], where the nonlinear Schur complement is given by −(C + BF −1 B T ). In [6], a globally convergent Newton method is proposed for this nonlinear Schur complement system which is later interpreted as a preconditioned Uzawa iteration.…”

“…By nonlinear Gaussian elimination of the u * variables, the system above could be reduced to a nonlinear Schur complement system in w * variables [6], where the nonlinear Schur complement is given by −(C + BF −1 B T ). In [6], a globally convergent Newton method is proposed for this nonlinear Schur complement system which is later interpreted as a preconditioned Uzawa iteration. Note that F (x) is a set valued mapping due to the presence of set-valued operator ∂I K ; to solve the inclusion F (x) ∋ y, or equivalently, x ∈ F −1 y corresponding to the quadratic obstacle problem, many methods have been proposed: projected block GaussSeidel [7], monotone multigrid method [8], [9], [10], truncated monotone multigrid [11], and recently introduced truncated Newton multigrid [11].…”

“…Assuming semi-implicit time discretizations [4] to alleviate the time step restrictions, most of the proposed methods essentially differ in the way the nonlinearity and non-smoothness are handled. Two of the main approaches for such problems are: regularization around the non-smooth region [5] or an active set approach [6] i.e., identify the active sets and solve a reduced problem which is linear, [6] unlike [5] also ensures global convergence of the Newton method by using proper damping parameter. The nonlinear problem corresponding to Cahn-Hilliard problem with Funded by EC math and Matheon obstacle potential could be written as a non-linear system in block 2 × 2 matrix form as follows:…”