Double Saddle-Point Preconditioning for Krylov Methods in the Inexact Sequential Homotopy Method

We derive an extension of the sequential homotopy method that allows for the application of inexact Krylov methods for the linear (double) saddle-point systems arising in the local semismooth Newton method for the homotopy subproblems. For the class of problems that exhibit (after suitable partitioning of the variables) a zero in the off-diagonal blocks of the Hessian of the Lagrangian, we propose and analyze an efficient, parallelizable, symmetric positive definite preconditioner based on a double Schur complement approach. For discretized optimal control problems with PDE constraints, this structure is often present with the canonical partitioning of the variables in states and controls. We conclude with numerical results for a badly conditioned and highly nonlinear benchmark optimization problem with elliptic partial differential equations and control bounds. The resulting method is faster than using direct linear algebra for the 2D benchmark and allows for the parallel solution of large 3D problems.