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We derive an extension of the sequential homotopy method that allows for theapplication of inexact solvers for the linear (double) saddle-point systemsarising in the local semismooth Newton method for the homotopy subproblems. Forthe class of problems that exhibit (after suitable partitioning of thevariables) a zero in the off-diagonal blocks of the Hessian of the Lagrangian,we propose and analyze an efficient, parallelizable, symmetric positivedefinite preconditioner based on a double Schur complement approach. Fordiscretized optimal control problems with PDE constraints, this structure isoften present with the canonical partitioning of the variables in states andcontrols. We conclude with numerical results for a badly conditioned and highlynonlinear benchmark optimization problem with elliptic partial differentialequations and control bounds. The resulting method allows for the parallelsolution of large 3D problems.

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