The inexact subspace preconditioning techniques of domain decomposition for PDEs, of Schur and Schwarz types, while originally motivated by results from analysis, have generalizations of broad applicability at the discrete algebraic level. A strict implementation of the Reduced Sequential Quadratic Programming (RSQP) technique for equality-constrained optimization requires factorizations of the reduced Hessian and the constraint Jacobian. For large-scale (e.g., millions of variables) PDE-constrained optimization, strict RSQP is therefore infeasible. Conventional approximations (quasi-Newton approximations, inexact Jacobian solves, omission of Hessian terms) lead to suboptimal departures from Newton's method applied to the optimality conditions.

Newton-like convergence can be preserved if the unreduced system is solved iteratively with a Krylov acceleration method requiring only matrix-vector products, which can be supplied by automatic differentiation for much less than the full-matrix costs in operations and storage. One challenge with this approach is preconditioning. This presentation describes preconditioners built from the principles of Schur and Schwarz, leading to a "Lagrange-Newton-Krylov-Schur-Schwarz" class of methods that are illustrated in a prototype problem in parameter identification.