Semi–Monotonic Augmented Lagrangians for Optimal Control and Parameter Identification

Optimization and inverse problems governed with partial differential equations are often formulated as constrained nonlinear programming problems via the Lagrange formalism. The nonlinearity is treated using the sequential quadratic programming. A numerical solution then hinges on an efficient iterative method for the resulting saddle–point systems. In this paper we apply a semi–monotonic augmented Lagrangians method, recently proposed and analyzed by the second author, for equality and simple–bound constrained quadratic programming subproblems arising from optimal control and parameter identification. Provided multigrid preconditioning of primal and dual space inner products and of the Hessian the algorithm converges at O(1) matrix–vector multiplications. Numerical results are given for applications in image segmentation and 2–dimensional magnetostatics discretized using lowest–order Lagrange finite elements.