Inexact Interior-Point Method for PDE-Constrained Nonlinear Optimization

Abstract

Starting from the inexact interior-point framework from Curtis, Schenk, and Wächter [SIAM J. Sci. Comput., 32 (2012), pp. 3447--3475], we propose an effective Schur-complement slack-control preconditioner for the full Lagrangian Hessian matrix needed at each Newton iteration. Together they yield a scalable, robust, and highly parallel method for the numerical solution of large-scale nonconvex PDE-constrained optimization problems with inequality constraints. Because it uses the full Hessian matrix, modifying it whenever needed, the method not only is globally convergent, but also converges fast locally. Our preconditioner is not tailored to any particular class of PDEs or constraints, but instead judiciously exploits the sparsity structure of the Hessian. Numerical examples from PDE-constrained optimal control, parameter estimation, and full-waveform inversion demonstrate the robustness and efficiency of the method, even in the presence of active inequality constraints.