Primal-dual interior-point methods for PDE-constrained optimization

This paper provides a detailed analysis of a primal-dual interior-point method for PDE-constrained optimization. Considered are optimal control problems with control constraints in Lp. It is shown that the developed primal-dual interior-point method converges globally and locally superlinearly. Not only the easier L∞-setting is analyzed, but also a more involved Lq-analysis, q < ∞, is presented. In L∞, the set of feasible controls contains interior points and the Fréchet differentiability of the perturbed optimality system can be shown. In the Lq-setting, which is highly relevant for PDE-constrained optimization, these nice properties are no longer available. Nevertheless, a convergence analysis is developed using refined techniques. In parti- cular, two-norm techniques and a smoothing step are required. The Lq-analysis with smoothing step yields global linear and local superlinear convergence, whereas the L∞-analysis without smoothing step yields only global linear convergence.