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.
Primal-dual interior point methods PDE-constraints Optimal control Control constraints Superlinear convergence Global convergence
Mathematics Subject Classification (2000)
90C51 90C48 49M15 65K10
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