Mathematical Programming

, Volume 117, Issue 1, pp 435–485

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


DOI: 10.1007/s10107-007-0168-7

Cite this article as:
Ulbrich, M. & Ulbrich, S. Math. Program. (2009) 117: 435. doi:10.1007/s10107-007-0168-7


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 

Copyright information

© Springer-Verlag 2007

Authors and Affiliations

  1. 1.Chair of Mathematical OptimizationZentrum Mathematik M1, TU MünchenGarching b. MünchenGermany
  2. 2.TU Darmstadt, Fachbereich MathematikAG10: Nonlinear Optimization and Optimal ControlDarmstadtGermany