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Primal-dual interior point methods for PDE-constrained optimization

Michael Ulbrich (mulbrich***at***ma.tum.de)
Stefan Ulbrich (ulbrich***at***mathematik.tu-darmstadt.de)

Abstract: 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 $L^p$. It is shown that the developed primal-dual interior-point method converges globally and locally superlinearly. Not only the easier $L^\infty$-setting is analyzed, but also a more involved $L^q$-analysis, $q<\infty$, is presented. In $L^\infty$, the set of feasible controls contains interior points and the Fr\'echet differentiability of the perturbed optimality system can be shown. In the $L^q$-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 particular, two-norm techniques and a smoothing step are required.

Keywords: interior point methods, PDE-constrained optimization, global convergence, superlinear local convergence, optimal control, infinite dimensional optimization

Category 1: Nonlinear Optimization (Systems governed by Differential Equations Optimization )

Category 2: Nonlinear Optimization (Constrained Nonlinear Optimization )

Category 3: Infinite Dimensional Optimization

Citation: Technical Report, Fachbereich Mathematik, TU Darmstadt, Schlossgartenstr. 7, D-64289 Darmstadt, Germany and Zentrum Mathematik, TU Muenchen, Boltzmannstr. 3, D-85747 Garching, Germany, March 2006.

Download: [Postscript][PDF]

Entry Submitted: 04/24/2006
Entry Accepted: 04/24/2006
Entry Last Modified: 04/24/2006

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