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

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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 ∞-setting is analyzed, but also a more involved L q-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 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 parti- cular, two-norm techniques and a smoothing step are required. The L q-analysis with smoothing step yields global linear and local superlinear convergence, whereas the L ∞-analysis without smoothing step yields only global linear convergence.

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Authors and Affiliations

  1. Chair of Mathematical Optimization, Zentrum Mathematik M1, TU München, Boltzmannstr. 3, 85747, Garching b. München, Germany

    Michael Ulbrich

  2. TU Darmstadt, Fachbereich Mathematik, AG10: Nonlinear Optimization and Optimal Control, Schlossgartenstr. 7, 64289, Darmstadt, Germany

    Stefan Ulbrich

Authors
  1. Michael Ulbrich
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  2. Stefan Ulbrich
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Correspondence to Michael Ulbrich.

Additional information

This paper is dedicated to Steve Robinson on the occasion of his 65th birthday.

Research of M. Ulbrich supported by DFG grant UL 348/2-1. Research of second author supported by DFG grant UL 158/6-1 and by SFB 666.

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Ulbrich, M., Ulbrich, S. Primal-dual interior-point methods for PDE-constrained optimization. Math. Program. 117, 435–485 (2009). https://doi.org/10.1007/s10107-007-0168-7

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  • DOI: https://doi.org/10.1007/s10107-007-0168-7

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