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$$L^1$$ L 1 penalization of volumetric dose objectives in optimal control of PDEs

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Listed:
  • Richard C. Barnard

    (Oak Ridge National Laboratory)

  • Christian Clason

    (University Duisburg-Essen)

Abstract

This work is concerned with a class of PDE-constrained optimization problems that are motivated by an application in radiotherapy treatment planning. Here the primary design objective is to minimize the volume where a functional of the state violates a prescribed level, but prescribing these levels in the form of pointwise state constraints leads to infeasible problems. We therefore propose an alternative approach based on $$L^1$$ L 1 penalization of the violation that is also applicable when state constraints are infeasible. We establish well-posedness of the corresponding optimal control problem, derive first-order optimality conditions, discuss convergence of minimizers as the penalty parameter tends to infinity, and present a semismooth Newton method for their efficient numerical solution. The performance of this method for a model problem is illustrated and contrasted with an alternative approach based on (regularized) state constraints.

Suggested Citation

  • Richard C. Barnard & Christian Clason, 2017. "$$L^1$$ L 1 penalization of volumetric dose objectives in optimal control of PDEs," Computational Optimization and Applications, Springer, vol. 67(2), pages 401-419, June.
    • Handle: RePEc:spr:coopap:v:67:y:2017:i:2:d:10.1007_s10589-017-9897-6
    DOI: 10.1007/s10589-017-9897-6
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    References listed on IDEAS

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    1. Georg Stadler, 2009. "Elliptic optimal control problems with L 1 -control cost and applications for the placement of control devices," Computational Optimization and Applications, Springer, vol. 44(2), pages 159-181, November.
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