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

Richard C. Barnard () and Christian Clason ()
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Richard C. Barnard: Oak Ridge National Laboratory
Christian Clason: University Duisburg-Essen

Computational Optimization and Applications, 2017, vol. 67, issue 2, No 6, 419 pages

Abstract: 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.

Keywords: Optimal control; $$L^1$$ L 1 penalization; Dose volume constraints; Semi-smooth Newton method; 49J52; 49J20; 49M15 (search for similar items in EconPapers)
Date: 2017
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DOI: 10.1007/s10589-017-9897-6

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