L1 penalization of volumetric dose objectives in optimal control of PDEs

TL;DR

This paper introduces an $L^1$ penalization method for PDE-constrained optimal control problems in radiotherapy, addressing infeasibility issues with pointwise state constraints and providing theoretical and numerical analysis.

Contribution

It proposes an $L^1$ penalization approach for PDE control problems, establishing well-posedness, deriving optimality conditions, and developing an efficient semismooth Newton method.

Findings

The $L^1$ penalization effectively manages state constraint violations.

The semismooth Newton method demonstrates computational efficiency.

Numerical results compare favorably with regularized state constraint approaches.

Abstract

This work is concerned with a class of optimal control problems governed by a partial differential equation that are motivated by an application in radiotherapy treatment planning, where 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 can lead to infeasible problems. We therefore propose an alternative approach based on $L^1$ penalization of the violation. We establish well-posedness of the corresponding optimal control problem, derive first-order optimality conditions, and present a semismooth Newton method for the efficient numerical solution of these problems. The performance of this method for a model problem is illustrated and contrasted with the alternative approach based on (regularized) state constraints.