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

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 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. Finally, the performance of this method for a model problem is illustrated and contrasted with an alternative approach based on (regularized) state constraints.

Authors:
 [1]; ORCiD logo [2]
  1. Oak Ridge National Lab. (ORNL), Oak Ridge, TN (United States)
  2. Univ. of Duisburg-Essen, Essen (Germany)
Publication Date:
Research Org.:
Oak Ridge National Lab. (ORNL), Oak Ridge, TN (United States)
Sponsoring Org.:
USDOE
OSTI Identifier:
1344291
Grant/Contract Number:
AC05-00OR22725
Resource Type:
Journal Article: Accepted Manuscript
Journal Name:
Computational Optimization and applications
Additional Journal Information:
Journal Volume: 67; Journal Issue: 2; Journal ID: ISSN 0926-6003
Publisher:
Springer
Country of Publication:
United States
Language:
English
Subject:
97 MATHEMATICS AND COMPUTING; optimal control; L1 penalization; dose volume constraints; semi-smooth Newton method

Citation Formats

Barnard, Richard C., and Clason, Christian. $L^1$ penalization of volumetric dose objectives in optimal control of PDEs. United States: N. p., 2017. Web. doi:10.1007/s10589-017-9897-6.
Barnard, Richard C., & Clason, Christian. $L^1$ penalization of volumetric dose objectives in optimal control of PDEs. United States. doi:10.1007/s10589-017-9897-6.
Barnard, Richard C., and Clason, Christian. Sat . "$L^1$ penalization of volumetric dose objectives in optimal control of PDEs". United States. doi:10.1007/s10589-017-9897-6. https://www.osti.gov/servlets/purl/1344291.
@article{osti_1344291,
title = {$L^1$ penalization of volumetric dose objectives in optimal control of PDEs},
author = {Barnard, Richard C. and Clason, Christian},
abstractNote = {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 L1 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. Finally, the performance of this method for a model problem is illustrated and contrasted with an alternative approach based on (regularized) state constraints.},
doi = {10.1007/s10589-017-9897-6},
journal = {Computational Optimization and applications},
number = 2,
volume = 67,
place = {United States},
year = {Sat Feb 11 00:00:00 EST 2017},
month = {Sat Feb 11 00:00:00 EST 2017}
}

Journal Article:
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  • We develop a viscosity solution theory for a system of nonlinear degenerate parabolic integro-partial differential equations (IPDEs) related to stochastic optimal switching and control problems or stochastic games. In the case of stochastic optimal switching and control, we prove via dynamic programming methods that the value function is a viscosity solution of the IPDEs. In our setting the value functions or the solutions of the IPDEs are not smooth, so classical verification theorems do not apply.
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