Abstract
This paper deals with numerical behavior of a recently presented column generation approach for optimization of so called step-and-shoot radiotherapy treatment plans. The approach and variants of it have been reported to be efficient in practice, finding near-optimal solutions by generating only a low number of columns.
The impact of different restrictions on the columns in a column generation method is studied, and numerical results are given for quadratic programs corresponding to three patient cases. In particular, it is noted that with a bound on the two-norm of the columns, the method is equivalent to the conjugate-gradient method. Further, the above-mentioned column generation approach for radiotherapy is obtained by employing a restriction based on the infinity-norm and non-negativity.
The column generation method has weak convergence properties if restricted to generating feasible step-and-shoot plans, with a “tailing-off” effect for the objective values. However, the numerical results demonstrate that, like the conjugate-gradient method, a rapid decrease of the objective value is obtained in the first few iterations. For the three patient cases, the restriction on the columns to generate feasible step-and-shoot plans has small effect on the numerical efficiency.
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Acknowledgements
We thank Göran Sporre and Henrik Rehbinder for their careful reading of the manuscript and for a number of helpful comments. We also thank the anonymous referees for many constructive comments that significantly improved the presentation. The research was supported by the Swedish Research Council (VR).
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Carlsson, F., Forsgren, A. On column generation approaches for approximate solutions of quadratic programs in intensity-modulated radiation therapy. Ann Oper Res 223, 471–481 (2014). https://doi.org/10.1007/s10479-013-1360-1
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DOI: https://doi.org/10.1007/s10479-013-1360-1