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A conjugate-gradient based approach for approximate solutions of quadratic programs
KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Optimization and Systems Theory.
KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Optimization and Systems Theory.ORCID iD: 0000-0002-6252-7815
2008 (English)Report (Other academic)
Abstract [en]

This paper deals with numerical behaviour and convergence properties 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.

Place, publisher, year, edition, pages
Stockholm: KTH Royal Institute of Technology, 2008. , 14 p.
Trita-MAT. OS, ISSN 1401-2294 ; 2008:02
Keyword [en]
column generation, conjugate-gradient method, intensity-modulated radiation therapy, step-and-shoot delivery
National Category
Computational Mathematics
URN: urn:nbn:se:kth:diva-92115OAI: diva2:512118
QC 20120326Available from: 2012-03-26 Created: 2012-03-26 Last updated: 2012-03-26Bibliographically approved

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Carlsson, FredrikForsgren, Anders
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