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Bokrantz, R. & Fredriksson, A. (2017). Necessary and sufficient conditions for Pareto efficiency in robust multiobjective optimization. European Journal of Operational Research, 262(2), 682-692
Open this publication in new window or tab >>Necessary and sufficient conditions for Pareto efficiency in robust multiobjective optimization
2017 (English)In: European Journal of Operational Research, ISSN 0377-2217, E-ISSN 1872-6860, Vol. 262, no 2, p. 682-692Article in journal (Refereed) Published
Abstract [en]

We provide necessary and sufficient conditions for robust efficiency (in the sense of Ehrgott et al., 2014) to multiobjective optimization problems that depend on uncertain parameters. These conditions state that a solution is robust efficient (under minimization) if it is optimal to a strongly increasing scalarizing function, and only if it is optimal to a strictly increasing scalarizing function. By counterexample, we show that the necessary condition cannot be strengthened to convex scalarizing functions, even for convex problems. We therefore define and characterize a subset of the robust efficient solutions for which an analogous necessary condition holds with respect to convex scalarizing functions. This result parallels the deterministic case where optimality to a convex and strictly increasing scalarizing function constitutes a necessary condition for efficiency. By a numerical example from the field of radiation therapy treatment plan optimization, we illustrate that the curvature of the scalarizing function influences the conservatism of an optimized solution in the uncertain case.

Place, publisher, year, edition, pages
ELSEVIER SCIENCE BV, 2017
Keywords
Multiobjective optimization, Robust optimization, Scalarization, Uncertainty, Convexity
National Category
Economics and Business Mathematics
Identifiers
urn:nbn:se:kth:diva-210329 (URN)10.1016/j.ejor.2017.04.012 (DOI)000403525300023 ()2-s2.0-85018278140 (Scopus ID)
Note

QC 20170705

Available from: 2017-07-05 Created: 2017-07-05 Last updated: 2017-07-05Bibliographically approved
Bokrantz, R. (2013). Multicriteria optimization for managing tradeoffs in radiation therapy treatment planning. (Doctoral dissertation). Stockholm: KTH Royal Institute of Technology
Open this publication in new window or tab >>Multicriteria optimization for managing tradeoffs in radiation therapy treatment planning
2013 (English)Doctoral thesis, comprehensive summary (Other academic)
Abstract [en]

Treatment planning for radiation therapy inherently involves tradeoffs, such as between tumor control and normal tissue sparing, between time-efficiency and dose quality, and between nominal plan quality and robustness. The purpose of this thesis is to develop methods that can facilitate decision making related to such tradeoffs. The main focus of the thesis is on multicriteria optimization methods where a representative set of treatment plans are first calculated and the most appropriate plan contained in this representation then selected by the treatment planner through continuous interpolation between the precalculated alternatives. These alternatives constitute a subset of the set of Pareto optimal plans, meaning plans such that no criterion can be improved without a sacrifice in another.

Approximation of Pareto optimal sets is first studied with respect to fluence map optimization for intensity-modulated radiation therapy. The approximation error of a discrete representation is minimized by calculation of points one at the time at the location where the distance between an inner and outer approximation of the Pareto set currently attains its maximum. A technique for calculating this distance that is orders of magnitude more efficient than the best previous method is presented. A generalization to distributed computational environments is also proposed.

Approximation of Pareto optimal sets is also considered with respect to direct machine parameter optimization. Optimization of this form is used to calculate representations where any interpolated treatment plan is directly deliverable. The fact that finite representations of Pareto optimal sets have approximation errors with respect to Pareto optimality is addressed by a technique that removes these errors by a projection onto the exact Pareto set. Projections are also studied subject to constraints that prevent the dose-volume histogram from deteriorating.

Multicriteria optimization is extended to treatment planning for volumetric-modulated arc therapy and intensity-modulated proton therapy. Proton therapy plans that are robust against geometric errors are calculated by optimization of the worst case outcome. The theory for multicriteria optimization is extended to accommodate this formulation. Worst case optimization is shown to be preferable to a previous more conservative method that also protects against uncertainties which cannot be realized in practice.

Abstract [sv]

En viktig aspekt av planering av strålterapibehandlingar är avvägningar mellan behandlingsmål vilka står i konflikt med varandra. Exempel på sådana avvägningar är mellan tumörkontroll och dos till omkringliggande frisk vävnad, mellan behandlingstid och doskvalitet, och mellan nominell plankvalitet och robusthet med avseende på geometriska fel. Denna avhandling syftar till att utveckla metoder som kan underlätta beslutsfattande kring motstridiga behandlingsmål. Primärt studeras en metod för flermålsoptimering där behandlingsplanen väljs genom kontinuerlig interpolation över ett representativt urval av förberäknade alternativ. De förberäknade behandlingsplanerna utgör en delmängd av de Paretooptimala planerna, det vill säga de planer sådana att en förbättring enligt ett kriterium inte kan ske annat än genom en försämring enligt ett annat.

Beräkning av en approximativ representation av mängden av Paretooptimala planer studeras först med avseende på fluensoptimering för intensitetsmodulerad strålterapi. Felet för den approximativa representationen minimeras genom att innesluta mängden av Paretooptimala planer mellan inre och yttre approximationer. Dessa approximationer förfinas iterativt genom att varje ny plan genereras där avståndet mellan approximationerna för tillfället är som störst. En teknik för att beräkna det maximala avståndet mellan approximationerna föreslås vilken är flera storleksordningar snabbare än den bästa tidigare kända metoden. En generalisering till distribuerade beräkningsmiljöer föreslås även.

Approximation av mängden av Paretooptimala planer studeras även för direkt maskinparameteroptimering, som används för att beräkna representationer där varje interpolerad behandlingsplan är direkt levererbar. Det faktum att en ändlig representation av mängden av Paretooptimala lösningar har ett approximationsfel till Paretooptimalitet hanteras via en metod där en interpolerad behandlingsplan projiceras på Paretomängden. Projektioner studeras även under bivillkor som förhindrar att den interpolerade planens dos-volym histogram kan försämras.

Flermålsoptimering utökas till planering av rotationsterapi och intensitetsmodulerad protonterapi. Protonplaner som är robusta mot geometriska fel beräknas genom optimering med avseende på det värsta möjliga utfallet av de föreliggande osäkerheterna. Flermålsoptimering utökas även teoretiskt till att innefatta denna formulering. Nyttan av värsta fallet-optimering jämfört med tidigare mer konservativa metoder som även skyddar mot osäkerheter som inte kan realiseras i praktiken demonstreras experimentellt.

Place, publisher, year, edition, pages
Stockholm: KTH Royal Institute of Technology, 2013. p. xvi, 54
Series
Trita-MAT. OS, ISSN 1401-2294 ; 13:07
Keywords
Optimization, multicriteria optimization, robust optimization, Pareto optimality, Pareto surface approximation, Pareto surface navigation, intensity-modulated radiation therapy, volumetric-modulated arc therapy, intensity-modulated proton therapy, Optimering, flermålsoptimering, robust optimering, Paretooptimalitet, Paretofrontsapproximation, Paretofrontsnavigering, intensitetsmodulerad strålterapi, rotationsterapi, intensitetsmodulerad protonterapi
National Category
Computational Mathematics
Identifiers
urn:nbn:se:kth:diva-122663 (URN)978-91-7501-790-7 (ISBN)
Public defence
2013-06-14, F3, Lindstedtsvägen 26, KTH, Stockholm, 10:00 (English)
Opponent
Supervisors
Note

QC 20130527

Available from: 2013-05-27 Created: 2013-05-24 Last updated: 2013-05-28Bibliographically approved
Bokrantz, R., Eriksson, K. & Hardemark, B. (2011). DOES DOSE RATE AND GANTRY SPEED PROVIDE SUFFICIENT DEGREES OF FREEDOM TO ALLOW FOR MULTI-CRITERIA VMAT PLANNING?. Radiotherapy and Oncology, 99, S99-S99
Open this publication in new window or tab >>DOES DOSE RATE AND GANTRY SPEED PROVIDE SUFFICIENT DEGREES OF FREEDOM TO ALLOW FOR MULTI-CRITERIA VMAT PLANNING?
2011 (English)In: Radiotherapy and Oncology, ISSN 0167-8140, E-ISSN 1879-0887, Vol. 99, p. S99-S99Article in journal, Meeting abstract (Other academic) Published
Place, publisher, year, edition, pages
ELSEVIER IRELAND LTD, 2011
National Category
Mathematics
Identifiers
urn:nbn:se:kth:diva-243214 (URN)10.1016/S0167-8140(11)70374-7 (DOI)000433476201154 ()
Note

QC 20190917

Available from: 2019-09-17 Created: 2019-09-17 Last updated: 2019-09-17Bibliographically approved
Organisations
Identifiers
ORCID iD: ORCID iD iconorcid.org/0000-0001-6642-3282

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