An Algorithm for Approximating Convex Pareto Surfaces Based on Dual Techniques

References

• Ahnesjö A, Hårdemark B, Isacsson U, Montelius A. The IMRT information process—Mastering the degrees of freedom in external beam therapy. Phys. Med. Biol. (2006) 51(13):R381–R402Crossref, Google Scholar
• Bellman R. Adaptive Control Processes: A Guided Tour (1961) (Princeton University Press, Princenton, NJ) Crossref, Google Scholar
• Benson H. An outer approximation algorithm for generating all efficient extreme points in the outcome set of a multiple objective linear programming problem. J. Global Optim. (1998) 13(1):1–24Crossref, Google Scholar
• Bortfeld T, Schlegel W, Rhein B. Decomposition of pencil beam kernels for fast dose calculations in three-dimensional treatment planning. Med. Phys. (1993) 20(2):311–318Crossref, Google Scholar
• Boyd S, Vandenberghe L. Convex Optimization (2004) (Cambridge University Press, Cambridge, UK) Crossref, Google Scholar
• Bremner D, Fukuda K, Marzetta A. Primal-dual methods for vertex and facet enumeration. Discrete Comput. Geom. (1998) 20(3):333–357Crossref, Google Scholar
• Carlsson F, Forsgren A, Rehbinder H, Eriksson K. Using eigenstructure of the Hessian to reduce the dimension of the intensity modulated radiation therapy optimization problem. Ann. Oper. Res. (2006) 148(1):81–94Crossref, Google Scholar
• Chernykh O. Approximating the Pareto-hull of a convex set by polyhedral sets. Comp. Math. Math. Phys. (1995) 35(8):1033–1039Google Scholar
• Cotrutz C, Lahanas M, Kappas C, Baltas D. A multiobjective gradient-based dose optimization algorithm for external beam conformal radiotherapy. Phys. Med. Biol. (2001) 46(8):2161–2175Crossref, Google Scholar
• Craft D. Calculating and controlling the error of discrete representations of Pareto surfaces in convex multi-criteria optimization. Phys. Medica (2010) 26(4):184–191Crossref, Google Scholar
• Craft D, Bortfeld T. How many plans are needed in an IMRT multi-objective plan database? Phys. Med. Biol. (2008) 53(11):2785–2796Crossref, Google Scholar
• Craft D, Halabi T, Bortfeld T. Exploration of tradeoffs in intensity-modulated radiotherapy. Phys. Med. Biol. (2005) 50(24):5857–5868Crossref, Google Scholar
• Craft D, Halabi T, Shih H, Bortfeld T. Approximating convex Pareto surfaces in multiobjective radiotherapy planning. Med. Phys. (2006) 33(9):3399–3407Crossref, Google Scholar
• Craft D, Halabi T, Shih H, Bortfeld T. An approach for practical multiobjective IMRT treatment planning. Int. J. Radiat. Oncol. (2007) 69(5):1600–1607Crossref, Google Scholar
• Craft D, Hong T, Shih H, Bortfeld T. Improved planning time and plan quality through multi-criteria optimization for intensity modulated radiation therapy. Int. J. Radiat. Oncol. (2012) 82(1):e83–e90Crossref, Google Scholar
• Das I, Dennis E. A closer look at drawbacks of minimizing weighted sums of objectives for Pareto set generation in multicriteria optimization problems. Struct. Multidiscip. O. (1997) 14(1):63–69Crossref, Google Scholar
• Dempe S. Nonconvex optimization and its applications. Foundation of Bilevel Programming (2002) (Kluwer Academic Publishers, Dordrecht, The Netherlands) Google Scholar
• Edelsbrunner H. Algorithms in Combinatorial Geometry (1987) (Springer-Verlag, Berlin) Crossref, Google Scholar
• Ehrgott M. Multicriteria Optimization (2005) 4912nd ed.(Springer-Verlag, Berlin) Lecture Notes in Economics and Mathematical SystemsGoogle Scholar
• Ehrgott M, Löhne A, Shao L. A dual variant of Benson's “outer approximation algorithm” for multiple objective linear programming. J. Global Optim. (2012) 52(4):757–778Crossref, Google Scholar
• Ehrgott M, Shao L, Schöbel A. An approximation algorithm for convex multi-objective programming problems. J. Global Optim. (2011) 50(3):397–416Crossref, Google Scholar
• Ehrgott M, Güler Ç, Hamacher H, Shao L. Mathematical optimization in intensity modulated radiation therapy. 4OR-Q. J. Oper. Res. (2008) 6(3):199–262Crossref, Google Scholar
• Engau A. Tradeoff-based decomposition and decision-making in multiobjective programming. Eur. J. Oper. Res. (2009) 199(3):883–891Crossref, Google Scholar
• Eskelinen P, Miettinen K, Klamroth K, Hakanen J. Pareto navigator for interactive nonlinear multiobjective optimization. OR Spectrum (2010) 32(1):211–227Crossref, Google Scholar
• Fiacco A, Kyparisis J. Convexity and concavity properties of the optimal value function in parametric nonlinear programming. J. Optim. Theory Appl. (1986) 48(1):95–126Crossref, Google Scholar
• Fruhwirth B, Bukkard R, Rote G. Approximation of convex curves with application to the bicriterial minimum cost flow problem. Eur. J. Oper. Res. (1989) 42(3):326–338Crossref, Google Scholar
• Grünbaum B. Convex Polytopes (2003) 2nd ed.(Springer-Verlag, New York) Crossref, Google Scholar
• Hong T, Craft D, Carlsson F, Bortfeld T. Multicriteria optimization in intensity-modulated radiation therapy treatment planning for locally advanced cancer of the pancreatic head. Int. J. Radiat. Oncol. (2008) 72(4):1208–1214Crossref, Google Scholar
• Hunt B, Wiecek M, Tanino T, Tanaka T, Inuiguchi M. Cones to aid decision making in multicriteria programming. Multi-Objective Programming and Goal Programming (2003) (Springer-Verlag, Berlin) 153–158Crossref, Google Scholar
• Hunt M, Hsiung C, Spirou S, Chui C, Amols H, Ling C. Evaluation of concave dose distributions created using an inverse planning system. Int. J. Radiat. Oncol. (2002) 54(3):953–962Crossref, Google Scholar
• Kendall M. Rank Correlation Methods (1962) (Griffin, London) Google Scholar
• Küfer K-H, Scherrer A, Monz M, Alonso F, Trinkaus H, Bortfeld T, Thieke C. Intensity-modulated radiotherapy—A large scale multi-criteria programming problem. OR Spectrum (2003) 25(2):223–249Crossref, Google Scholar
• Klamroth K, Tind J, Wiecek M. Unbiased approximation in multicriteria optimization. Math. Method Oper. Res. (2003) 56(3):413–437Crossref, Google Scholar
• Lahanas M, Schreibmann E, Baltas D. Multiobjective inverse planning for intensity modulated radiotherapy with constraint-free gradient-based optimization algorithms. Phys. Med. Biol. (2003) 48(17):2843–2871Crossref, Google Scholar
• McMullen P. The maximum number of faces of a convex polytope. Mathematika (1970) 17(2):179–184Crossref, Google Scholar
• Miettinen K. Nonlinear Multiobjective Optimization (1999) (Kluwer Academic Publishers, Dordrecht, The Netherlands) Google Scholar
• Monz M. Pareto navigation—Interactive multiobjective optimisation and its application in radiotherapy planning. (2006) . Ph.D. thesis, Kaiserslautern University of Technology, Kaiserslautern, GermanyGoogle Scholar
• Monz M, Kufer K-H, Bortfeld T, Thieke C. Pareto navigation—Algorithmic foundation of interactive multi-criteria IMRT planning. Phys. Med. Biol. (2008) 53(4):985–998Crossref, Google Scholar
• Pennanen T, Eckstein J. Generalized Jacobians of vector-valued convex functions. (1997) . Technical Report RRR 6-97, Rutgers University, New Brunswick, NJGoogle Scholar
• Preparata F, Shamos M. Computational Geometry—An Introduction (1985) (Springer-Verlag, New York) Crossref, Google Scholar
• Rennen G, van Dam E, den Hertog D. Enhancement of sandwich algorithms for approximating higher-dimensional convex Pareto sets. INFORMS J. Comput. (2011) 23(4):493–517Link, Google Scholar
• Rockafellar R. Convex Analysis (1970) (Princeton University Press, Princeton, NJ) Crossref, Google Scholar
• Romeijn E, Dempsey J. Intensity modulated radiation therapy treatment plan optimization. TOP (2008) 16(2):215–243Crossref, Google Scholar
• Romeijn E, Dempsey J, Li J. A unifying framework for multi-criteria fluence map optimization models. Phys. Med. Biol. (2004) 49(10):1991–2013Crossref, Google Scholar
• Rote G. The convergence rate of the sandwich algorithm for approximating convex functions. Computing (1992) 48(3):337–361Crossref, Google Scholar
• Ruzika S, Wiecek M. Approximation methods in multiobjective programming. J. Optimiz. Theory App. (2005) 126(3):473–501Crossref, Google Scholar
• Sayin S. Measuring the quality of discrete representations of efficient sets in multiple objective mathematical programming. Math. Program. (2000) 87(3):543–560Crossref, Google Scholar
• Schandl B, Klamroth K, Wiecek M. Norm-based approximation in multicriteria programming. Comput. Math. Appl. (2002) 44(7):925–942Crossref, Google Scholar
• Serna J, Monz M, Küfer K-H, Thieke C. Trade-off bounds for the Pareto surface approximation in multi-criteria IMRT planning. Phys. Med. Biol. (2009) 54(20):6299–6311Crossref, Google Scholar
• Shao L, Ehrgott M. Approximating the nondominated set of an MOLP by approximately solving its dual problem. Math. Method Oper. Res. (2008a) 68(2):469–492Crossref, Google Scholar
• Shao L, Ehrgott M. Approximately solving multiobjective linear programmes in objective space and an application in radiotherapy treatment planning. Math. Method Oper. Res. (2008b) 68(3):257–276Crossref, Google Scholar
• Siem A, den Hertog D, Hoffmann A. The effect of transformations on the approximation of univariate (convex) functions with applications to Pareto curves. Eur. J. Oper. Res. (2008) 189(2):347–362Crossref, Google Scholar
• Solanki R, Appino P, Cohon J. Approximating the noninferior set in multiobjective linear programming problems. Eur. J. Oper. Res. (1993) 68(3):356–373Crossref, Google Scholar
• Spalke T, Craft D, Bortfeld T. Analyzing the main trade-offs in multiobjective radiation therapy treatment planning databases. Phys. Med. Biol. (2009) 54(12):3741–3754Crossref, Google Scholar
• Stoer J, Witzgall C. Convexity and Optimization in Finite Dimensions I (1970) (Springer-Verlag, Berlin) Crossref, Google Scholar
• Thieke C, Küfer K-H, Monz M, Scherrer A, Alonso F, Oelfke U, Huber P, Debus J, Bortfeld T. A new concept for interactive radiotherapy planning with multicriteria optimization: First clinical evaluation. Radiother. Oncol. (2007) 85(2):292–298Crossref, Google Scholar