An algorithm for approximating convex pareto surfaces based on dual techniques

INFORMS Journal on Computing

Volumn 25, Issue 2, 2013, Pages 377-393

  Bokrantz, Rasmus ^a,b   Forsgren, Anders ^a  

^a ROYAL INSTITUTE OF TECHNOLOGY   (Sweden)

^b RaySearch Laboratories   (Sweden)

Author keywords

Hausdorff distance; Multicriteria optimization; Pareto optimality; Radiation therapy; Sandwich algorithms

Indexed keywords

ALGORITHM FOR SOLVING; CONFLICTING OBJECTIVES; HAUSDORFF DISTANCE; MULTICRITERIA OPTIMIZATION; NONCONVEX OPTIMIZATION PROBLEM; PARETO OPTIMAL SOLUTIONS; PARETO-OPTIMALITY; RADIATION THERAPY OPTIMIZATION;

APPROXIMATION ALGORITHMS; RADIOTHERAPY;

PARETO PRINCIPLE;

EID: 84877962755     PISSN: 10919856     EISSN: 15265528     Source Type: Journal    
DOI: 10.1287/ijoc.1120.0508     Document Type: Article

References (56)

1. Ahnesjö A, Hårdemark B, Isacsson U, Montelius A (2006) The IMRT information process - Mastering the degrees of freedom in external beam therapy. Phys. Med. Biol. 51(13):R381-R402.
2. Bellman R (1961) Adaptive Control Processes: A Guided Tour (Princeton University Press, Princenton, NJ).
3. Benson H (1998) An outer approximation algorithm for generating all efficient extreme points in the outcome set of a multiple objective linear programming problem. J. Global Optim. 13(1): 1-24.
4. Bortfeld T, Schlegel W, Rhein B (1993) Decomposition of pencil beam kernels for fast dose calculations in three-dimensional treatment planning. Med. Phys. 20(2): 311-318.
5. Boyd S, Vandenberghe L (2004) Convex Optimization (Cambridge University Press, Cambridge, UK).
6. Bremner D, Fukuda K, Marzetta A (1998) Primal-dual methods for vertex and facet enumeration. Discrete Comput. Geom. 20(3): 333-357.
7. Carlsson F, Forsgren A, Rehbinder H, Eriksson K (2006) Using eigenstructure of the Hessian to reduce the dimension of the intensity modulated radiation therapy optimization problem. Ann. Oper. Res. 148(1): 81-94.
8. Chernykh O (1995) Approximating the Pareto-hull of a convex set by polyhedral sets. Comp. Math. Math. Phys. 35(8): 1033-1039.
9. Cotrutz C, Lahanas M, Kappas C, Baltas D (2001) A multiobjective gradient-based dose optimization algorithm for external beam conformal radiotherapy. Phys. Med. Biol. 46(8): 2161-2175.
10. Craft D (2010) Calculating and controlling the error of discrete representations of Pareto surfaces in convex multi-criteria optimization. Phys. Medica 26(4): 184-191.
11. Craft D, Bortfeld T (2008) How many plans are needed in an IMRT multi-objective plan database? Phys. Med. Biol. 53(11): 2785-2796.
12. Craft D, Halabi T, Bortfeld T (2005) Exploration of tradeoffs in intensity-modulated radiotherapy. Phys. Med. Biol. 50(24): 5857-5868.
13. Craft D, Halabi T, Shih H, Bortfeld T (2006) Approximating convex Pareto surfaces in multiobjective radiotherapy planning. Med. Phys. 33(9): 3399-3407.
14. Craft D, Halabi T, Shih H, Bortfeld T (2007) An approach for practical multiobjective IMRT treatment planning. Int. J. Radiat. Oncol. 69(5): 1600-1607.
15. Craft D, Hong T, Shih H, Bortfeld T (2012) Improved planning time and plan quality through multi-criteria optimization for intensity modulated radiation therapy. Int. J. Radiat. Oncol. 82(1):e83-e90.
16. Das I, Dennis E (1997) A closer look at drawbacks of minimizing weighted sums of objectives for Pareto set generation in multicriteria optimization problems. Struct. Multidiscip. O. 14(1): 63-69.
17. Dempe S (2002) Nonconvex optimization and its applications. Foundation of Bilevel Programming (Kluwer Academic Publishers, Dordrecht, The Netherlands).
18. Edelsbrunner H (1987) Algorithms in Combinatorial Geometry (Springer-Verlag, Berlin).
19. Ehrgott M (2005) Multicriteria Optimization, Lecture Notes in Economics and Mathematical Systems, Vol. 491, 2nd ed. (Springer- Verlag, Berlin).
20. Ehrgott M, Löhne A, Shao L (2012) A dual variant of Benson's "outer approximation algorithm" for multiple objective linear programming. J. Global Optim. 52(4): 757-778.
21. Ehrgott M, Shao L, Schöbel A (2011) An approximation algorithm for convex multi-objective programming problems. J. Global Optim. 50(3): 397-416.
22. Ehrgott M, Güler Ç, Hamacher H, Shao L (2008) Mathematical optimization in intensity modulated radiation therapy. 4OR-Q. J. Oper. Res. 6(3): 199-262.
23. Engau A (2009) Tradeoff-based decomposition and decisionmaking in multiobjective programming. Eur. J. Oper. Res. 199(3): 883-891.
24. Eskelinen P, Miettinen K, Klamroth K, Hakanen J (2010) Pareto navigator for interactive nonlinear multiobjective optimization. OR Spectrum 32(1): 211-227.
25. Fiacco A, Kyparisis J (1986) Convexity and concavity properties of the optimal value function in parametric nonlinear programming. J. Optim. Theory Appl. 48(1): 95-126.
26. Fruhwirth B, Bukkard R, Rote G (1989) Approximation of convex curves with application to the bicriterial minimum cost flow problem. Eur. J. Oper. Res. 42(3): 326-338.
27. Grünbaum B (2003) Convex Polytopes, 2nd ed. (Springer-Verlag, New York).
28. Hong T, Craft D, Carlsson F, Bortfeld T (2008) Multicriteria optimization in intensity-modulated radiation therapy treatment planning for locally advanced cancer of the pancreatic head. Int. J. Radiat. Oncol. 72(4): 1208-1214.
29. Hunt B, Wiecek M (2003) Cones to aid decision making in multicriteria programming. Tanino T, Tanaka T, Inuiguchi M, eds. Multi-Objective Programming and Goal Programming (Springer- Verlag, Berlin), 153-158.
30. Hunt M, Hsiung C, Spirou S, Chui C, Amols H, Ling C (2002) Evaluation of concave dose distributions created using an inverse planning system. Int. J. Radiat. Oncol. 54(3): 953-962.
31. Kendall M (1962) Rank Correlation Methods (Griffin, London).
32. Küfer K-H, Scherrer A, Monz M, Alonso F, Trinkaus H, Bortfeld T, Thieke C (2003) Intensity-modulated radiotherapy - A large scale multi-criteria programming problem. OR Spectrum 25(2): 223-249.
33. Klamroth K, Tind J, Wiecek M (2003) Unbiased approximation in multicriteria optimization. Math. Method Oper. Res. 56(3): 413-437.
34. Lahanas M, Schreibmann E, Baltas D (2003) Multiobjective inverse planning for intensity modulated radiotherapy with constraintfree gradient-based optimization algorithms. Phys. Med. Biol. 48(17): 2843-2871.
35. McMullen P (1970) The maximum number of faces of a convex polytope. Mathematika 17(2): 179-184.
36. Miettinen K (1999) Nonlinear Multiobjective Optimization (Kluwer Academic Publishers, Dordrecht, The Netherlands).
37. Monz M (2006) Pareto navigation - Interactive multiobjective optimisation and its application in radiotherapy planning. Ph.D. thesis, Kaiserslautern University of Technology, Kaiserslautern, Germany.
38. Monz M, Kufer K-H, Bortfeld T, Thieke C (2008) Pareto navigation - Algorithmic foundation of interactive multicriteria IMRT planning. Phys. Med. Biol. 53(4): 985-998.
39. Pennanen T, Eckstein J (1997) Generalized Jacobians of vectorvalued convex functions. Technical Report RRR 6-97, Rutgers University, New Brunswick, NJ.
40. Preparata F, Shamos M (1985) Computational Geometry - An Introduction (Springer-Verlag, New York).
41. Rennen G, van Dam E, den Hertog D (2011) Enhancement of sandwich algorithms for approximating higher-dimensional convex Pareto sets. INFORMS J. Comput. 23(4): 493-517.
42. Rockafellar R (1970) Convex Analysis (Princeton University Press, Princeton, NJ).
43. Romeijn E, Dempsey J (2008) Intensity modulated radiation therapy treatment plan optimization. TOP 16(2): 215-243.
44. Romeijn E, Dempsey J, Li J (2004) A unifying framework for multicriteria fluence map optimization models. Phys. Med. Biol. 49(10): 1991-2013.
45. Rote G (1992) The convergence rate of the sandwich algorithm for approximating convex functions. Computing 48(3): 337-361.
46. Ruzika S, Wiecek M (2005) Approximation methods in multiobjective programming. J. Optimiz. Theory App. 126(3): 473-501.
47. Sayin S (2000) Measuring the quality of discrete representations of efficient sets in multiple objective mathematical programming. Math. Program. 87(3): 543-560.
48. Schandl B, Klamroth K, Wiecek M (2002) Norm-based approximation in multicriteria programming. Comput. Math. Appl. 44(7): 925-942.
49. Serna J, Monz M, Küfer K-H, Thieke C (2009) Trade-off bounds for the Pareto surface approximation in multi-criteria IMRT planning. Phys. Med. Biol. 54(20): 6299-6311.
50. Shao L, Ehrgott M (2008a) Approximating the nondominated set of an MOLP by approximately solving its dual problem. Math. Method Oper. Res. 68(2): 469-492.
51. Shao L, Ehrgott M (2008b) Approximately solving multiobjective linear programmes in objective space and an application in radiotherapy treatment planning. Math. Method Oper. Res. 68(3): 257-276.
52. Siem A, den Hertog D, Hoffmann A (2008) The effect of transformations on the approximation of univariate (convex) functions with applications to Pareto curves. Eur. J. Oper. Res. 189(2): 347-362.
53. Solanki R, Appino P, Cohon J (1993) Approximating the noninferior set in multiobjective linear programming problems. Eur. J. Oper. Res. 68(3): 356-373.
54. Spalke T, Craft D, Bortfeld T (2009) Analyzing the main trade-offs in multiobjective radiation therapy treatment planning databases. Phys. Med. Biol. 54(12): 3741-3754.
55. Stoer J, Witzgall C (1970) Convexity and Optimization in Finite Dimensions I (Springer-Verlag, Berlin).
56. Thieke C, Küfer K-H, Monz M, Scherrer A, Alonso F, Oelfke U, Huber P, Debus J, Bortfeld T (2007) A new concept for interactive radiotherapy planning with multicriteria optimization: First clinical evaluation. Radiother. Oncol. 85(2): 292-298.