Skip to main content
SpringerLink
Log in

Approximation Methods in Multiobjective Programming

  • Survey Paper
  • Published:
Journal of Optimization Theory and Applications Aims and scope Submit manuscript

Abstract

Approaches to approximate the efficient set and Pareto set of multiobjective programs are reviewed. Special attention is given to approximating structures, methods generating Pareto points, and approximation quality. The survey covers more than 50 articles published since 1975.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. K. Deb (2001) Multiobjective Optimization Using Evolutionary Algorithms. Series in Systems and Optimization John Wiley and Sons Chichester England

    Google Scholar 

  2. C. Coello Coello D. Van Veldhuizen G. Lamont (2002) Evolutionary Algorithms for Solving Multiobjective Problems Kluwer Academic Publishers Boston, Massachusetts

    Google Scholar 

  3. M. Ehrgott X. Gandibleux (2002) Multiobjective Combinatorial Optimization Multiple-Criteria Optimization: State of the Art Annotated Bibliographic Surveys Kluwer Academic Publishers Boston, Massachusetts

    Google Scholar 

  4. Ehrgott, M.. and Wiecek, M. M. Multiobjective Programming, Multiple-Criteria Decision Analysis: State of the Art Surveys, Edited by J. Figueira, S. Greco, and M. Ehrgott, Springer, Berlin, Germany, 2005.

  5. H. J. Payne E. Polak D. C. Collins W. S. Meisel (1975) ArticleTitleAn Algorithm for Bicriteria Optimization Based on the Sensitivity Function IEEE Transactions on Automatic Control 20 546–648 Occurrence Handle10.1109/TAC.1975.1101018

    Article  Google Scholar 

  6. E. Polak (1976) ArticleTitleOn the Approximation of Solutions to Multiple-Criteria Decision Making Problems Lecture Notes in Economics and Mathematical Systems, Springer, New York ,NY, 123 271–282

    Google Scholar 

  7. J. Jahn A. Merkel (1992) ArticleTitleReference-Point Approximation Method for the Solution of Bicriterial Nonlinear Optimization Problems Journal of Optimization Theory and Applications 78 87–103 Occurrence Handle10.1007/BF00939894

    Article  Google Scholar 

  8. S. Helbig (1994) ArticleTitleOn a Constructive Approximation of the Efficient Outcomes in Bicriterion Vector Optimization Journal of Global Optimization 5 35–48 Occurrence Handle10.1007/BF01097002

    Article  Google Scholar 

  9. B. Schandl K. Klamroth M. M. Wiecek (2001) ArticleTitleNorm-Based Approximation in Bicriteria Programming Computational Optimization and Applications 20 23–42 Occurrence Handle10.1023/A:1011267305444

    Article  Google Scholar 

  10. Das, I. An Improved Technique for Choosing Parameters for Pareto Surface Generation Using Normal-Boundary Intersection, Proceedings of The 3rd World Congress of Structural and Multidisciplinary Optimization, Buffalo, NY, 1999.

  11. J. Cohon R. Church D. Sheer (1979) ArticleTitleGenerating Multiobjective Tradeoffs: An Algorithm for Bicriterion Problems Water Resources Research 15 1001–1010

    Google Scholar 

  12. B. Fruhwirth R. E. Burkard G. Rote (1989) ArticleTitleApproximation of Convex Curves with Application to the Bicriterial Minimum Cost Flow Problem European Journal of Operational Research 42 326–338 Occurrence Handle10.1016/0377-2217(89)90443-8

    Article  Google Scholar 

  13. X. Yang C. Goh (1997) ArticleTitleA Method for Convex Curve Approximation European Journal of Operational Research 97 205–212 Occurrence Handle10.1016/0377-2217(95)00368-1

    Article  Google Scholar 

  14. R. S. Solanki J. L. Cohon (1989) ArticleTitleApproximating the Noninferior Set in Linear Biobjective Programs Using Multiparametric Decomposition European Journal of Operational Research 41 355–366 Occurrence Handle10.1016/0377-2217(89)90256-7

    Article  Google Scholar 

  15. G. Ruhe B. Fruhwirth (1990) ArticleTitleɛ-Optimality for Bicriteria Programs and Its Application to Minimum Cost Flows Computing 44 21–34

    Google Scholar 

  16. A. N. Payne E. Polak (1980) ArticleTitleAn Interactive Rectangle Elimination Method for Biobjective Decision Making IEEE Transactions on Automatic Control 25 421–432 Occurrence Handle10.1109/TAC.1980.1102344

    Article  Google Scholar 

  17. R. S. Solanki (1991) ArticleTitleGenerating the Noninferior Set in MixedInteger Biobjective Linear Programs: An Application to a Location Problem Computers and Operations Research 18 1–15 Occurrence Handle10.1016/0305-0548(91)90037-R

    Article  Google Scholar 

  18. A. N. Payne (1993) ArticleTitleEfficient Approximate Representation of Biobjective Tradeoff Sets Journal of the Franklin Institute 330 1219–1233 Occurrence Handle10.1016/0016-0032(93)90073-4

    Article  Google Scholar 

  19. M. Wiecek W. Chen J. Zhang (2001) ArticleTitlePiecewise Quadratic Approximation of the Nondominated Set for Bicriteria Programs Journal of Multicriteria Decision Analysis 10 35–47 Occurrence Handle10.1002/mcda.287

    Article  Google Scholar 

  20. Y. Liu K. Teo X. Yang (1999) ArticleTitleApproximation Methods for Nonconvex Curves European Journal of Operational Research 117 125–135 Occurrence Handle10.1016/S0377-2217(98)90203-X

    Article  Google Scholar 

  21. Y. Li G. M. Fadel M. M. Wiecek (2003) ArticleTitleMinimum-Effort Approximation of the Pareto Space of Convex Bicriteria Problems Optimization and Engineering 4 231–261 Occurrence Handle10.1023/A:1023993231971

    Article  Google Scholar 

  22. G. Fadel Y. Li (2002) ArticleTitleApproximating the Pareto Curve to Help Solve Biobjective Design Problems Structural Multidisciplinary Optimization 23 280–296 Occurrence Handle10.1007/s00158-002-0186-2

    Article  Google Scholar 

  23. N. Popov (1982) ArticleTitleApproximation of a Pareto Set by the Convolution Method Moscow University, Computational Mathematics and Cybernetics 2 41–48

    Google Scholar 

  24. V. N. Nefëdov (1986) ArticleTitleApproximation of a Set of Pareto-Optimal Solutions USSR Computational Mathematics and Mathematical Physics 26 99–107 Occurrence Handle10.1016/0041-5553(86)90192-8

    Article  Google Scholar 

  25. V. N. Nefëdov (1984) ArticleTitleOn the Approximation of a Pareto Set USSR Computational Mathematics and Mathematical Physics 24 19–28 Occurrence Handle10.1016/0041-5553(84)90225-8

    Article  Google Scholar 

  26. I. Das (1999) ArticleTitleOn Characterizing the “Knee” of the Pareto Curve Based on Normal–Boundary Intersection Structural Optimization 18 107–115

    Google Scholar 

  27. R. E. Steuer F. W. Harris (1980) ArticleTitleIntra-set Point Generation and Filtering in Decision and Criterion Space Computers and Operations Research 7 41–58 Occurrence Handle10.1016/0305-0548(80)90013-1

    Article  Google Scholar 

  28. H. Reuter (1990) ArticleTitleAn Approximation Method for the Efficiency Set of Multiobjective Programming Problems Optimization 21 905–911

    Google Scholar 

  29. S. Sayin (2003) ArticleTitleA Procedure to Find Discrete Representations of the Efficient Set with Specified Coverage Errors Operations Research 51 427–436 Occurrence Handle10.1287/opre.51.3.427.14951

    Article  Google Scholar 

  30. M. Smirnov (1996) ArticleTitleThe Logical Convolution of the Criterion Vector in the Problem of Approximating a Pareto Set Computational Mathematics and Mathematical Physics 36 605–614

    Google Scholar 

  31. J. Buchanan L. Gardiner (2003) ArticleTitleA Comparison of Two Reference-Point Methods in Multiple-Objective Mathematical Programming European Journal of Operational Research 149 17–34 Occurrence Handle10.1016/S0377-2217(02)00487-3

    Article  Google Scholar 

  32. J. Fliege A. Heseler (2002) Constructing Approximations to the Efficient Set of Convex Quadratic Multiobjective Problems Technical Report, Ergebnisberichte in Angewandte Mathematik, Fachbereich Mathematik Universität Dortmund Dortmund Germany

    Google Scholar 

  33. I. Das J. Dennis (1998) ArticleTitleNormal Boundary Intersection: A New Method for Generating the Pareto Surface in Nonlinear Multicriteria Optimization Problems SIAM Journal on Optimization 8 631–657 Occurrence Handle10.1137/S1052623496307510

    Article  Google Scholar 

  34. S. Y. Churkina (2002) ArticleTitleSearch for Efficient Solutions of Multicriterion Problems by Target-Level Method Computational Mathematics and Modeling 13 208–214 Occurrence Handle10.1023/A:1015227216248

    Article  Google Scholar 

  35. S. Helbig (1991) ArticleTitleApproximation of the Efficient Point Set by Perturbation of the Ordering Cone ZOR: Methods and Models of Operations Research 35 197–220 Occurrence Handle10.1007/BF01415907

    Article  Google Scholar 

  36. M. Abramova (1986) ArticleTitleApproximation of a Pareto Set on the Basis of Inexact Information Moscow University, Computational Mathematics and Cybernetics 2 62–69

    Google Scholar 

  37. R. Armann (1989) ArticleTitleSolving Multiobjective Programming Problems by Discrete Representations Optimization 20 483–492

    Google Scholar 

  38. M. M. Kostreva Q. Zheng D. A Zhuang (1995) ArticleTitleMethod for Approximating Solutions of Multicriteria Nonlinear Optimization Problems Optimization Methods and Software 5 209–226

    Google Scholar 

  39. H. P. Benson S. Sayin (1997) ArticleTitleToward Finding Global Representations of the Efficient Set in Multiple-Objective Mathematical Programming Naval Research Logistics, 44 47–67

    Google Scholar 

  40. E. K. Karaskal M. Köksalan (2001) Generatinga Representative Subset of the Efficient Frontier in Multiple-Criteria Decision Analysis University of Ottawa Ottawa, Ontario, Canada

    Google Scholar 

  41. B. Wilson D. J. Capelleri T. W. Simpson M. I. Frecker (2000) Efficient Pareto Frontier Exploration Using Surrogate Approximations Long Beach California

    Google Scholar 

  42. A. Messac C. A. Mattson (2002) ArticleTitleGenerating Well-Distributed Sets of Pareto Points for Engineering Using Physical Programming Optimization and Engineering 3 431–450 Occurrence Handle10.1023/A:1021179727569

    Article  Google Scholar 

  43. Ismail-Yahaya, A. and Messac, A. Effective Generation of the Pareto Frontier: The Normalized Normal-Constraint Method. Paper AIAA-2002-1232, AIAA/ASME/ASCE/AHS Structures, Structural Dynamics, and Mateial Conference, Denver, Colorado, 2002.

  44. Mattson, C., Mullur, A.. and Messac, A. Minimal Representation of Multiobjective Design Space Using a Smart Pareto Filter, 9th AIAA/ISSMO Symposium on Multidisciplinary Analysis and Optimization, Atlanta, Georgia, 2002.

  45. O. Chernykh (1995) ArticleTitleApproximating the Pareto Hull of a Convex Set by Polyhedral Sets Computational Mathematics and Mathematical Physics 35 1033–1039 Occurrence HandleMR1355312

    MathSciNet  Google Scholar 

  46. B. Schandl K. Klamroth M. M. Wiecek (2002) ArticleTitleNorm-Based Approximation in Multicriteria Programming Computers and Mathematics with Applications 44 925–942 Occurrence Handle10.1016/S0898-1221(02)00204-3

    Article  Google Scholar 

  47. H. Minkowski (1911) Gesammelte Abhandlungen. 2 Teubner Verlag Leipzig Germany

    Google Scholar 

  48. V. Voinalovich (1984) ArticleTitleExternal Approximation to the Pareto Set in Criterion Space for Multicriterion Linear Programming Tasks Cybernetics and Computing Technology 1 135–142

    Google Scholar 

  49. H. P. Benson (1998) ArticleTitleAn Outer-Approximation Algorithm for Generating All Efficient Extreme Points in the Outcome Set of a Multiple-Objective Linear Programming Problem Journal of Global Optimization 13 1–24 Occurrence Handle10.1023/A:1008215702611

    Article  Google Scholar 

  50. I. Kaliszewski (1994) Quantitative Pareto Analysis by Cone Separation Technique Kluwer Academic Publishers Dordrecht, Netherlands

    Google Scholar 

  51. R. S. Solanki P. A. Appino J. L. Cohon (1993) ArticleTitleApproximating the Noninferior Set in Multiobjective Linear Programming Problems European Journal of Operational Research 68 356–373 Occurrence Handle10.1016/0377-2217(93)90192-P

    Article  Google Scholar 

  52. K. Klamroth J. Tind M.M. Wiecek (2002) ArticleTitleUnbiased Approximation in Multicriteria Optimization Mathematical Methods of Operations Research 56 413–437

    Google Scholar 

  53. A. Mateos S. Rios-Insua (1997) ArticleTitleApproximation of Value Efficient Solutions Journal of Multicriteria Decision Analysis 6 227–232 Occurrence Handle10.1002/(SICI)1099-1360(199707)6:4<227::AID-MCDA149>3.0.CO;2-J

    Article  Google Scholar 

  54. A. Mateos S. Rios-Insua L. Prieto (1999) ArticleTitleComputational Study of the Relationships between Feasible and Efficient Sets and an Approximation Computational Optimization and Applications 14 241–260 Occurrence Handle10.1023/A:1008751400773 Occurrence HandleMR1714073

    Article  MathSciNet  Google Scholar 

  55. S. Sayin (2000) ArticleTitleMeasuring the Quality of Discrete Representations of Efficient Sets in Multiple-Objective Mathematical Programming Mathematical Programming 87 543–560 Occurrence Handle10.1007/s101070050128

    Article  Google Scholar 

  56. Beauzamy B. Approximation des Optima de Pareto.Seminaire Choquet 1977/1978; Initiation a l’Analyse, No. 2, Exposé 17, 1978 (in French).

  57. Karyakin, A. A. Methods of Construction and Approximation of the Pareto Boundary of Linear Multicriteria Problems. Software of Economic Studies, Collection of Scientific Works, Novosibirsk, Russia:25–36, 1985 (in Russian).

  58. A. Lotov G. Kamenev V. Berezkin (2002) ArticleTitleApproximation and Visualization of the Pareto Frontier for Nonconvex Multicriteria Problems Doklady Mathematics 66 260–262

    Google Scholar 

  59. Polishchuk, L. A Piecewise Linear Approximation of the Pareto Boundary for Convex Two-Criteria Problems. Models and Methods of Investigating Economic Systems, Collection of Scientific Works, Novosibirsk, Russia:108–116, 1979 (in Russian).

  60. Popov, N. Estimation of the Computational Complexity of Multicriteria Optimization. Computing Complexes and Modeling of Complex Systems, Collection of Scientific Works, Moscow, Russia:142–152, 1989 (in Russian).

  61. N. Popovici (1998) ArticleTitleOn the Approximation of Efficiency Sets Revue d’Analyse Numérique et de Théorie del’Approximation 27 321–329

    Google Scholar 

  62. V. Postolica A. Scarelli (2000) ArticleTitleSome Connections between Best Approximation, Vectorial Optimization, and Multicriteria Analysis Nonlinear Analysis Forum 5 111–123 Occurrence HandleMR1798688

    MathSciNet  Google Scholar 

  63. M. Smirnov (1996) ArticleTitleMethods for Approximating Faces of a Pareto Set in Linear Multicriterion Problems Vestnik Moskovskogo Universiteta, Matematika i Kibernetika 15 37–43

    Google Scholar 

  64. Yannakakis, M. Approximation of Multiobjective Optimization Problems. Algorithms and Data Structures: 7th International Workshop, Edited by F. Dehne, J. R. Sack, and R. Tamassia, 2001.

Download references

Author information

Authors and Affiliations

  1. Diplommathematiker, Fachbereich Mathematik, Universität Kaiserslautern, Kaiserslautern, Germany

    S. Ruzika

  2. Professor, Department of Mathematical Sciences, Clemson University, Clemson, SC, USA

    M. M. Wiecek

Authors
  1. S. Ruzika
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. M. M. Wiecek
    View author publications

    You can also search for this author in PubMed Google Scholar

Additional information

H.P. Benson

His work was supported by Deutsche Forschungsgemeinschaft, Grant HA 1795/7-2.

Her work was done while on a sabbatical leave at the University of Kaiserslautern with support of Deutsche Forschungsgemeinschaft, Grant Ka 477/24-1.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Ruzika, S., Wiecek, M.M. Approximation Methods in Multiobjective Programming. J Optim Theory Appl 126, 473–501 (2005). https://doi.org/10.1007/s10957-005-5494-4

Download citation

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10957-005-5494-4

Keywords

Search

Navigation