Journal of Optimization Theory and Applications

, Volume 126, Issue 3, pp 473–501

Approximation Methods in Multiobjective Programming

  • S. Ruzika
  • M. M. Wiecek
Survey Paper


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.


Multiobjective programming approximation efficient sets Pareto sets nondominated sets. 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Deb, K. 2001Multiobjective Optimization Using Evolutionary Algorithms. Series in Systems and OptimizationJohn Wiley and Sons ChichesterEnglandGoogle Scholar
  2. 2.
    Coello Coello, C., Van Veldhuizen, D., Lamont, G. 2002Evolutionary Algorithms for Solving Multiobjective ProblemsKluwer Academic PublishersBoston, MassachusettsGoogle Scholar
  3. 3.
    Ehrgott, M., Gandibleux, X. 2002Multiobjective Combinatorial Optimization Multiple-Criteria Optimization: State of the Art Annotated Bibliographic SurveysKluwer Academic PublishersBoston, MassachusettsGoogle Scholar
  4. 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.Google Scholar
  5. 5.
    Payne, H. J., Polak, E., Collins, D. C., Meisel, W. S. 1975An Algorithm for Bicriteria Optimization Based on the Sensitivity FunctionIEEE Transactions on Automatic Control20546648CrossRefGoogle Scholar
  6. 6.
    Polak, E. 1976On the Approximation of Solutions to Multiple-Criteria Decision Making ProblemsLecture Notes in Economics and Mathematical Systems, Springer, New York ,NY,123271282Google Scholar
  7. 7.
    Jahn, J., Merkel, A. 1992Reference-Point Approximation Method for the Solution of Bicriterial Nonlinear Optimization ProblemsJournal of Optimization Theory and Applications7887103CrossRefGoogle Scholar
  8. 8.
    Helbig, S. 1994On a Constructive Approximation of the Efficient Outcomes in Bicriterion Vector OptimizationJournal of Global Optimization53548CrossRefGoogle Scholar
  9. 9.
    Schandl, B., Klamroth, K., Wiecek, M. M. 2001Norm-Based Approximation in Bicriteria ProgrammingComputational Optimization and Applications202342CrossRefGoogle Scholar
  10. 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.Google Scholar
  11. 11.
    Cohon, J., Church, R., Sheer, D. 1979Generating Multiobjective Tradeoffs: An Algorithm for Bicriterion ProblemsWater Resources Research1510011010Google Scholar
  12. 12.
    Fruhwirth, B., Burkard, R. E., Rote, G. 1989Approximation of Convex Curves with Application to the Bicriterial Minimum Cost Flow ProblemEuropean Journal of Operational Research42326338CrossRefGoogle Scholar
  13. 13.
    Yang, X., Goh, C. 1997A Method for Convex Curve ApproximationEuropean Journal of Operational Research97205212CrossRefGoogle Scholar
  14. 14.
    Solanki, R. S., Cohon, J. L. 1989Approximating the Noninferior Set in Linear Biobjective Programs Using Multiparametric DecompositionEuropean Journal of Operational Research41355366CrossRefGoogle Scholar
  15. 15.
    Ruhe, G., Fruhwirth, B. 1990ɛ-Optimality for Bicriteria Programs and Its Application to Minimum Cost FlowsComputing442134Google Scholar
  16. 16.
    Payne, A. N., Polak, E. 1980An Interactive Rectangle Elimination Method for Biobjective Decision MakingIEEE Transactions on Automatic Control25421432CrossRefGoogle Scholar
  17. 17.
    Solanki, R. S. 1991Generating the Noninferior Set in MixedInteger Biobjective Linear Programs: An Application to a Location ProblemComputers and Operations Research18115CrossRefGoogle Scholar
  18. 18.
    Payne, A. N. 1993Efficient Approximate Representation of Biobjective Tradeoff SetsJournal of the Franklin Institute33012191233CrossRefGoogle Scholar
  19. 19.
    Wiecek, M., Chen, W., Zhang, J. 2001Piecewise Quadratic Approximation of the Nondominated Set for Bicriteria ProgramsJournal of Multicriteria Decision Analysis103547CrossRefGoogle Scholar
  20. 20.
    Liu, Y., Teo, K., Yang, X. 1999Approximation Methods for Nonconvex CurvesEuropean Journal of Operational Research117125135CrossRefGoogle Scholar
  21. 21.
    Li, Y., Fadel, G. M., Wiecek, M. M. 2003Minimum-Effort Approximation of the Pareto Space of Convex Bicriteria ProblemsOptimization and Engineering4231261CrossRefGoogle Scholar
  22. 22.
    Fadel, G., Li, Y. 2002Approximating the Pareto Curve to Help Solve Biobjective Design ProblemsStructural Multidisciplinary Optimization23280296CrossRefGoogle Scholar
  23. 23.
    Popov, N. 1982Approximation of a Pareto Set by the Convolution MethodMoscow University, Computational Mathematics and Cybernetics24148Google Scholar
  24. 24.
    Nefëdov, V. N. 1986Approximation of a Set of Pareto-Optimal SolutionsUSSR Computational Mathematics and Mathematical Physics2699107CrossRefGoogle Scholar
  25. 25.
    Nefëdov, V. N. 1984On the Approximation of a Pareto SetUSSR Computational Mathematics and Mathematical Physics241928CrossRefGoogle Scholar
  26. 26.
    Das, I. 1999On Characterizing the “Knee” of the Pareto Curve Based on Normal–Boundary IntersectionStructural Optimization18107115Google Scholar
  27. 27.
    Steuer, R. E., Harris, F. W. 1980Intra-set Point Generation and Filtering in Decision and Criterion SpaceComputers and Operations Research74158CrossRefGoogle Scholar
  28. 28.
    Reuter, H. 1990An Approximation Method for the Efficiency Set of Multiobjective Programming ProblemsOptimization21905911Google Scholar
  29. 29.
    Sayin, S. 2003A Procedure to Find Discrete Representations of the Efficient Set with Specified Coverage ErrorsOperations Research51427436CrossRefGoogle Scholar
  30. 30.
    Smirnov, M. 1996The Logical Convolution of the Criterion Vector in the Problem of Approximating a Pareto SetComputational Mathematics and Mathematical Physics36605614Google Scholar
  31. 31.
    Buchanan, J., Gardiner, L. 2003A Comparison of Two Reference-Point Methods in Multiple-Objective Mathematical ProgrammingEuropean Journal of Operational Research1491734CrossRefGoogle Scholar
  32. 32.
    Fliege, J., Heseler, A. 2002Constructing Approximations to the Efficient Set of Convex Quadratic Multiobjective Problems Technical Report, Ergebnisberichte in Angewandte Mathematik, Fachbereich MathematikUniversität DortmundDortmund GermanyGoogle Scholar
  33. 33.
    Das, I., Dennis, J. 1998Normal Boundary Intersection: A New Method for Generating the Pareto Surface in Nonlinear Multicriteria Optimization ProblemsSIAM Journal on Optimization8631657CrossRefGoogle Scholar
  34. 34.
    Churkina, S. Y. 2002Search for Efficient Solutions of Multicriterion Problems by Target-Level MethodComputational Mathematics and Modeling13208214CrossRefGoogle Scholar
  35. 35.
    Helbig, S. 1991Approximation of the Efficient Point Set by Perturbation of the Ordering ConeZOR: Methods and Models of Operations Research35197220CrossRefGoogle Scholar
  36. 36.
    Abramova, M. 1986Approximation of a Pareto Set on the Basis of Inexact InformationMoscow University, Computational Mathematics and Cybernetics26269Google Scholar
  37. 37.
    Armann, R. 1989Solving Multiobjective Programming Problems by Discrete RepresentationsOptimization20483492Google Scholar
  38. 38.
    Kostreva, M. M., Zheng, Q., Zhuang, D. A 1995Method for Approximating Solutions of Multicriteria Nonlinear Optimization ProblemsOptimization Methods and Software5209226Google Scholar
  39. 39.
    Benson, H. P., Sayin, S. 1997Toward Finding Global Representations of the Efficient Set in Multiple-Objective Mathematical ProgrammingNaval Research Logistics,444767Google Scholar
  40. 40.
    Karaskal, E. K., Köksalan, M. 2001Generatinga Representative Subset of the Efficient Frontier in Multiple-Criteria Decision AnalysisUniversity of OttawaOttawa, Ontario, CanadaWorking Paper 01–20, Faculty of AdministrationGoogle Scholar
  41. 41.
    Wilson, B., Capelleri, D. J., Simpson, T. W., Frecker, M. I. 2000Efficient Pareto Frontier Exploration Using Surrogate ApproximationsLong BeachCalifornia8th AIAA/USAF/NASA/ISSMO Symposium on Multidisciplinary Analysis and OptimizationGoogle Scholar
  42. 42.
    Messac, A., Mattson, C. A. 2002Generating Well-Distributed Sets of Pareto Points for Engineering Using Physical ProgrammingOptimization and Engineering3431450CrossRefGoogle Scholar
  43. 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.Google Scholar
  44. 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.Google Scholar
  45. 45.
    Chernykh, O. 1995Approximating the Pareto Hull of a Convex Set by Polyhedral SetsComputational Mathematics and Mathematical Physics3510331039MathSciNetGoogle Scholar
  46. 46.
    Schandl, B., Klamroth, K., Wiecek, M. M. 2002Norm-Based Approximation in Multicriteria ProgrammingComputers and Mathematics with Applications44925942CrossRefGoogle Scholar
  47. 47.
    Minkowski, H. 1911Gesammelte Abhandlungen.2 Teubner Verlag LeipzigGermanyin GermanGoogle Scholar
  48. 48.
    Voinalovich, V. 1984External Approximation to the Pareto Set in Criterion Space for Multicriterion Linear Programming TasksCybernetics and Computing Technology1135142Google Scholar
  49. 49.
    Benson, H. P. 1998An Outer-Approximation Algorithm for Generating All Efficient Extreme Points in the Outcome Set of a Multiple-Objective Linear Programming ProblemJournal of Global Optimization13124CrossRefGoogle Scholar
  50. 50.
    Kaliszewski, I. 1994Quantitative Pareto Analysis by Cone Separation TechniqueKluwer Academic PublishersDordrecht, NetherlandsGoogle Scholar
  51. 51.
    Solanki, R. S., Appino, P. A., Cohon, J. L. 1993Approximating the Noninferior Set in Multiobjective Linear Programming ProblemsEuropean Journal of Operational Research68356373CrossRefGoogle Scholar
  52. 52.
    Klamroth, K., Tind, J., Wiecek, M.M. 2002Unbiased Approximation in Multicriteria OptimizationMathematical Methods of Operations Research56413437Google Scholar
  53. 53.
    Mateos, A., Rios-Insua, S. 1997Approximation of Value Efficient SolutionsJournal of Multicriteria Decision Analysis6227232CrossRefGoogle Scholar
  54. 54.
    Mateos, A., Rios-Insua, S., Prieto, L. 1999Computational Study of the Relationships between Feasible and Efficient Sets and an ApproximationComputational Optimization and Applications14241260CrossRefMathSciNetGoogle Scholar
  55. 55.
    Sayin, S. 2000Measuring the Quality of Discrete Representations of Efficient Sets in Multiple-Objective Mathematical ProgrammingMathematical Programming87543560CrossRefGoogle Scholar
  56. 56.
    Beauzamy B. Approximation des Optima de Pareto.Seminaire Choquet 1977/1978; Initiation a l’Analyse, No. 2, Exposé 17, 1978 (in French).Google Scholar
  57. 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).Google Scholar
  58. 58.
    Lotov, A., Kamenev, G., Berezkin, V. 2002Approximation and Visualization of the Pareto Frontier for Nonconvex Multicriteria ProblemsDoklady Mathematics66260262in RussianGoogle Scholar
  59. 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).Google Scholar
  60. 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).Google Scholar
  61. 61.
    Popovici, N. 1998On the Approximation of Efficiency SetsRevue d’Analyse Numérique et de Théorie del’Approximation27321329in FrenchGoogle Scholar
  62. 62.
    Postolica, V., Scarelli, A. 2000Some Connections between Best Approximation, Vectorial Optimization, and Multicriteria AnalysisNonlinear Analysis Forum5111123MathSciNetGoogle Scholar
  63. 63.
    Smirnov, M. 1996Methods for Approximating Faces of a Pareto Set in Linear Multicriterion ProblemsVestnik Moskovskogo Universiteta, Matematika i Kibernetika153743in RussianGoogle Scholar
  64. 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.Google Scholar

Copyright information

© Springer Science+Business Media, Inc. 2005

Authors and Affiliations

  • S. Ruzika
    • 1
  • M. M. Wiecek
    • 2
  1. 1.Diplommathematiker, Fachbereich MathematikUniversität KaiserslauternKaiserslauternGermany
  2. 2.Professor, Department of Mathematical SciencesClemson UniversityClemsonUSA

Personalised recommendations