Multiobjective Optimization via Parametric Optimization: Models, Algorithms, and Applications

  • Oleksandr Romanko
  • Alireza Ghaffari-Hadigheh
  • Tamás Terlaky
Conference paper
Part of the Springer Proceedings in Mathematics & Statistics book series (PROMS, volume 21)


In this chapter we highlight the relationships between multiobjective optimization and parametric optimization that is used to solve such problems. Solution of a multiobjective problem is the set of Pareto efficient points, known in the literature as Pareto efficient frontier or Pareto front. Pareto points can be obtained by using either weighting the objectives or by ε-constrained (hierarchical) method for solving multiobjective optimization models. Using those methods we can formulate them as parametric optimization problems and compute their efficient solution set numerically. We present a methodology for conic quadratic optimization that allows tracing the Pareto efficient frontier without discretization of the objective space and without solving the corresponding optimization problem at each discretization point.


Pareto Front Multiobjective Optimization Linear Optimization Invariancy Region Optimal Partition 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.



The authors’ research was partially supported by the NSERC Discovery Grant #48923, the Canada Research Chair Program, and MITACS. The third author was also supported by a Start-up Grant of Lehigh University, and the Hungarian National Development Agency and the European Union within the frame of the project TAMOP 4.2.2-08/1-2008-0021 at the Széchenyi István University, entitled “Simulation and Optimization — basic research in numerical mathematics.” We are grateful to Helmut Mausser, Alexander Kreinin, and Ian Iscoe from Algorithmics Incorporated, an IBM Company, for valuable discussions on the practical examples of multiobjective optimization problems in finance. We would like to thank Antoine Deza from McMaster University, Imre Pólik from SAS Institute and Yuri Zinchenko from University of Calgary for their helpful suggestions.


  1. 1.
    Alexandrov, N.M., Hussaini, M.Y. (eds.): Multidisciplinary Design Optimization: State of the Art. Proceedings in Applied Mathematics Series, No. 80. SIAM, Philadelphia (1997)Google Scholar
  2. 2.
    Alizadeh, F., Goldfarb, D.: Second-order cone programming. Math. Program. Ser. B 95(1), 3–51 (2003)MathSciNetMATHCrossRefGoogle Scholar
  3. 3.
    Bank, B., Guddat, J., Klatte, D., Kummer, B., Tammer, K.: Non-Linear Parametric Optimization. Birkhäuser Verlag, Basel (1983)MATHGoogle Scholar
  4. 4.
    Ben-Tal, A., Nemirovski, A.: Lectures on Modern Convex Optimization: Analysis, Algorithms, and Engineering Applications. MPS-SIAM Series on Optimization. MPS/SIAM, Philadephia (2001)MATHCrossRefGoogle Scholar
  5. 5.
    Berkelaar, A.B., Roos, C., Terlaky, T.: The optimal set and optimal partition approach to linear and quadratic programming. In: Gal, T., Greenberg, H.J. (eds.) Advances in Sensitivity Analysis and Parametric Programming, Chapter 6, pp. 6-1–6-44. Kluwer Academic Publishers, Boston (1997)Google Scholar
  6. 6.
    Bonnans, J.F., Ramírez C., H.: Perturbation analysis of second-order cone programming problems. Math. Program. Ser. B 104(2–3), 205–227 (2005)Google Scholar
  7. 7.
    Borrelli, F., Bemporad, A., Morari, M.: Geometric algorithm for multiparametric linear programming. J. Optim. Theory Appl. 118(3), 515–540 (2003)MathSciNetMATHCrossRefGoogle Scholar
  8. 8.
    Boyd, S.P., Vandenberghe, L.: Convex Optimization. Cambridge University Press, Cambridge (2004)MATHGoogle Scholar
  9. 9.
    Ceria, S.: Robust portfolio construction. Presentation at Workshop on Mixed Integer Programming, University of Miami (June 5–8, 2006).
  10. 10.
    Ceria, S., Stubbs, R.A.: Incorporating estimation errors into portfolio selection: robust portfolio construction. J. Asset Manage. 7(1), 109–127 (2006)CrossRefGoogle Scholar
  11. 11.
    Chankong, V., Haimes, Y.Y. (eds.): Multiobjective Decision Making: Theory and Methodology. Elsevier Science Publishing Co., New York (1983)MATHGoogle Scholar
  12. 12.
    Craft, D.L., Halabi, T.F., Shih, H.A., Bortfeld, T.R.: Approximating convex Pareto surfaces in multiobjective radiotherapy planning. Med. Phys. 33(9), 3399–3407 (2006)CrossRefGoogle Scholar
  13. 13.
    de Berg, M., van Kreveld, M., Overmars, M., Schwarzkopf, O.: Computational Geometry: Algorithms and Applications, 2nd edn. Springer, Berlin (2000)MATHGoogle Scholar
  14. 14.
    Ehrgott, M.: Multicriteria Optimization, 2nd edn. Springer, Berlin (2005)MATHGoogle Scholar
  15. 15.
    Ehrgott, M.: Multiobjective optimization. AI Mag. 29(4), 47–57 (2008)Google Scholar
  16. 16.
    Ehrgott, M., Wiecek, M.M.: Mutiobjective programming. In: Figueira, J., Greco, S., Ehrgott, M. (eds.) Multiple Criteria Decision Analysis: State of the Art Surveys. International Series in Operations Research & Management Science, vol. 78, Chapter 17, pp. 667–708. Springer, New York (2005)Google Scholar
  17. 17.
    Fabozzi, F.J., P.N. Kolm, D.A. Pachamanova, S.M. Focardi: Robust Portfolio Optimization and Management. Wiley, Hoboken (2007)Google Scholar
  18. 18.
    Fliege, J.: An efficient interior-point method for convex multicriteria optimization problems. Math. Oper. Res. 31(4), 825–845 (2006)MathSciNetMATHCrossRefGoogle Scholar
  19. 19.
    Fliege, J., Heseler, A.: Constructing Approximations to the Efficient Set of Convex Quadratic Multiobjective Problems. Ergebnisberichte Angewandte Mathematik 211, Fachbereich Mathematik, Universitat Dortmund, Dortmund (2002)Google Scholar
  20. 20.
    Gal, T., Nedoma, J.: Multiparametric linear programming. Manage. Sci. 18(7), 406–422 (1972)MathSciNetMATHCrossRefGoogle Scholar
  21. 21.
    Ghaffari-Hadigheh, A., Ghaffari-Hadigheh, H., Terlaky, T.: Bi-parametric optimal partition invariancy sensitivity analysis in linear optimization. Central Eur. J. Oper. Res. 16(2), 215–238 (2008)MathSciNetMATHCrossRefGoogle Scholar
  22. 22.
    Ghaffari-Hadigheh, A., Romanko, O., Terlaky, T.: Sensitivity analysis in convex quadratic optimization: simultaneous perturbation of the objective and right-hand-side vectors. Algorithmic Oper. Res. 2(2), 94–111 (2007)MathSciNetMATHGoogle Scholar
  23. 23.
    Ghaffari-Hadigheh, A., Romanko, O., Terlaky, T.: Bi-parametric convex quadratic optimization. Optim. Methods Softw. 25(2), 229–245 (2010)MathSciNetMATHCrossRefGoogle Scholar
  24. 24.
    Goh, C.J., Yang, X.Q.: Analytic efficient solution set for multi-criteria quadratic programs. Eur. J. Oper. Res. 92, 166–181 (1996)MATHCrossRefGoogle Scholar
  25. 25.
    Goldfarb, D., Scheinberg, K.: On parametric semidefinite programming. Appl. Numer. Math. 29(3), 361–377 (1999)MathSciNetMATHCrossRefGoogle Scholar
  26. 26.
    Grodzevich, O., Romanko, O.: Normalization and other topics in multi-objective optimization. In: Aruliah, D.A., Lewis, G.M. (eds.) Proceedings of the First Fields-MITACS Industrial Problems Workshop, pp. 89–101. Fields Institute for Research in Mathematical Sciences, Toronto (2006)Google Scholar
  27. 27.
    Guddat, J., Vasquez, F.G., Tammer, K., Wendler, K.: Multiobjective and stochastic optimization based on parametric optimization. In: Mathematical Research, vol. 26. Akademie-Verlag, Berlin (1985)Google Scholar
  28. 28.
    Hladík, M.: Multiparametric linear programming: support set and optimal partition invariancy. Eur. J. Oper. Res. 202(1), 25–31 (2010)MATHCrossRefGoogle Scholar
  29. 29.
    ILOG Inc.: ILOG CPLEX User’s Manual (2008). CPLEX 11.2
  30. 30.
    Kheirfam, B., Mirnia, K.: Quaternion parametric optimal partition invariancy sensitivity analysis in linear optimization. Adv. Model. Optim. 10(1), 39–40 (2008)MathSciNetMATHGoogle Scholar
  31. 31.
    Kim, I.Y., de Weck, O.L.: Adaptive weighted-sum method for bi-objective optimization: Pareto front generation. Struct. Multidisciplinary Optim. 29(2), 149–158 (2005)CrossRefGoogle Scholar
  32. 32.
    Kvasnica, M.: Real-Time Model Predictive Control via Multi-Parametric Programming: Theory and Tools. VDM Verlag, Saarbrücken (2009).
  33. 33.
    Lee, G.M., Tam, N.N., Yen, N.D.: Continuity of the solution map in quadratic programs under linear perturbations. J. Optim. Theory Appl. 129(3), 415–423 (2006)MathSciNetMATHCrossRefGoogle Scholar
  34. 34.
    Lin, J.G.: Three methods for determining Pareto-optimal solutions of multiple-objective problems. In: Ho, Y.C., Mitter, S.K. (eds.) Directions in Large-Scale Systems, pp. 117–138. Plenum, New York (1975)Google Scholar
  35. 35.
    Lin, J.G.: Proper inequality constraints and maximization of index vectors. J. Optim. Theory Appl. 21(4), 505–521 (1977)MATHCrossRefGoogle Scholar
  36. 36.
    Lobo, M.S., Vandenberghe, L., Boyd, S.P., Lebret, H.: Applications of second-order cone programming. Linear Algebra Appl. 284, 193–228 (1998)MathSciNetMATHCrossRefGoogle Scholar
  37. 37.
    Löfberg, J.: YALMIP: a toolbox for modeling and optimization in MATLAB. In: Proceedings of the CACSD Conference. Taipei, Taiwan (2004)Google Scholar
  38. 38.
    Markowitz, H.M.: Portfolio selection. J. Finance 7(1), 77–91 (1952)Google Scholar
  39. 39.
    Miettinen, K.M.: Nonlinear Multiobjective Optimization. Kluwer Academic Publishers, Boston (1999)MATHGoogle Scholar
  40. 40.
    MOSEK ApS: The MOSEK Optimization Tools Manual. Version 5.0 (2008). Revision 105
  41. 41.
    O’Rourke, J.: Computational Geometry in C, 2nd edn. Cambridge University Press, Cambridge (2001)Google Scholar
  42. 42.
    Pistikopoulos, E.N., Georgiadis, M.C., Dua, V. (eds.): Multi-Parametric Programming: Theory, Algorithms, and Applications, vol. 1. Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim (2007)Google Scholar
  43. 43.
    Romanko, O.: Parametric and multiobjective optimization with applications in finance. Ph.D. Thesis, Department of Computing and Software, McMaster University, Hamilton (2010)Google Scholar
  44. 44.
    Roos, C., Terlaky, T., Vial, J.P.: Interior Point Methods for Linear Optimization. Springer Science, New York (2006)MATHGoogle Scholar
  45. 45.
    Siem, A.Y.D., den Hertog, D., Hoffmann, A.L.: The effect of transformations on the approximation of univariate (convex) functions with applications to Pareto curves. Eur. J. Oper. Res. 189(2), 347–362 (2008)MATHCrossRefGoogle Scholar
  46. 46.
    Yildirim, E.A.: An interior-point perspective on sensitivity analysis in linear programming and semidefinite programming. Ph.D. Thesis, School of Operations Research and Information Engineering, Cornell University, Ithaca (2001)Google Scholar
  47. 47.
    Yildirim, E.A.: Unifying optimal partition approach to sensitivity analysis in conic optimization. J. Optim. Theory Appl. 122(2), 405–423 (2004)MathSciNetMATHCrossRefGoogle Scholar
  48. 48.
    Zopounidis, C., Despotis, D.K., Kamaratou, I.: Portfolio selection using the ADELAIS multiobjective linear programming system. Comput. Econ. 11(3), 189–204 (1998)MATHCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2012

Authors and Affiliations

  • Oleksandr Romanko
    • 1
  • Alireza Ghaffari-Hadigheh
    • 2
  • Tamás Terlaky
    • 3
  1. 1.Department of Computing and SoftwareMcMaster UniversityHamiltonCanada
  2. 2.Department of MathematicsAzarbaijan University of Tarbiat MoallemTabrizIran
  3. 3.Department of Industrial and Systems EngineeringLehigh University, Harold S. Mohler LaboratoryBethlehemUSA

Personalised recommendations