Advertisement

Computational Optimization and Applications

, Volume 52, Issue 3, pp 845–867 | Cite as

PAINT: Pareto front interpolation for nonlinear multiobjective optimization

  • Markus Hartikainen
  • Kaisa Miettinen
  • Margaret M. Wiecek
Article

Abstract

A method called PAINT is introduced for computationally expensive multiobjective optimization problems. The method interpolates between a given set of Pareto optimal outcomes. The interpolation provided by the PAINT method implies a mixed integer linear surrogate problem for the original problem which can be optimized with any interactive method to make decisions concerning the original problem. When the scalarizations of the interactive method used do not introduce nonlinearity to the problem (which is true e.g., for the synchronous NIMBUS method), the scalarizations of the surrogate problem can be optimized with available mixed integer linear solvers. Thus, the use of the interactive method is fast with the surrogate problem even though the problem is computationally expensive. Numerical examples of applying the PAINT method for interpolation are included.

Keywords

Multiobjective optimization Interactive decision making Computationally expensive problems Approximation 

Notes

Acknowledgements

We thank Mr. Karthik Sindhya for providing the Pareto optimal outcomes for the Viennet’s test problem. We also thank Dr. Jussi Hakanen for his help with the wastewater treatment planning problem.

This research was partly supported by the Academy of Finland (grant number 128495) and Jenny and Antti Wihuri Foundation.

References

  1. 1.
    Barber, C.B., Dobkin, D.P., Huhdanpää, H.: The Quickhull algorithm for convex hulls. ACM Trans. Math. Softw. 22, 469–483 (1996) zbMATHCrossRefGoogle Scholar
  2. 2.
    Bezerkin, V.E., Kamenev, G.K., Lotov, A.V.: Hybrid adaptive methods for approximating a nonconvex multidimensional Pareto frontier. Comput. Math. Math. Phys. 46, 1918–1931 (2006) MathSciNetCrossRefGoogle Scholar
  3. 3.
    Brown, K.Q.: Voronoi diagrams from convex hulls. Inf. Process. Lett. 9, 223–228 (1979) zbMATHCrossRefGoogle Scholar
  4. 4.
    Deb, K., Thiele, L., Laumanns, M., Zitzler, E.: Scalable multi-objective optimization test problems. In: IEEE International Conference on E-Commerce Technology, vol. 1, pp. 825–830 (2002) Google Scholar
  5. 5.
    Eaton, J.W.: GNU Octave Manual. Network Theory Limited (2002) Google Scholar
  6. 6.
    Edelsbrunner, H., Mücke, E.P.: Simulation of simplicity: a technique to cope with degenerate cases in geometric algorithms. ACM Trans. Graph. 9, 66–104 (1990) zbMATHCrossRefGoogle Scholar
  7. 7.
    Efremov, R.V., Kamenev, G.K.: Properties of a method for polyhedral approximation of the feasible criterion set in convex multiobjective problems. Ann. Oper. Res. 166, 271–279 (2009) MathSciNetzbMATHCrossRefGoogle Scholar
  8. 8.
    Eskelinen, P., Miettinen, K., Klamroth, K., Hakanen, J.: Pareto navigator for interactive nonlinear multiobjective optimization. OR Spektrum 32, 211–227 (2010) MathSciNetzbMATHCrossRefGoogle Scholar
  9. 9.
    Goel, T., Vaidyanathan, R., Haftka, R.T., Shyy, W., Queipo, N.V., Tucker, K.: Response surface approximation of Pareto optimal front in multi-objective optimization. Comput. Methods Appl. Mech. Eng. 196, 879–893 (2007) zbMATHCrossRefGoogle Scholar
  10. 10.
    Grünbaum, B.: Convex Polytopes. Interscience, London (1967) zbMATHGoogle Scholar
  11. 11.
    Hakanen, J., Miettinen, K., Sahlstedt, K.: Wastewater treatment: new insight provided by interactive multiobjective optimization. Decis. Support Syst. 51, 328–337 (2011) CrossRefGoogle Scholar
  12. 12.
    Hartikainen, M., Miettinen, K., Wiecek, M.M.: Constructing a Pareto front approximation for decision making. Math. Methods Oper. Res. 73, 209–234 (2011) MathSciNetzbMATHCrossRefGoogle Scholar
  13. 13.
    Hartikainen, M., Miettinen, K., Wiecek, M.M.: Pareto front approximations for decision making with inherent non-dominance. In: Shi, Y., Wang, S., Kou, G., Wallenius, J. (eds.) New State of MCDM in the 21st Century, Selected Papers of the 20th International Conference on Multiple Criteria Decision Making 2009, pp. 35–46. Springer, Berlin (2011) Google Scholar
  14. 14.
    Hasenjäger, M., Sendhoff, B.: Crawling along the Pareto front: tales from the practice. In: The 2005 IEEE Congress on Evolutionary Computation (IEEE CEC 2005), pp. 174–181. IEEE Press, Piscataway (2005) Google Scholar
  15. 15.
    Hwang, C., Masud, A.S.M.: Multiple Objective Decision Making—Methods and Applications: a State-of-the-Art Survey. Springer, Berlin (1979) zbMATHCrossRefGoogle Scholar
  16. 16.
    Kamenev, G.K.: Study of an adaptive single-phase method for approximating the multidimensional Pareto frontier in nonlinear systems. Comput. Math. Math. Phys. 49, 2103–2113 (2009) MathSciNetCrossRefGoogle Scholar
  17. 17.
    Keeney, R.L.: Value-Focused Thinking: a Path to Creative Decisionmaking. Harward University Press, Harward (1996) Google Scholar
  18. 18.
    Laukkanen, T., Tveit, T.-M., Ojalehto, V., Miettinen, K., Fogelholm, C.-J.: An interactive multi-objective approach to heat exchanger network synthesis. Comput. Chem. Eng. 34, 943–952 (2010) CrossRefGoogle Scholar
  19. 19.
    Lotov, A.V., Bushenkov, V.A., Kamenev, G.A.: Interactive Decision Maps. Kluwer Academic, Boston (2004) zbMATHGoogle Scholar
  20. 20.
    Luque, M., Ruiz, F., Miettinen, K.: Global formulation for interactive multiobjective optimization. OR Spektrum 33, 27–48 (2011) MathSciNetzbMATHCrossRefGoogle Scholar
  21. 21.
    Martin, J., Bielza, C., Insua, D.R.: Approximating nondominated sets in continuous multiobjective optimization problems. Nav. Res. Logist. 52, 469–480 (2005) zbMATHCrossRefGoogle Scholar
  22. 22.
    McMullen, P.: The maximum number of faces of a convex polytope. Mathematika 17, 179–184 (1970) MathSciNetzbMATHCrossRefGoogle Scholar
  23. 23.
    Miettinen, K.: Nonlinear Multiobjective Optimization. Kluwer Academic, Boston (1999) zbMATHGoogle Scholar
  24. 24.
    Miettinen, K.: IND-NIMBUS for demanding interactive multiobjective optimization. In: Trzaskalik, T. (ed.) Multiple Criteria Decision Making’05, pp. 137–150. The Karol Adamiecki University of Economics in Katowice, Katowice (2006) Google Scholar
  25. 25.
    Miettinen, K., Mäkelä, M.: Interactive bundle-based method for nondifferentiable multiobjective optimization: NIMBUS. Optimization 34, 231–246 (1995) MathSciNetzbMATHCrossRefGoogle Scholar
  26. 26.
    Miettinen, K., Mäkelä, M.M.: On scalarizing functions in multiobjective optimization. OR Spektrum 24, 193–213 (2002) zbMATHCrossRefGoogle Scholar
  27. 27.
    Miettinen, K., Mäkelä, M.M.: Synchronous approach in interactive multiobjective optimization. Eur. J. Oper. Res. 170, 909–922 (2006) zbMATHCrossRefGoogle Scholar
  28. 28.
    Miettinen, K., Ruiz, F., Wierzbicki, A.P.: Introduction to multiobjective optimization: interactive approaches. In: Branke, J., Deb, K., Miettinen, K., Slowinski, R. (eds.) Multiobjective Optimization: Interactive and Evolutionary Approaches, pp. 27–57. Springer, Berlin (2008) Google Scholar
  29. 29.
    Monz, M.: Pareto navigation—algorithmic foundation of interactive multi-criteria IMRT planning. PhD thesis, University of Kaiserslautern (2006) Google Scholar
  30. 30.
    Ruzika, S., Wiecek, M.M.: Approximation methods in multiobjective programming. J. Optim. Theory Appl. 126, 473–501 (2005) MathSciNetzbMATHCrossRefGoogle Scholar
  31. 31.
    Sindhya, K., Deb, K., Miettinen, K.: Improving convergence of evolutionary multi-objective optimization with local search: a concurrent-hybrid algorithm. Nat. Comp. (to appear). doi: 10.1007/s11047-011-9250-4
  32. 32.
    Viennet, R., Fonteix, C., Marc, I.: Multicriteria optimization using a genetic algorithm for determining a Pareto set. Int. J. Syst. Sci. 27, 255–260 (1996) zbMATHCrossRefGoogle Scholar
  33. 33.
    Wierzbicki, A.P.: On the completeness and constructiveness of parametric characterizations to vector optimization problems. OR Spektrum 8, 73–87 (1986) MathSciNetzbMATHCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2011

Authors and Affiliations

  • Markus Hartikainen
    • 1
  • Kaisa Miettinen
    • 1
  • Margaret M. Wiecek
    • 2
  1. 1.Department of Mathematical Information TechnologyUniversity of JyväskyläJyväskyläFinland
  2. 2.Department of Mathematical SciencesClemson UniversityClemsonUSA

Personalised recommendations