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Approximation of Multi-Dimensional Edgeworth-Pareto Hull in Non-linear Multi-Objective Problems

  • Alexander V. LotovEmail author
  • Andrey I. Ryabikov
Conference paper
Part of the Lecture Notes in Computational Science and Engineering book series (LNCSE, volume 131)

Abstract

The paper is devoted to approximating the multi-dimensional Edgeworth-Pareto Hull, which is a tool for decision support in multi-objective optimization problems. The notion of the Edgeworth-Pareto Hull is introduced. It is demonstrated how the effective hull of a non-convex multi-dimensional set given by a mapping can be approximated by the product (intersection) of a finite number of Edgeworth-Pareto Hulls. Then, a new numerical technique for approximating the non-convex EPH for complicated problems is proposed and its properties are discussed.

Notes

Acknowledgement

This research was supported by the Russian Foundation for Basic Research, project no. 17-29-05108 ofi m.

References

  1. 1.
    Lotov, A.V., Bushenkov, V.A., Kamenev, G.K.: Interactive Decision Maps. Approximation and Visualization of Pareto Frontier. Kluwer, Boston (2004)Google Scholar
  2. 2.
    Lotov, A.V., Miettinen, K.: Visualizing the Pareto Frontier Multiobjective Optimization. Interactive and Evolutionary Approaches. Lecture Notes in Computer Science, vol. 5252, pp. 213–244. Springer, Berlin (2008)Google Scholar
  3. 3.
    Evtushenko, Y.G., Posypkin, M.A.: Effective hull of a set and its approximation. Dokl. Math. 90(3), 791–794 (2014)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Kamenev, G.K., Lotov, A.V.: Approximation of the effective hull of a nonconvex multidimensional set given by a nonlinear mapping. Dokl. Math. 97(1), 104–108 (2018)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Miettinen, K.: Nonlinear Multiobjective Optimization. Kluwer, Boston (1999)zbMATHGoogle Scholar
  6. 6.
    Berezkin, V.E., Kamenev, G.K., Lotov, A.V.: Hybrid adaptive methods for approximating a nonconvex multidimensional Pareto Frontier. Comput. Math. Math. Phys. 46(11), 1918–1931 (2006)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Lotov, A.V., Riabikov, A.I.: Multiobjective feedback control and its application to the construction of control rules for a cascade of hydroelectric power stations. Trudy Inst. Math. Mech. 20(4), 187–203 (2014) (in Russian)Google Scholar
  8. 8.
    Riabikov, A.I.: Ersatz function method for minimizing a finite-valued function on a compact set. Comput. Math. Math. Phys. 54(2), 206–1218 (2014)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Lotov, A., Berezkin, V., Kamenev, G., Miettinen, K.: Optimal control of cooling process in continuous casting of steel using a visualization-based multi-criteria approach. Appl. Math. Model. 29(7), 653–672 (2005)CrossRefGoogle Scholar
  10. 10.
    Berezkin, V.E., Lotov, A.V., Lotova, E.A.: Study of hybrid methods for approximating the Edgeworth-Pareto Hull in nonlinear multicriteria optimization problems. Comput. Math. Math. Phys. 54(6), 919–930 (2014)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Deb, K.: Multi-objective Optimization Using Evolutionary Algorithms, p. 515. Wiley, Chichester (2001)Google Scholar
  12. 12.
    Deb, K., Pratap, A., Agarwal, S., Meyarivan, T.: A fast and elitist multiobjective genetic algorithm: NSGA-II. IEEE Trans. Evol. Comput. 6(2), 182–197 (2002)CrossRefGoogle Scholar
  13. 13.
    Horst, R., Pardalos, P.M.: Handbook on Global Optimization. Kluwer, Dordrecht (1995)CrossRefGoogle Scholar
  14. 14.
    Berezkin, V.E., Lotov, A.V.: Comparison of two Pareto Frontier approximations. Comput. Math. Math. Phys. 54(9), 1402–1410 (2014)MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.Dorodnicyn Computing Center of FRC Computer Science and Control RASMoscowRussia

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