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Solution of the Portfolio Optimization Model as a Fuzzy Bilevel Programming Problem

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Book cover Applied Mathematics and Computational Intelligence (FIM 2015)

Part of the book series: Advances in Intelligent Systems and Computing ((AISC,volume 730))

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Abstract

In this chapter, we consider a mixed-integer bilevel linear programming problem with one parameter in the right-hand side of the constraints in the lower level (or, the follower’s) problem. Motivated by an application to the fuzzy portfolio optimization model, we consider a particular case that consists in maximizing the investor’s expected return. The functions are linear at both the upper and lower levels, and the proposed algorithm is based upon an approximation of the optimal value function using the branch-and-bound method. Therefore, at every node of this tree-type structure, we apply a new branch-and-bound procedure to deal with the integrity restriction.

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References

  1. Bard, J.F.: An algorithm for solving the general bilevel programming problem. Math. Oper. Res. 8, 260–282 (1983)

    Article  MathSciNet  MATH  Google Scholar 

  2. Dempe, S.: A simple algorithm for the linear bilevel programming problem. Optimization 18, 373–385 (1987)

    Article  MathSciNet  MATH  Google Scholar 

  3. Dempe, S.: Foundations of Bilevel Programming. Kluwer Academic Publishers, Dordrecht, London, Boston (2002)

    MATH  Google Scholar 

  4. Dempe, S., Kalashnikov, V.V.: Discrete Bilevel Programming with Linear Lower Level Problems. Preprint, TU Bergakademie Freiberg (2005)

    Google Scholar 

  5. Dempe, S., Schreier, H.: Operations Research - Deterministische Modelle und Methoden. Teubner Verlag, Wiesbaden (2006)

    MATH  Google Scholar 

  6. Dempe, S., Zemkoho, A.B.: A Bilevel Approach to Optimal Toll Setting in Capacitated Networks. Preprint, TU Bergakademie Freiberg (2008)

    Google Scholar 

  7. Dempe, S., Kalashnikov, V.V., Kalashnykova, N.I., Arévalo Franco, A.: A new approach to solving bi-level programming problems with integer upper level variables. ICIC Express Lett. 3, 1281–1286 (2009)

    Google Scholar 

  8. Dempe, S., Kalashnikov, V.V., Pérez-Valdés, G.A., Kalashnykova, N.I.: Bilevel Programming Problems: Theory, Algorithms and Applications to Energy Networks. Springer, Heidelberg, New York, Dordrecht, London (2015)

    Book  MATH  Google Scholar 

  9. Dempe, S., Kalashnikov, V.V., Ríos-Mercado, R.Z.: Discrete bilevel programming: application to a natural gas cash-out problem. Eur. J. Oper. Res. 166, 469–488 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  10. Faísca, N.P., Dua, V., Rustem, B., Saraiva, P.M., Pistikopoulos, E.N.: Parametric global optimization for bilevel programming. J. Glob. Optim. 38, 609–623 (2007)

    Article  MATH  Google Scholar 

  11. Floudas, C.A., Gümüş, Z.H., Ierapetritou, M.G.: Global optimization in design under uncertainty: feasibility test and flexibility index problem. Ind. Eng. Chem. Res. 40, 4267–4282 (2001)

    Article  Google Scholar 

  12. Gao, D.Y.: Canonical duality theory and solutions to constrained nonconvex quadratic programming. J. Glob. Optim. 29, 377–399 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  13. Gao, D.Y.: Solutions and optimality criteria to box constrained nonconvex minimization problems. J. Ind. Manag. Optim. 3, 293–304 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  14. Grygarová, L.: Qualitative Untersuchung des I. Optimierungsproblems in mehrparametrischer Programmierung. Appl. Math. 15, 276–295 (1970)

    MathSciNet  MATH  Google Scholar 

  15. Gümüş, Z.H., Floudas, C.A.: Global optimization of mixed-integer-bilevel programming problems. Comput. Manag. Sci. 2, 181–212 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  16. Hu, X.P., Li, Y.X., Guo, J.W., Sun, L.J., Zeng, A.Z.: A simulation optimization algorithms with heuristic transformation and its application to vehicle routing problems. Int. J. Innov. Comput. Inf. Control 4, 1169–1182 (2008)

    Google Scholar 

  17. Jan, R.H., Chern, M.S.: Non-linear integer bilevel programming. Eur. J. Oper. Res. 72, 574–587 (1994)

    Article  MATH  Google Scholar 

  18. Kalashnikov, V.V., Kalashnykova, N.I., Castillo-Pérez, F.J.: Finding equilibrium in a financial model by solving a variational inequality problem. In: Le Thi, H.A., et al. (eds.) Modelling, Computation and Optimization in Information Systems and Management Sciences, Proceedings of the 3rd International Conference on Modelling Computation and Optimization in Information Systems and Management Sciences (MCO 2015), Metz, France, Part I, pp. 281–291. Springer, Cham, Heidelberg, New York, Dordrecht, London (2015)

    Google Scholar 

  19. Kalashnikov, V.V., Matis, T., Pérez-Valdés, G.A.: Time series analysis applied to construct US natural gas price functions for groups of states. Energy Econ. 32, 887–900 (2010)

    Article  Google Scholar 

  20. Kalashnikov, V.V., Pérez-Valdés, G.A., Kalashnykova, N.I.: A linearization approach to solve the natural gas cash-out bilevel problem. Ann. Oper. Res. 181, 423–442 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  21. Kalashnikov, V.V., Pérez-Valdés, G.A., Kalashnykova, N.I., Tomasgard, A.: Natural gas cash-out problem: bilevel stochastic optimization approach. Eur. J. Oper. Res. 206, 18–33 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  22. Kalashnikov, V.V., Ríos-Mercado, R.Z.: A natural gas cash-out problem: a bilevel programming framework and a penalty function method. Optim. Eng. 7, 403–420 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  23. Liu, C., Wang, Y.: A new evolutionary algorithm for multi-objective optimization problems. ICIC Express Lett. 1, 93–98 (2007)

    Google Scholar 

  24. Markowitz, H.: Portfolio selection. J. Financ. 7, 77–91 (1952)

    Google Scholar 

  25. Moore, J.T., Bard, J.F.: The mixed integer linear bilevel programming problem. Oper. Res. 38, 911–921 (1990)

    Article  MathSciNet  MATH  Google Scholar 

  26. Nishizaki, I., Sakawa, M., Kan, T.: Computational methods through genetic algorithms for obtaining Stackelberg solutions to two-level integer programming problems. Electron. Commun. Jpn. Part 3 86, 1251–1257 (2003)

    Google Scholar 

  27. Rockafellar, R.T.: Convex Analysis. Princeton University Press, Princeton (1970)

    Book  MATH  Google Scholar 

  28. Saharidis, G.K., Ierapetritou, M.G.: Resolution method for mixed integer bi-level linear problems based on decomposition techinque. J. Glob. Optim. 44, 29–51 (2009)

    Article  MATH  Google Scholar 

  29. Sahin, K.H., Ciric, A.R.: A dual temperature simulated annealing approach for solving bilevel programming problems. Comput. Chem. Eng. 23, 11–25 (1998)

    Article  Google Scholar 

  30. Sharpe, W.: Portfolio Theory and Capital Markets. McGrow Hill Book Company, New York (1970)

    Google Scholar 

  31. Sharpe, W., Alexander, G., Bailey, J.: Investments. Prentice Hall, England Cliffs (1999)

    Google Scholar 

  32. von Stackelberg, H.: Marktform und Gleichgewicht. Julius Springer, Vienna (1934.) English translation: The Theory of the Market Economy. Oxford University Press, Oxford (1952)

    Google Scholar 

  33. Wen, U.P., Yang, Y.H.: Algorithms for solving the mixed integer two level linear programming problem. Comput. Oper. Res. 17, 133–142 (1990)

    Article  MathSciNet  MATH  Google Scholar 

  34. Wendell, R.E.: A preview of a tolerance approach to sensitivity analysis in linear programming. Discrete Math. 38, 121–124 (1982)

    Article  MathSciNet  MATH  Google Scholar 

  35. Ye, J.J., Zhu, D.L.: Optimality conditions for bilevel programming problems. Optimization 33, 9–27 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  36. Zhang, R., Wu, C.: A decomposition-based optimization algorithm for scheduling large-scale job shops. Int. J. Innov. Comput. Inf. Control 5, 2769–2780 (2009)

    Google Scholar 

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Acknowledgment

The research activity of the first author was partially funded by the R&D Department of the Tecnológico de Monterrey (ITESM), Campus Monterrey, Mexico, and by the SEP-CONACYT grant CB-2013-01-221676 (Mexico), while the second and third authors were financially supported by the SEP-CONACYT grant FC-2016-01-1938 (Mexico).

Author information

Authors and Affiliations

  1. Tecnológico de Monterrey (ITESM), 64849, Monterrey, NL, Mexico

    Vyacheslav Kalashnikov

  2. Central Economics and Mathematics Institute (CEMI), Moscow, 117418, Russia

    Vyacheslav Kalashnikov

  3. Sumy State University, Sumy, 40007, Ukraine

    Vyacheslav Kalashnikov

  4. Universidad Autónoma de Nuevo León (UANL), 66455, San Nicolás de los Garza, NL, Mexico

    Nataliya Kalashnykova & José G. Flores-Muñiz

Authors
  1. Vyacheslav Kalashnikov
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  2. Nataliya Kalashnykova
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  3. José G. Flores-Muñiz
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Corresponding author

Correspondence to Vyacheslav Kalashnikov .

Editor information

Editors and Affiliations

  1. Department of Business, University of Barcelona, Barcelona, Spain

    Anna M. Gil-Lafuente

  2. Department of Management Control and Information Systems, University of Chile, Santiago, Chile

    José M. Merigó

  3. Department of Mathematics, University of Delhi, Delhi, India

    Bal Kishan Dass

  4. Department of Applied Sciences, Delhi Technical Campus, Greater Noida, Uttar Pradesh, India

    Rajkumar Verma

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Kalashnikov, V., Kalashnykova, N., Flores-Muñiz, J.G. (2018). Solution of the Portfolio Optimization Model as a Fuzzy Bilevel Programming Problem. In: Gil-Lafuente, A., Merigó, J., Dass, B., Verma, R. (eds) Applied Mathematics and Computational Intelligence. FIM 2015. Advances in Intelligent Systems and Computing, vol 730. Springer, Cham. https://doi.org/10.1007/978-3-319-75792-6_14

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  • DOI: https://doi.org/10.1007/978-3-319-75792-6_14

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  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-319-75791-9

  • Online ISBN: 978-3-319-75792-6

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