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Deterministic solution approach for some classes of nonlinear multilevel programs with multiple followers

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Abstract

In this paper we investigate multilevel programming problems with multiple followers in each hierarchical decision level. It is known that such type of problems are highly non-convex and hard to solve. A solution algorithm have been proposed by reformulating the given multilevel program with multiple followers at each level that share common resources into its equivalent multilevel program having single follower at each decision level. Even though, the reformulated multilevel optimization problem may contain non-convex terms at the objective functions at each level of the decision hierarchy, we applied multi-parametric branch-and-bound algorithm to solve the resulting problem that has polyhedral constraints. The solution procedure is implemented and tested for a variety of illustrative examples.

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References

  1. Adjiman, S.C., Dallwing, S., Floudas, A.C., Neumaier, A.: A global optimization method, \(\alpha \)BB, for general twice-defferentiable constrained NLPs; I. Theoretical advances. Comput. Chem. Eng. 22, 1137–1158 (1998)

    Article  Google Scholar 

  2. Al-Khayyal, A.F.: Jointly constrained bilinear programms and related problems: an overview. Comput. Math. Appl. 19, 53–62 (1990)

    Article  MathSciNet  MATH  Google Scholar 

  3. Androulakis, P.I., Maranas, D.C., Floudas, A.C.: \(\alpha \)BB: a global optimization method for general constrained nonconvex problems. J. Glob. Optim. 7, 337–363 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  4. Bard, J.F.: Optimality conditions for the bilevel programming problem. Nav. Res. Logist. Q. 31, 13–26 (1984)

    Article  MathSciNet  MATH  Google Scholar 

  5. Bard, J.F.: Convex two-level optimization. Math. Program. 40, 15–27 (1988)

    Article  MathSciNet  MATH  Google Scholar 

  6. Bard, J.F.: Practical Bilevel Optimization, Algorithms and Applications. Kluwer Academic Publishers, Dordrecht (1998)

    Book  MATH  Google Scholar 

  7. Başar, T., Srikant, R.: A stackelberg network game with a large number of followers. J. Optim. Theory Appl. 115, 479–490 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  8. Blair, C.: The computational complexity of multi-level linear programs. Ann. Oper. Res. 34, 13–19 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  9. Calvete, H.I., Galé, C.: Linear bilevel multi-follower programming with independent followers. J. Glob. Optim. 39, 409–417 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  10. Cassidy, R.G., Kirby, M.J.L., Raike, W.M.: Efficient distribution of resources through three levels of government. Manag. Sci. 17, 462–473 (1971)

    Article  Google Scholar 

  11. Dempe, S.: A necessary and a sufficient optimality condition for bilevel programming problems. Optimization 25, 341–354 (1992)

    Article  MathSciNet  MATH  Google Scholar 

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

    MATH  Google Scholar 

  13. Dempe, S.: Annotated bibliography on bilevel programming and mathematical programs with equilibrium constraints. J. Math. Program. Oper. Res. 52, 333–359 (2003)

    MathSciNet  MATH  Google Scholar 

  14. Dempe, S., Dutta, J.: Is bilevel programming a special case of a mathematical program with complementarity constraints? Math. Program. 131, 37–48 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  15. Dua, V., Pistikopoulos, N.E.: An algorithm for the solution of multiparametric mixed integer linear programming problems. Ann. Oper. Res. 99, 123–139 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  16. Faísca, N.P., Dua, V., Rustem, B., Saraiva, M.P., Pistikopoulos, N.E.: Parametric global optimisation for bilevel programming. J. Glob. Optim. 38, 609–623 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  17. Faísca, N.P., Saraiva, P.M., Rustem, B., Pistikopoulos, N.E.: A multiparametric programming approach for multilevel hierarchical and decentralized optimization problems. Comput. Manag. Sci. 6, 377–397 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  18. Fiacco, A.V.: Sensitivity analysis for nonlinear programming using penalty methods. Math. Program. 10, 287–311 (1976)

    Article  MathSciNet  MATH  Google Scholar 

  19. Fiacco, A.V.: Introduction to Sensitivity and Stability Analysis in Nonlinear Programming. Acadamic Press, New York (1983)

    MATH  Google Scholar 

  20. Guignard, M.: Generalized Kuhn–Tucker conditions for mathematical programming problems in a Banach space. SIAM J. Control 7(2), 232–241 (1969)

    Article  MathSciNet  MATH  Google Scholar 

  21. Gümüs, Z.H., Floudas, C.A.: Global optimization of nonlinear bilevel programming problems. J. Glob. Optim. 20, 1–31 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  22. Han, J., Lu, J., Hu, Y., Zhang, G.: Tri-level decision-making with multiple followers: model, algorithm and case study. Inf. Sci. 311, 182–204 (2015)

    Article  Google Scholar 

  23. Hansen, P., Jaumard, B., Savard, G.: New branch-and-bound rules for linear bilevel programming. SIAM J. Sci. Stat. Comput. 13, 1194–1217 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  24. Herskovits, J., Filho, M.T., Leontiev, A.: An interior point technique for solving bilevel programming problems. Optim. Eng. 14, 381–394 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  25. Kassa, A.M., Kassa, S.M.: A multi-parametric programming algorithm for special classes of non-convex multilevel optimization problems. Int. J. Optim. Control Theor. Appl. 3(2), 133–144 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  26. Kassa, A.M., Kassa, S.M.: Approximate solution algorithm for multi-parametric non-convex programming problems with polyhedral constraints, in press. Int. J. Optim. Control Theor. Appl. 4(2), 89–98 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  27. Kassa, A.M., Kassa, S.M.: A branch-and-bound multi-parametric programming approach for general non-convex multilevel optimization with polyhedral constraints. J. Glob. Optim. 64, 745–764 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  28. Li, P., Lin, G.: Solving a class of generalized Nash equilibrium problems. J. Math. Res. Appl. 33, 372–378 (2013)

    MathSciNet  MATH  Google Scholar 

  29. Liu, B.: Stackelberg-Nash equilibrium for multilevel programming with multiple followers using genetic algorithms. Comput. Math. Appl. 36, 79–89 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  30. Lv, Y.: An exact penalty function method for solving a class of nonlinear bilevel programs. J. Appl. Math. Inform. 29, 1533–1539 (2011)

    MathSciNet  MATH  Google Scholar 

  31. Mersha, A.G., Dempe, S.: Feasible direction method for bilevel programming problem. Optimization 61, 597–616 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  32. Pistikopoulos, N.E., Georgiadis, M.C., Dua, V. (eds.): Multiparametric Programming: Theory, Algorithm and Application. WILEY-VCH Verlag GmbH and Co. KGaA, Weinheim (2007)

    Google Scholar 

  33. Tilahun, S.L., Kassa, S.M., Ong, H.C.: A new algorithm for multilevel optimization problems using evolutionary strategy, inspired by natural selection. In: Anthony, A., Ishizuka, M., Lukose, D. (eds.) PRICAI 2012, LNAI 7458, pp. 577–588. Springer, Berlin (2012)

    Google Scholar 

  34. Vicente, N.L., Calamai, H.P.: Bilevel and multilevel programming: a bibliography review. J. Glob. Optim. 5, 1–9 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  35. Wang, S.Y., Wang, Q., Romano-Rodriquez, S.: Optimality conditions and an algorithm for linear-quadratic programs. Optimization 31, 127–139 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  36. Wang, Q., Yang, F., Liu, Y.: Bilevel programs with multiple followers. Syst. Sci. Math. Sci. 13, 265–276 (2000)

    MathSciNet  MATH  Google Scholar 

  37. Wang, Y., Jiao, Y., Li, H.: An evolutionary algorithm for solving nonlinear bilevel programming based on a new constraint-handling scheme. IEEE 35, 221–232 (2005)

    Google Scholar 

  38. Zhang, G., Lu, J., Dillon, T.: An approximation branch-and-bound algorithm for fuzzy bilevel decision making problems. In: Proceedings of the International Multiconference on Computer Science and Information Technology, PIPS, pp. 223–231 (2006)

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Acknowledgements

This work is partially supported by the International Science Program (ISP) of Sweden, a research project at the Department of Mathematics, Addis Ababa University. The authors would like to thank an anonymous referee who indicated to the authors valuable comments and suggestions to improve the earlier version of the manuscript.

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Correspondence to Semu Mitiku Kassa.

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Kassa, A.M., Kassa, S.M. Deterministic solution approach for some classes of nonlinear multilevel programs with multiple followers. J Glob Optim 68, 729–747 (2017). https://doi.org/10.1007/s10898-017-0502-4

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