A branch-and-bound multi-parametric programming approach for non-convex multilevel optimization with polyhedral constraints
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In this paper we develop a general but smooth global optimization strategy for nonlinear multilevel programming problems with polyhedral constraints. At each decision level successive convex relaxations are applied over the non-convex terms in combination with a multi-parametric programming approach. The proposed algorithm reaches the approximate global optimum in a finite number of steps through the successive subdivision of the optimization variables that contribute to the non-convexity of the problem and partitioning of the parameter space. The method is implemented and tested for a variety of bilevel, trilevel and fifth level problems which have non-convexity formulation at their inner levels.
KeywordsMultilevel optimization Multi-parametric programming Convex relaxation
Mathematics Subject Classification90C26 90C31 91A10 90C99 65K05
This work is in part supported by the Swedish International Science Program (ISP), through the project at the Department of Mathematics, Addis Ababa University. The authors also would like to thank the anonymous referees from whom we received valuable comments and suggestions to improve the earlier version of the manuscript.
- 4.Bialas, W.F., Karwan, M.H.: Multilevel Otimization: A Mathematical Programming Perspective. M.Sc. thesis, State University of New York (1980)Google Scholar
- 6.Dua, V., Bozinis, N.A., Pistikopoulos, N.E.: A multiparametric programming approach for mixed-integer quadratic engineering problem. Comput. Chem. Eng. 26(45), 715733 (2002)Google Scholar
- 14.Kassa, A.M., Kassa, S.M.: Approximate solution algorithm for multi-parametric non-convex programming problems with polyhedral constraints. Int. J. Optim. Control Theor. Appl. 4(2), 89–98 (2014)Google Scholar
- 15.Lakie, E.: Linear Three Level Programming Problem with the Application to Hierarchical Organizations. M.Sc. thesis, Department of mathematics, Addis Ababa University (2007)Google Scholar
- 17.Migdalas, A., Värbrand, P.: Multilevel Optimization: Algorithm, Theory and Applications. Kluwer, Dordrecht (1992)Google Scholar
- 18.Pistikopoulos, N.E., Georgiadis, M.C., Dua, V. (eds.): Multiparametric Programming: Theory, Algorithm and Applications. Wiley-VCH, Weinheim (2007)Google Scholar
- 21.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, vol. 7458, pp. 577–588. Springer, Berlin (2012)Google Scholar