V. V. Kalashnikov, N. I. Kalashnykova, M. A. LEAL-CORONADO


In this paper, we consider a mixed-integer bi-level linear programming (or a leader’s) problem with one parameter in the right-hand side of the constraints in the lower level (or a follower’s) problem. Motivated by the application to a fuzzy portfolio optimization model, we consider a particular case that consists in maximizing the investor’s expected return. The functions are linear at the upper level and quadratic at the lower level, 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 branch-and-bound structure, we apply a new branch-and-bound technique to process the integrity condition.

Ключові слова

fuzzy portfolio optimization, integer programming, parametric programming, branch-and-bound approach.


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

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

Kalashnikov, V.V., Kalashnykova, N.I., and Castillo-Pérez, F.J. (2015) Finding equilibrium in a financial model by solving a variational inequality problem. – In: Hoai An Le Thi 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, 2015, pp. 281–291; Springer: Cham/Heidelberg/New York/Dordrecht/London.

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

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

S. Dempe, V. V. Kalashnikov and R. Z. Ríos-Mercado, Discrete bilevel programming: Application to a natural gas cash-out problem, European Journal of Operational Research, vol. 166, no. 2, pp. 469-488, 2005.

V. V. Kalashnikov and R. Z. Ros-Mercado, A natural gas cash-out problem: A bilevel programming framework and a penalty function method, Optimization and Engineering, vol. 7, pp. 403-420, 2006.

V. V. Kalashnikov, T. Matis and G. A. Pérez-Valdés, Time series analysis applied to construct US natural gas price functions for groups of states, to appear in Energy Economics, ISSN 0140-9883, doi:10.1016/j.eneco.2009.11.006. - 15 p.

V. V. Kalashnikov, G. A. Pérez-Valdés, N. I. Kalashnykova and A. Tomasgard, Natural gas cash-out problem: Bilevel stochastic optimization approach, to appear in European J. Operational Research, ISSN 0377-2217, doi: 10.1016/j.ejor.2010.02.018. - 39 p.

V. V. Kalashnikov, G. A. Pérez-Valdés and N. I. Kalashnykova, A linearization approach to solve the natural gas cash-out bilevel problem, to appear in Annals of Operations Research, 2010. - 20 p.

Z. H. Gümü , C. A. Floudas, Global optimization of mixed-integer-bilevel programming problems, Computational Management Science, vol. 2, pp. 181-212, 2005.

J. T. Moore and J. F. Bard, The mixed integer linear bilevel programming problem, Operations Research, vol. 38, pp. 911-921, 1990.

I. Nishizaki, M. Sakawa and T. Kan, Computational methods through genetic algorithms for obtaining Stackelberg solutions to two-level integer programming problems, Electronics and Communications in Japan, Part 3, vol. 86, pp. 1251-1257, 2003.

U. P. Wen and Y. H. Yang, Algorithms for solving the mixed integer two level linear programming problem, Computers and Operations Research, vol. 17, no. 2, pp. 133-142, 1990.

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

S. Dempe, A simple algorithm for the linear bilevel programming problem, Optimization, vol. 18, no. 3, pp. 373-385, 1987.

J. Bard, An algorithm for solving the general bilevel programming problem, Mathematics of Operations Research, vol. 8, no. 2, pp. 260-282, 1983.

G. K. Saharidis and M. G. Ierapetritou, Resolution method for mixed integer bi-level linear problems based on decomposition techinque, J. Global Optimization, vol. 44, pp. 29-51, 2009.

R. Zhang and C. Wu, A decomposition-based optimization algorithm for scheduling large-scale job shops, International Journal of Innovative Computing, Information and Control, vol. 5, no. 9, pp. 2769-2780, 2009.

R. H. Jan and M. S. Chern, Non-linear integer bilevel programming, European J. Operational Research, vol. 72, pp. 574-587, 1994.

K. H. Sahin and A. R. Ciric, A dual temperature simulated annealing approach for solving bilevel programming problems, Computers and Chemical Engineering, vol. 23, pp. 11-25, 1998.

N. P. Fasca, V. Dua, B. Rustem, P. M. Saraiva and E. N. Pistikopoulos, Parametric global optimization for bilevel programming, J. Global Optimization, vol. 38, pp. 609-623, 2007.

C. A. Floudas, Z. H. Gümü , M. G. Ierapetritou, Global optimization in design under uncertainty: Feasibility test and flexibility index problem, Industrial and Engineering Chemical Research, vol. 40, pp. 4267-4282, 2001.

R.E. Wendell, A preview of a tolerance approach to sensitivity analysis in linear programming, Discrete Mathematics, vol. 38, no. 1, pp. 121-124, 1982.

S. Dempe and V. V. Kalashnikov, Discrete Bilevel Programming with Linear Lower Level Problems, Preprint, TU Bergakademie Freiberg, 2005.

S. Dempe, V. V. Kalashnikov, N. I. Kalashnykova and A. Arévalo Franco, A new approach to solving bi-level programming problems with integer upper level variables, ICIC Express Letters, vol. 3, no. 4 (B), pp. 1281-1286, 2009.

C. Liu and Y. Wang, A new evolutionary algorithm for multi-objective optimization problems, ICIC Express Letters, vol. 1, no. 1, pp. 93-98, 2007.

X. P. Hu, Y. X. Li, J. W. Guo, L. J. Sun and A. Z. Zeng, A simulation optimization algorithms with heuristic transformation and its application to vehicle routing problems, International Journal of Innovative Computing, Information and Control, vol. 4, no. 5, pp. 1169-1182, 2008.

J. J. Ye and D. L. Zhu, Optimality conditions for bilevel programming problems, Optimization, vol. 33, no. 1, pp. 9-27, 1995.

L. Grygarová, Qualitative Untersuchung des I. Optimierungsproblems in mehrparametrischer Programmierung, Applications of Mathematics, vol. 15, no. 4, pp. 276-295, 1970.

S. Dempe and H. Schreier, Operations Research – Deterministische Modelle und Methoden, Teubner Verlag, Wiesbaden, 2006.

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

R. T. Rockafellar, Convex Analysis, Princeton University Press, Princeton, New Jersey, 1970.

D. Y. Gao, Canonical duality theory and solutions to constrained nonconvex quadratic programming, Journal of Global Optimization, vol. 29, pp. 377-399, 2004.

D. Y. Gao, Solutions and optimality criteria to box constrained nonconvex minimization problems, Journal of Industry and Management Optimization, vol. 3, no. 2, pp. 293-304, 2007.

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