Abstract
This paper addresses one of the classes of bilevel optimization problems in their optimistic statement. The reduction of the bilevel problem to a series of nonconvex mathematical optimization problems, together with the specialized Global Search Theory, is used for developing methods of local and global searches to find optimistic solutions. Illustrative examples show that the approach proposed is prospective and performs well.
Keywords
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
References
Bard, J.F.: Convex two-level optimization. Math. Prog. 40, 15–27 (1988)
Bazara, M.S., Shetty, C.M.: Nonlinear Programming. Theory and Algorithms. Wiley, New York (1979)
Bonnans, J.-F., Gilbert, J.C., Lemarechal, C., Sagastizabal, C.A.: Numerical Optimization: Theoretical and Practical Aspects. Springer, Heidelberg (2006)
Calamai, P., Vicente, L.: Generating quadratic bilevel programming test problems. ACM Trans. Math. Softw. 20, 103–119 (1994)
Colson, B., Marcotte, P., Savard, G.: A trust-region method for nonlinear bilevel programming: algorithm and computational experience. Comput. Optim. Appl. 30, 211–227 (2005)
Colson, B., Marcotte, P., Savard, G.: An overview of bilevel optimization. Ann. Oper. Res. 153, 235–256 (2007)
Dempe, S.: Foundations of Bilevel Programming. Kluwer Academic Publishers, Dordrecht (2002)
Dempe, S.: Bilevel programming. In: Audet, C., Hansen, P., Savard, G. (eds.) Essays and Surveys in Global Optimization, pp. 165–193. Springer, Boston (2005)
Dempe, S., Kalashnikov, V.V., Perez-Valdes, G.A., Kalashnykova, N.: Bilevel Programming Problems: Theory, Algorithms and Applications to Energy Networks. Springer, Heidelberg (2015)
Etoa, J.B.E.: Solving quadratic convex bilevel programming problems using a smoothing method. Appl. Math. Comput. 217, 6680–6690 (2011)
Gruzdeva, T.V., Petrova, E.G.: Numerical solution of a linear bilevel problem. Comp. Math. Math. Phys. 50, 1631–1641 (2010)
Gumus, Z.H., Floudas, C.A.: Global optimization of nonlinear bilevel programming problems. J. Glob. Optim. 20, 1–31 (2001)
MATLAB—The language of technical computing. http://www.mathworks.com/products/matlab/
Muu, L.D., Quy, N.V.: A global optimization method for solving convex quadratic bilevel programming problems. J. Glob. Optim. 26, 199–219 (2003)
Orlov, A.V.: Numerical solution of bilinear programming problems. Comput. Math. Math. Phys. 48, 225–241 (2008)
Orlov, A.V., Strekalovsky, A.S.: Numerical search for equilibria in bimatrix games. Comput. Math. Math. Phys. 45, 947–960 (2005)
Pang, J.-S.: Three modeling paradigms in mathematical programming. Math. Prog. Ser. B. 125, 297–323 (2010)
Pistikopoulos, E.N., Dua, V., Ryu, J.-H.: Global optimization of bilevel programming problems via parametric programming. In: Floudas, C.A., Pardalos, P.M. (eds.) Frontiers in Global Optimization, pp. 457–476. Kluwer Academic Publishers, Dordrecht (2004)
Saboia, C.H., Campelo, M., Scheimberg, S.: A computational study of global algorithms for linear bilevel programming. Numer. Algorithms 35, 155–173 (2004)
Strekalovsky, A.S.: Elements of Nonconvex Optimization. Nauka, Novosibirsk (2003). [in Russian]
Strekalovsky, A.S.: On solving optimization problems with hidden nonconvex structures. In: Rassias, T.M., Floudas, C.A., Butenko, S. (eds.) Optimization in Science and Engineering, pp. 465–502. Springer, New York (2014). doi:10.1007/978-1-4939-0808-0_23
Strekalovsky, A.S., Orlov, A.V.: Bimatrix Games and Bilinear Programming. FizMatLit, Moscow (2007). [in Russian]
Strekalovsky, A.S., Orlov, A.V., Malyshev, A.V.: On computational search for optimistic solution in bilevel problems. J. Glob. Optim. 48, 159–172 (2010)
Acknowledgments
This work has been supported by the Russian Science Foundation (Project no. 15-11-20015).
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2017 Springer International Publishing AG
About this paper
Cite this paper
Orlov, A. (2017). A Nonconvex Optimization Approach to Quadratic Bilevel Problems. In: Battiti, R., Kvasov, D., Sergeyev, Y. (eds) Learning and Intelligent Optimization. LION 2017. Lecture Notes in Computer Science(), vol 10556. Springer, Cham. https://doi.org/10.1007/978-3-319-69404-7_16
Download citation
DOI: https://doi.org/10.1007/978-3-319-69404-7_16
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-69403-0
Online ISBN: 978-3-319-69404-7
eBook Packages: Computer ScienceComputer Science (R0)