Geodesic
Numerical solution of a~class of bilevel programming problems
Sibirskij žurnal vyčislitelʹnoj matematiki, Tome 13 (2010) no. 2, pp. 201-212.

Voir la notice de l'article provenant de la source Math-Net.Ru

The quadratic-linear bilevel programming problem is considered. Its optimistic statement is reduced to a series of non-convex mathematical programming problems. An approximate algorithm of the global search in the problems obtained is proposed. Numerical solutions of randomly generated test problems are given and analyzed. 
@article{SJVM_2010_13_2_a4,
     author = {A. S. Strekalovsky and A. V. Orlov and A. V. Malyshev},
     title = {Numerical solution of a~class of bilevel programming problems},
     journal = {Sibirskij \v{z}urnal vy\v{c}islitelʹnoj matematiki},
     pages = {201--212},
     publisher = {mathdoc},
     volume = {13},
     number = {2},
     year = {2010},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/SJVM_2010_13_2_a4/}
}
TY  - JOUR
AU  - A. S. Strekalovsky
AU  - A. V. Orlov
AU  - A. V. Malyshev
TI  - Numerical solution of a~class of bilevel programming problems
JO  - Sibirskij žurnal vyčislitelʹnoj matematiki
PY  - 2010
SP  - 201
EP  - 212
VL  - 13
IS  - 2
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/SJVM_2010_13_2_a4/
LA  - ru
ID  - SJVM_2010_13_2_a4
ER  - 
%0 Journal Article
%A A. S. Strekalovsky
%A A. V. Orlov
%A A. V. Malyshev
%T Numerical solution of a~class of bilevel programming problems
%J Sibirskij žurnal vyčislitelʹnoj matematiki
%D 2010
%P 201-212
%V 13
%N 2
%I mathdoc
%U http://geodesic.mathdoc.fr/item/SJVM_2010_13_2_a4/
%G ru
%F SJVM_2010_13_2_a4
A. S. Strekalovsky; A. V. Orlov; A. V. Malyshev. Numerical solution of a~class of bilevel programming problems. Sibirskij žurnal vyčislitelʹnoj matematiki, Tome 13 (2010) no. 2, pp. 201-212. http://geodesic.mathdoc.fr/item/SJVM_2010_13_2_a4/

[1] Germeier Yu. B., Igry s neprotivopolozhnymi interesami, Nauka, M., 1976 | MR | Zbl

[2] Gorelik V. A., Kononenko A. F., Teoretiko-igrovye modeli prinyatiya reshenii v ekologo-ekonomicheskikh sistemakh, Radio i svyaz, M., 1982 | MR | Zbl

[3] Vasin A. A., Morozov V. V., Teoriya igr i modeli matematicheskoi ekonomiki, MAKS Press, M., 2005

[4] Bard J. F., Practical Bilevel Optimization, Kluwer Academic Publishers, Dordrecht, The Netherlands, 1998 | MR

[5] Dempe S., Foundations of Bilevel Programming, Kluwer Academic Publishers, Dordrecht, The Netherlands, 2002 | MR | Zbl

[6] Colson B., Marcotte P., Savard G., “An overview of bilevel optimization”, Annals of operations research, 153:1 (2007), 235–256 | DOI | MR | Zbl

[7] Saboia C. H., Campelo M., Scheimberg S., “A computational study of global algorithms for linear bilevel programming”, Numerical Algorithms, 35:2–4 (2004), 155–173 | DOI | MR | Zbl

[8] Audet C., Savard G., Zghal W., “New branch-and-cut algorithm for bilevel linear programming”, J. of Optimization Theory and Applications, 134:2 (2007), 53–370 | DOI | MR

[9] Muu L. D., Quy N. V., “A global optimization method for solving convex quadratic bilevel programming problems”, J. of Global Optimization, 26:2 (2003), 199–219 | DOI | MR | Zbl

[10] Li H., Wang Y., “A hybrid genetic algorithm for solving a class of nonlinear bilevel programming problems”, Proc. of 6th International Conference Simulated Evolution and Learning (Hetei, China, October 15–18, 2006), eds. Wang T. D., Li X. et al., Springer, 408–415

[11] Bard J. F., “Convex two-level optimization”, Mathematical Programming, 40:1 (1988), 15–27 | DOI | MR | Zbl

[12] Gumus Z. H., Floudas C. A., “Global optimization of nonlinear bilevel programming problems”, J. of Global Optimization, 20:1 (2001), 1–31 | DOI | MR

[13] Colson B., Marcotte P., Savard G., “A trust-region method for nonlinear bilevel programming: algorithm and computational experience”, Computational Optimization and Applications, 30:3 (2005), 211–227 | DOI | MR | Zbl

[14] Strekalovskii A. S., Orlov A. V., Malyshev A. V., “Lokalnyi poisk v kvadratichno-lineinoi zadache dvukhurovnevogo programmirovaniya”, Sib. zhurn. vychisl. matem., 13:1 (2010), 75–88

[15] Vasilev F. P., Metody optimizatsii, Faktorial-press, M., 2002

[16] Bazara M., Shetti K., Nelineinoe programmirovanie. Teoriya i algoritmy, Mir, M., 1982 | MR | Zbl

[17] Strekalovskii A. S., Elementy nevypukloi optimizatsii, Nauka, Novosibirsk, 2003

[18] Strekalovskii A. C., “O minimizatsii raznosti dvukh vypuklykh funktsii na dopustimom mnozhestve”, Zhurn. vychisl. matem. i mat. fiziki, 43:3 (2003), 399–409 | MR | Zbl

[19] Strekalovskii A. S., Orlov A. V., Bimatrichnye igry i bilineinoe programmirovanie, Fizmatlit, M., 2007

[20] Strekalovsky A. S., Orlov A. V., “A new approach to nonconvex optimization”, Vychislitelnye metody i programmirovanie, 8:2 (2007), 11–27

[21] Orlov A. V., Strekalovskii A. S., “O chislennom poiske situatsii ravnovesiya v bimatrichnykh igrakh”, Zhurn. vychisl. matem. i mat. fiziki, 45:6 (2005), 983–997 | MR | Zbl

[22] Vasilev I. L., Klimentova K. B., Orlov A. V., “Parallelnyi globalnyi poisk ravnovesnykh situatsii v bimatrichnykh igrakh”, Vychislitelnye metody i programmirovanie, 8:2 (2007), 84–94

[23] Orlov A. V., “Chislennoe reshenie zadach bilineinogo programmirovaniya”, Zhurn. vychisl. matem. i mat. fiziki, 48:2 (2008), 237–254 | MR | Zbl

[24] Calamai P., Vicente L., “Generating linear and linear-quadratic bilevel programming problems”, SIAM J. on Scientific Computing, 14:4 (1993), 770–782 | DOI | MR | Zbl

[25] Calamai P., Vicente L., “Generating quadratic bilevel programming test problems”, ACM Transactions on Mathematical Software, 20 (1994), 103–119 | DOI | MR | Zbl