Geodesic
An extension of the Jacobi algorithm for the complementarity problem in the presence of multivalence
Žurnal vyčislitelʹnoj matematiki i matematičeskoj fiziki, Tome 45 (2005) no. 7, pp. 1167-1173 
I. V. Konnov. An extension of the Jacobi algorithm for the complementarity problem in the presence of multivalence. Žurnal vyčislitelʹnoj matematiki i matematičeskoj fiziki, Tome 45 (2005) no. 7, pp. 1167-1173. http://geodesic.mathdoc.fr/item/ZVMMF_2005_45_7_a2/
@article{ZVMMF_2005_45_7_a2,
     author = {I. V. Konnov},
     title = {An extension of the {Jacobi} algorithm for the complementarity problem in the presence of multivalence},
     journal = {\v{Z}urnal vy\v{c}islitelʹnoj matematiki i matemati\v{c}eskoj fiziki},
     pages = {1167--1173},
     year = {2005},
     volume = {45},
     number = {7},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/ZVMMF_2005_45_7_a2/}
}
TY  - JOUR
AU  - I. V. Konnov
TI  - An extension of the Jacobi algorithm for the complementarity problem in the presence of multivalence
JO  - Žurnal vyčislitelʹnoj matematiki i matematičeskoj fiziki
PY  - 2005
SP  - 1167
EP  - 1173
VL  - 45
IS  - 7
UR  - http://geodesic.mathdoc.fr/item/ZVMMF_2005_45_7_a2/
LA  - ru
ID  - ZVMMF_2005_45_7_a2
ER  - 
%0 Journal Article
%A I. V. Konnov
%T An extension of the Jacobi algorithm for the complementarity problem in the presence of multivalence
%J Žurnal vyčislitelʹnoj matematiki i matematičeskoj fiziki
%D 2005
%P 1167-1173
%V 45
%N 7
%U http://geodesic.mathdoc.fr/item/ZVMMF_2005_45_7_a2/
%G ru
%F ZVMMF_2005_45_7_a2

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

The complementarity problem is examined in the case where the basic mapping is the sum of a finite number of superpositions of a univalent off-diagonal antitone mapping and a multivalent diagonal monotone one. An extension is proposed for the Jacobi algorithm, which constructs a sequence converging to a point solution. With the use of this property, the existence of a solution to the original problem is also established. Under certain additional conditions, the minimal element in the feasible set of this problem is one of its solutions.

[1] Berschanskii Ya. M., Meerov M. V., “Teoriya i metody resheniya zadach dopolnitelnosti”, Avtomatika i telemekhan., 1983, no. 6, 5–31

[2] Cottle R. W., Pang J. S., Stone R. E., The linear complementarity problem, Academic Press, Boston, 1992 | MR | Zbl

[3] Ferris M. C., Pang J.-S., “Engineering and economic applications of complementarity problems”, SIAM Review, 39 (1997), 669–713 | DOI | MR | Zbl

[4] Isac G., Complementarity problems, Springer, Berlin, 1992 | MR | Zbl

[5] Facchinei F., Pang J.-S., Finite-dimensional variational inequalities and complementarity problems, v. 1, 2, Springer, Berlin, 2003

[6] Lapin A., “Geometric convergence of iterative methods for variational inequalities with $M$-matrices and diagonal monotone operators”, Numerical methods for continuous casting and related problems, DAS, Kazan, 2001, 63–72 | MR | Zbl

[7] More J. J., “Classes of functions and feasibility conditions in nonlinear complementarity problems”, Math. Programming, 6:3 (1974), 327–338 | DOI | MR | Zbl

[8] Tamir A., “Minimality and complementarity problems associated with $Z$-functions and $M$-functions”, Math. Programming, 7:1 (1974), 17–31 | DOI | MR | Zbl

[9] Polterovich V. M., Spivak V. A., “Otobrazheniya s valovoi zamenimostyu v teorii ekonomicheskogo ravnovesiya”, Itogi nauki i tekhn. Sovremennye problemy matem., 19, VINITI, M., 1992, 111–154 | MR

[10] Nikaido Kh., Vypuklye struktury i matematicheskaya ekonomika, Mir, M., 1972

[11] Ershov E. B., Teoriya klyuvov i mezhotraslevoe modelirovanie, Preprint WP2/2002/03, Gos. un-t-Vyssh. shkola ekonomiki, M., 2002