Solution method for monotone mixed variational inequalities

I. V. Konnov   ; O. V. Pinyagina

Učënye zapiski Kazanskogo universiteta. Seriâ Fiziko-matematičeskie nauki, Uchenye Zapiski Kazanskogo Universiteta. Seriya Fiziko-Matematicheskie Nauki, Tome 153 (2011) no. 1, pp. 221-230

Résumé

In this article, we propose a method, which combines the techniques of regularization and descent over a gap (merit) function, for solving a monotone mixed variational inequality. The same uniformly convex auxiliary function is used for the construction of both regularized problems and gap functions. To solve the regularized problems, we apply the method of descent over a gap function with inexact line search.

Détail

Keywords: mixed variational inequality, gap function, descent method, uniformly convex function.

Bibliographie

[1] Patriksson M., Nonlinear programming and variational inequality problems: a unified approach, Kluwer Acad. Publ., Dordrecht, The Netherlands, 1999, 348 pp. | MR | Zbl

[2] Konnov I. V., Pinyagina O. V., “$D$-gap functions for a class of equilibrium problems in Banach spaces”, Comput. Methods Appl. Math., 3 (2003), 274–286 | DOI | MR | Zbl

[3] Tikhonov A. N., “O reshenii nekorrektno postavlennykh zadach i metode regulyarizatsii”, Dokl. AN SSSR, 151:3 (1963), 501–504 | MR | Zbl

[4] Tikhonov A. N., Arsenin V. Ya., Metody resheniya nekorrektnykh zadach, Nauka, M., 1974, 224 pp. | MR | Zbl

[5] Browder F. E., “Existence and approximation of solutions of nonlinear variational inequalities”, Proc. Natl. Acad. Sci. USA, 56 (1966), 1080–1086 | DOI | MR | Zbl

[6] Konnov I. V., Kum S., “Descent methods for mixed variational inequalities in a Hilbert space”, Nonlinear Anal. Theory Meth. Appl., 47 (2001), 561–572 | DOI | MR | Zbl

[7] Konnov I. V., Kum S., Lee G. M., “On convergence of descent methods for variational inequalities in a Hilbert space”, Math. Meth. Oper. Res., 55 (2002), 371–382 | DOI | MR | Zbl

[8] Konnov I. V., Pinyagina O. V., “$D$-gap functions and descent methods for a class of monotone equilibrium problems”, Lobachevskii J. Math., 13 (2003), 57–65 | MR | Zbl

[9] Kaplan A., Tichatschke R., “Auxiliary problem principle and the approximation of variational inequalities with non-symmetric multi-valued operators”, Constructive, experimental, and nonlinear analysis, CRC Math. Model. Ser., 27, 2000, 185–209 | MR | Zbl

[10] Pinyagina O. V., Ali M. S. S., “Descent method for monotone mixed variational inequalities”, Calcolo, 45 (2008), 1–15 | DOI | MR | Zbl

[11] Baiokki K., Kapelo A., Variatsionnye i kvazivariatsionnye neravenstva. Prilozheniya k zadacham so svobodnoi granitsei, Nauka, M., 1988, 448 pp. | MR

[12] Blum E., Oettli W., “From optimization and variational inequalities to equilibrium problems”, The Math. Student, 63 (1994), 123–145 | MR | Zbl

[13] Konnov I. V., Combined relaxation methods for variational inequalities, Springer-Verlag, Berlin, 2001, 181 pp. | MR

[14] Vasilev F. P., Chislennye metody resheniya ekstremalnykh zadach, Nauka, M., 1988, 549 pp. | MR

[15] Chadli O., Konnov I. V., Yao J. C., “Descent method for equilibrium problems in a Banach space”, Comp. Math. Appl., 48 (2004), 609–616 | DOI | MR | Zbl

[16] Bakushinskii A. B., Goncharskii A. V., Iteratsionnye metody resheniya nekorrektnykh zadach, Nauka, M., 1989, 128 pp. | MR

[17] Panagiotopoulos P. D., Inequality problems in mechanics and applications, Birkhauser, Boston, 1985 | MR | Zbl

[18] Konnov I. V., “Iterative solution methods for mixed equilibrium problems and variational inequalities with non-smooth functions”, Game Theory: Strategies, Equilibria, and Theorems, Chapter 4, eds. I. N. Haugen, A. S. Nilsen, NOVA, Hauppauge, 2008, 117–160

[19] Konnov I. V., “Metod spuska s netochnym lineinym poiskom dlya smeshannykh variatsionnykh neravenstv”, Izv. vuzov. Matem., 2009, no. 8, 37–44 | MR | Zbl

[20] Demyanov V. F., Rubinov A. I., Osnovy negladkogo analiza i kvazidifferentsialnoe ischislenie, Nauka, M., 1990, 432 pp. | MR