×

zbMATH — the first resource for mathematics

Iterative methods for nonlinear complementarity problems on isotone projection cones. (English) Zbl 1162.47050
Author’s abstract: We present a recursion related to a nonlinear complementarity problem defined by a closed convex cone in a Hilbert space and a continuous mapping defined on the cone. If the recursion is convergent, then its limit is a solution of the nonlinear complementarity problem. In the case of isotone projection cones, sufficient conditions are given for the mapping, so that the recursion is convergent.

MSC:
47J25 Iterative procedures involving nonlinear operators
47H10 Fixed-point theorems
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Isac, G.; Németh, A.B., Isotone projection cones in Hilbert spaces and the complementarity problem, Boll. unione mat. ital. B, 7, 4, 773-802, (1990) · Zbl 0719.46011
[2] Isac, G.; Németh, A.B., Projection methods, isotone projection cones, and the complementarity problem, J. math. anal. appl., 153, 1, 258-275, (1990) · Zbl 0711.47030
[3] Auslender, A., Optimization Méthodes numériques, (1976), Masson Paris · Zbl 0326.90057
[4] Bertsekas, D.P.; Tsitsiklis, J.N., Parallel and distributed computation: numerical methods, (1989), Prentice-Hall, Inc. Englewood Cliffs, NJ · Zbl 0743.65107
[5] Iusem, A.N.; Svaiter, B.F., A variant of Korpelevich’s method for variational inequalities with a new search strategy, Optimization, 42, 4, 309-321, (1997) · Zbl 0891.90135
[6] Khobotov, E.N., A modification of the extragradient method for solving variational inequalities and some optimization problems, Zh. vychisl. mat. mat. fiz., 27, 10, 1462-1473, (1987) · Zbl 0665.90078
[7] Korpelevich, G.M., The extragradient method for finding saddle points and other problems, Matekon, 12, 747-756, (1976) · Zbl 0342.90044
[8] Marcotte, P., Application of Khobotov’s algorithm to variational inequalities and network equilibrium problems, INFOR inf. syst. oper. res., 29, 258-270, (1991) · Zbl 0781.90086
[9] Nagurney, A., Network economics—A variational inequality approach, (1993), Kluwer Acad. Publ. Dordrecht, The Netherlands · Zbl 0873.90015
[10] Sibony, M., Méthodes itératives pour LES équations et inéquations aux dérives partielles non linéaires de type monotone, Calcolo, 7, 65-183, (1970) · Zbl 0225.35010
[11] Solodov, M.V.; Svaiter, B.F., A new projection method for variational inequality problems, SIAM J. control optim., 37, 3, 765-776, (1999) · Zbl 0959.49007
[12] Solodov, M.V.; Tseng, P., Modified projection-type methods for monotone variational inequalities, SIAM J. control optim., 34, 5, 1814-1830, (1996) · Zbl 0866.49018
[13] Sun, D., A class of iterative methods for nonlinear projection equations, J. optim. theory appl., 91, 1, 123-140, (1996) · Zbl 0871.90091
[14] Kachurovskii, R., On monotone operators and convex functionals, Uspekhi mat. nauk, 15, 4, 213-215, (1960)
[15] Minty, G., Monotone operators in Hilbert spaces, Duke math. J., 29, 341-346, (1962) · Zbl 0111.31202
[16] Minty, G., On a “monotonicity” method for the solution of non-linear equations in Banach spaces, Proc. natl. acad. sci. USA, 50, 1038-1041, (1963) · Zbl 0124.07303
[17] Browder, F.E., Continuity properties of monotone non-linear operators in Banach spaces, Bull. amer. math. soc., 70, 551-553, (1964) · Zbl 0123.10702
[18] Isac, G.; Németh, A.B., Monotonicity of metric projections onto positive cones of ordered Euclidean spaces, Arch. math., 46, 6, 568-576, (1986) · Zbl 0574.41037
[19] Isac, G.; Németh, A.B., Every generating isotone projection cone is latticial and correct, J. math. anal. appl., 147, 1, 53-62, (1990) · Zbl 0725.46002
[20] Isac, G.; Németh, A.B., Isotone projection cones in Euclidean spaces, Ann. sci. math. Québec, 16, 1, 35-52, (1992) · Zbl 0760.52003
[21] Borwein, J.M.; Wolkowicz, H., Regularizing the abstract convex program, J. math. anal. appl., 83, 2, 495-530, (1981) · Zbl 0467.90076
[22] Karamardian, S.; Schaible, S., Seven kinds of monotone maps, J. optim. theory appl., 66, 1, 37-46, (1990) · Zbl 0679.90055
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.