×

New extragradient-type methods for general variational inequalities. (English) Zbl 1033.49015

In this paper, the author suggested and analyzed extragradient-type methods for solving general variational inequalities by using the projection technique and the Wiener-Hopf equations technique, respectively. As applications, some results concerned with a class of quasi-variational inequalities are derived.

MSC:

49J40 Variational inequalities
PDF BibTeX XML Cite
Full Text: DOI

References:

[1] Bertsekas, D.P.; Tsitsiklis, J., Parallel and distributed computation: numerical methods, (1989), Prentice-Hall Englewood Cliffs · Zbl 0743.65107
[2] Giannessi, F.; Maugeri, A., Variational inequalities and network equilibrium problems, (1995), Plenum Press New York · Zbl 0834.00044
[3] He, B.S., A class of projection and contraction methods for variational inequalities, Appl. math. optim., 35, 69-76, (1997) · Zbl 0865.90119
[4] He, B.S.; Liao, L.Z., Improvement of some projection methods for monotone nonlinear variational inequalities, J. optim. theory appl., 112, 111-128, (2002) · Zbl 1025.65036
[5] Iusem, A.N.; Svaiter, B.F., A variant of Korpelech’s method for variational inequalities with a new strategy, Optimization, 42, 309-321, (1997) · Zbl 0891.90135
[6] Lions, J.L.; Stampacchia, G., Variational inequalities, Comm. pure appl. math., 20, 493-512, (1967) · Zbl 0152.34601
[7] Noor, M.A., General variational inequalities, Appl. math. lett., 1, 119-121, (1988)
[8] Noor, M.A., An extragradient method for general monotone variational inequalities, Adv. nonlinear var. inequal., 2, 25-31, (1999) · Zbl 1007.49507
[9] Noor, M.A., Wiener – hopf equations and variational inequalities, J. optim. theory appl., 79, 197-206, (1993) · Zbl 0799.49010
[10] Noor, M.A., Some iterative techniques for general variational inequalities, Optimization, 46, 391-401, (1999) · Zbl 0966.49010
[11] Noor, M.A., Generalized wiener – hopf equations and nonlinear quasi variational inequalities, Panamer. math. J., 2, 51-70, (1992) · Zbl 0842.49011
[12] Noor, M.A., Wiener – hopf equations techniques for variational inequalities, Korean J. comput. appl. math., 7, 581-599, (2000) · Zbl 0978.49011
[13] Noor, M.A., Some recent advances in variational inequalities, part I, basic concepts, New Zealand J. math., 26, 53-80, (1997) · Zbl 0886.49004
[14] Noor, M.A., Some recent advances in variational inequalities, part II, other concepts, New Zealand J. math., 26, 229-255, (1997) · Zbl 0889.49006
[15] Noor, M.A., Generalized quasi variational inequalities and implicit wiener – hopf equations, Optimization, 45, 197-222, (1999) · Zbl 0939.49009
[16] Noor, M.A., New approximation schemes for general variational inequalities, J. math. anal. appl., 251, 217-229, (2000) · Zbl 0964.49007
[17] Noor, M.A., Extragradient method for pseudomonotone variational inequalities, J. optim. theory appl., 117, (2003) · Zbl 1049.49009
[18] M.A. Noor, Implicit dynamical systems and quasi variational inequalities, Appl. Math. Comput. (2002)
[19] Noor, M.A., A modified projection method for monotone variational inequalities, Appl. math. lett., 12, 83-87, (1999) · Zbl 0941.49006
[20] Noor, M.A.; Al-Said, E., Wiener – hopf equations technique for quasimonotone variational inequalities, J. optim. theory appl., 103, 704-714, (1999) · Zbl 0953.65050
[21] Noor, M.A.; Al-Said, E., Change of variable method for generalized complementarity problems, J. optim. theory appl., 100, 389-395, (1999) · Zbl 0915.90244
[22] M.A. Noor, T.M. Rassias, A class of projection methods for general variational inequalities, J. Math. Anal. Appl. (2002) · Zbl 1038.49017
[23] Noor, M.A.; Noor, K.I.; Rassias, Th.M., Some aspects of variational inequalities, J. comput. appl. math., 47, 285-312, (1993) · Zbl 0788.65074
[24] Noor, M.A.; Wang, Y.J.; Xiu, N.H., Some projection methods for variational inequalities, Appl. math. comput., (2003)
[25] Patriksson, M., Nonlinear programming and variational inequalities: A unified approach, (1998), Kluwer Academic Dordrecht
[26] Robinson, S.M., Normal maps induced by linear transformations, Math. oper. res., 17, 691-714, (1992) · Zbl 0777.90063
[27] Shi, P., Equivalence of variational inequalities with wiener – hopf equations, Proc. amer. math. soc., 111, 339-346, (1991) · Zbl 0881.35049
[28] Solodov, M.V.; Svaiter, B.F., A new projection method for variational inequality problems, SIAM J. control optim., 42, 309-321, (1997)
[29] Solodov, M.V.; Tseng, P., Modified projection type methods for monotone variational inequalities, SIAM J. control optim., 34, 1814-1830, (1996) · Zbl 0866.49018
[30] Sun, D., A class of iterative methods for solving nonlinear projection equations, J. optim. theory appl., 91, 123-140, (1996) · Zbl 0871.90091
[31] Sun, D., A projection and contraction method for the nonlinear complementarity problem and its extensions, Math. numer. sinica, 16, 183-194, (1994) · Zbl 0900.65188
[32] Stampacchia, G., Formes bilineaires coercitives sur LES ensembles convexes, C. R. acad. sci. Paris, 258, 4413-4416, (1964) · Zbl 0124.06401
[33] Wang, Y.J.; Xiu, N.H.; Wang, C.Y., Unified framework of projection methods for pseudomonotone variational inequalities, J. optim. theory appl., 111, 643-658, (2001) · Zbl 1039.49014
[34] Wang, Y.J.; Xiu, N.H.; Wang, C.Y., A new version of extragradient projection method for variational inequalities, Comput. math. appl., 42, 969-979, (2001) · Zbl 0993.49005
[35] Xiu, N.; Zhang, J.; Noor, M.A., Tangent projection equations and general variational equalities, J. math. anal. appl., 258, 755-762, (2001) · Zbl 1008.49010
[36] N.H. Xiu, J. Zhang, Some recent advances in projection-type methods for variational inequalities, Preprint (2001) · Zbl 1018.65083
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.