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An extragradient algorithm for solving general nonconvex variational inequalities. (English) Zbl 1193.49008

Summary: We suggest and analyze an extragradient method for solving general nonconvex variational inequalities using the technique of the projection operator. We prove that the convergence of the extragradient method requires only pseudomonotonicity, which is a weaker condition than requiring monotonicity. In this sense, our result can be viewed as an improvement and refinement of the previously known results. Our method of proof is very simple as compared with other techniques.

MSC:

49J40 Variational inequalities
49M15 Newton-type methods
47J20 Variational and other types of inequalities involving nonlinear operators (general)
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[1] Stampacchia, G., Formes bilin√™aires coercitives sur LES ensembles convexes, C. R. acad. sci, Paris, 258, 4413-4416, (1964) · Zbl 0124.06401
[2] Bounkhel, M.; Tadji, L.; Hamdi, A., Iterative schemes to solve nonconvex variational problems, J. inequal. pure appl. math., 4, 1-14, (2003) · Zbl 1045.58014
[3] Clarke, F.H.; Ledyaev, Y.S.; Wolenski, P.R., Nonsmooth analysis and control theory, (1998), Springer-Verlag Berlin · Zbl 1047.49500
[4] Cristescu, G.; Lupsa, L., Non-connected convexities and applications, (2002), Kluwer Academic Publishers Dordrecht · Zbl 1037.52008
[5] ()
[6] Kinderlehrer, D.; Stampacchia, G., An introduction to variational inequalities and their applications, (2000), SIAM Philadelphia · Zbl 0988.49003
[7] Noor, M.A., General variational inequalities, Appl. math. lett., 1, 119-121, (1988)
[8] Noor, M.A., Some developments in general variational inequalities, Appl. math. comput., 152, 199-277, (2004) · Zbl 1134.49304
[9] Noor, M.A., Merit functions for general variational inequalities, J. math. anal. appl., 316, 736-752, (2006) · Zbl 1085.49011
[10] Noor, M.A., Some iterative methods for general nonconvex variational inequalities, Comput. math. model., 21, 92-109, (2010)
[11] Noor, M.A., Projection methods for nonconvex variational inequalities, Optim. lett., 3, 411-418, (2009) · Zbl 1171.58307
[12] Noor, M.A., Iterative methods for general nonconvex variational inequalities, Albanian J. math., 3, 117-127, (2009) · Zbl 1213.49017
[13] Noor, M.A., Nonconvex quasi variational inequalities, J. adv. math. studies, 3, 59-72, (2010) · Zbl 1206.49011
[14] Noor, M.A.; Noor, K.I.; Rassias, T.M., Some aspects of variational inequalities, J. comput. appl. math., 47, 285-312, (1993) · Zbl 0788.65074
[15] Noor, M.A.; Rassias, T.M., On nonconvex equilibrium problems, J. math. anal. appl., 312, 289-299, (2005) · Zbl 1087.49009
[16] Poliquin, R.A.; Rockafellar, R.T.; Thibault, L., Local differentiability of distance functions, Trans. amer. math. soc., 352, 5231-5249, (2000) · Zbl 0960.49018
[17] Singer, I., Duality for nonconvex approximation and optimization, (2006), Springer New York · Zbl 1119.90002
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