×

Iterative schemes for nonconvex variational inequalities. (English) Zbl 1062.49009

Summary: We suggest and analyze some iterative methods for solving nonconvex variational inequalities using the auxiliary principle technique, the convergence of which requires either only pseudomonotonicity or partially relaxed strong monotonicity. Our proofs of convergence are very simple. As special cases, we obtain earlier results for solving general variational inequalities involving convex sets.

MSC:

49J40 Variational inequalities
47J20 Variational and other types of inequalities involving nonlinear operators (general)
47J25 Iterative procedures involving nonlinear operators
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Stampacchia, G., Formes Bilineaires Coercitives sur les Ensembles Convexes, Comptes Rendus de l’Academie des Sciences, Paris, Vol. 258, pp. 4413–4416, 1964. · Zbl 0124.06401
[2] Baiocchi, C., and Capelo, A., Variational and Quasivariational Inequalities, John Wiley and Sons, New York, NY, 1984. · Zbl 0551.49007
[3] Giannessi, F., and MAUGERI, A., Variational Inequalities and Network Equilibrium Problems, Plenum Press, New York, NY, 1995. · Zbl 0834.00044
[4] Giannessi, F., MAUGERI, A. and PARDALOS, M. S., Equilibrium Problems: Nonsmooth Optimization and Variational Inequality Models, Kluwer Academic Publishers, Dordrecht, Holland, 2001. · Zbl 0979.00025
[5] Glowinski, R., Lions, J. L., and Tremolieres, R., Numerical Analysis of Variational Inequalities, North-Holland, Amsterdam, Holland, 1981. · Zbl 0463.65046
[6] Noor, M. A., Mixed Quasivariational Inequalities, Applied Mathematics and Computation, Vol. 146, pp. 553–578, 2003. · Zbl 1035.65063 · doi:10.1016/S0096-3003(02)00605-7
[7] Patriksson, M., Nonlinear Programming and Variational Inequalities: A Unified Approach, Kluwer Academic Publishers, Dordrecht, Holland, 1998. · Zbl 0912.90261
[8] Noor, M. A., Some Recent Advances in Variational Inequalities, Part 1: Basic Concepts, New Zealand Journal of Mathematics, Vol. 26, pp. 53–80, 1997. · Zbl 0886.49004
[9] Noor, M. A., Some Recent Advances in Variational Inequalities, Part 2: Other Concepts, New Zealand Journal of Mathematics, Vol. 26, pp. 229–255, 1997. · Zbl 0889.49006
[10] Noor, M. A., New Extragradient-Type Methods for General Variational Inequalities, Journal of Mathematical Analysis and Applications, Vol. 277, pp. 379–395, 2003. · Zbl 1033.49015
[11] Noor, M. A., Some Developments in General Variational Inequalities, Applied Mathematics and Computation, Vol. 152, pp. 197–277, 2004. · Zbl 1134.49304
[12] Noor, M. A., Extragradient Methods for Pseudomonotone Variational Inequalities, Journal of Optimization Theory and Applications, Vol. 177, pp. 475–488, 2003. · Zbl 1049.49009 · doi:10.1023/A:1023989403613
[13] Clarke, F. H., Ledyaev, Y. S., Stern, R. J., and Wolenski, P.R., Nonsmooth Analysis and Control Theory, Springer Verlag, New York, NY, 1998. · Zbl 1047.49500
[14] Poliquin, R.A., Rockafellar, R.T., and Thibault, L., Local Differentiability of Distance Functions, Transactions of the American Mathematical Society, Vol. 352, pp. 5231–5249, 2000. · Zbl 0960.49018 · doi:10.1090/S0002-9947-00-02550-2
[15] Bounkhel, M., Tadj, L., and Hamdi, A., Iterative Schemes to Solve Nonconvex Variational Problems, Journal of Inequalities in Pure and Applied Mathematics, Vol. 4, pp. 1–14, 2003. · Zbl 1045.58014
[16] Noor, M. A., Theory of General Variational Inequalities, Preprint, Etisalat College of Engineering, Sharjah, United Arab Emirates, 2003.
[17] Moudafi, A.,An Algorithmic Approach to Prox-Regular Variational Inequalities, Preprint, UniversitéAntilles Guyane, France, 2003. · Zbl 1062.65071
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. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.