We use the auxiliary principle technique in conjunction with the Bregman function to suggest and analyze a three-step predictor-corrector method for solving mixed quasi-equilibrium-like problems. We also study the convergence criteria of this new method under some mild conditions. As special cases, we obtain various new and known methods for solving variational-like inequalities and related optimization problems.
E. Blum and W. Oettli, “From optimization and variational inequalities to equilibrium problems,” The Mathematics Student, vol. 63, no. 1–4, pp. 123–145, 1994.
View at: Google Scholar | Zentralblatt MATH | MathSciNetF. Giannessi and A. Maugeri, Variational Inequalities and Network Equilibrium Problems, Plenum Press, New York, 1995.
F. Giannessi, A. Maugeri, and P. M. Pardalos, Equilibrium Problems: Nonsmooth Optimization and Variational Inequality Models, Kluwer Academic, Dordrecht, 2001.
View at: Zentralblatt MATHR. Glowinski, J.-L. Lions, and R. Trémolières, Numerical Analysis of Variational Inequalities, North-Holland, Amsterdam, 1981.
View at: Zentralblatt MATH | MathSciNetM. A. Hanson, “On sufficiency of the Kuhn-Tucker conditions,” Journal of Mathematical Analysis and Applications, vol. 80, no. 2, pp. 545–550, 1981.
View at: Publisher Site | Google Scholar | MathSciNetD. Kinderlehrer and G. Stampacchia, An Introduction to Variational Inequalities and Their Applications, vol. 31 of Classics in Applied Mathematics, Academic Press, New York, 1980.
View at: Zentralblatt MATH | MathSciNetM. A. Noor, “Variational-like inequalities,” Optimization, vol. 30, pp. 323–330, 1944.
View at: Google ScholarM. A. Noor, “Nonconvex functions and variational inequalities,” Journal of Optimization Theory and Applications, vol. 87, no. 3, pp. 615–630, 1995.
View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNetM. A. Noor, “Multivalued general equilibrium problems,” Journal of Mathematical Analysis and Applications, vol. 283, no. 1, pp. 140–149, 2003.
View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNetM. A. Noor, “Fundamentals of mixed quasi variational inequalities,” International Journal of Pure and Applied Mathematics, vol. 15, no. 2, pp. 137–258, 2004.
View at: Google Scholar | Zentralblatt MATH | MathSciNetM. A. Noor, “On generalized preinvex functions and monotonicities,” Journal of Inequalities in Pure and Applied Mathematics, vol. 5, no. 4, pp. 1–9, 2004, article 110.
View at: Google Scholar | MathSciNetM. A. Noor, “Some developments in general variational inequalities,” Applied Mathematics and Computation, vol. 152, no. 1, pp. 199–277, 2004.
View at: Publisher Site | Google Scholar | MathSciNetM. A. Noor, “Invex equilibrium problems,” Journal of Mathematical Analysis and Applications, vol. 302, no. 2, pp. 463–475, 2005.
View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNetM. A. Noor and W. Oettli, “On general nonlinear complementarity problems and quasi-equilibria,” Le Matematiche, vol. 49, no. 2, pp. 313–331, 1994.
View at: Google Scholar | Zentralblatt MATH | MathSciNetM. Patriksson, Nonlinear Programming and Variational Inequalities Problems: A Unified Approach, vol. 23 of Applied Optimization, Kluwer, Dordrecht, 1999.
View at: MathSciNetT. Weir and B. Mond, “Pre-invex functions in multiobjective optimization,” Journal of Mathematical Analysis and Applications, vol. 136, no. 1, pp. 29–38, 1988.
View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNetX. Q. Yang and G. Y. Chen, “A class of nonconvex functions and prevariational inequalities,” Journal of Mathematical Analysis and Applications, vol. 169, no. 2, pp. 359–373, 1992.
View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNetD. L. Zhu and P. Marcotte, “Co-coercivity and its role in the convergence of iterative schemes for solving variational inequalities,” SIAM Journal on Optimization, vol. 6, no. 3, pp. 714–726, 1996.
View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet