Stochastic approximation results for variational inequality problem using random-type iterative schemes

References

• Alber, Y. I. and S. Guerre-Delabriere. 1997. Principle of weakly contractive maps in Hilbert spaces. In: New Results in Operator Theory and Its Applications. I., Gohberg, Lyubich (eds) 7–22. Birkhauser, Basel.
   Google Scholar
• Baillon, J. B. and G. Haddad. 1977. Quelques proprieties des operateurs angle-bornes et n- cycliquement monotones. Isr Journal of Mathematics 26:137–50.
   Web of Science ®Google Scholar
• Ceng, L. C., Q. H. Ansari and J. C. Yao. 2011. Some iterative methods for finding fixed points and for solving constrained convex minimization problems. Nonlinear Analysis 74:5286–02.
   Web of Science ®Google Scholar
• Chen, P., W. Liu and J. Feng. 2006. Sufficient and necessary condition for the convergence of stochastic approximation algorithms. Statistics and Probability Letters 76 (2):203–10.
   Web of Science ®Google Scholar
• Chen, X., R. J.-B. Wets and Y. Zhang. 2012. Stochastic variational inequalities: residual minimization smoothing/sample average approximations SIAM. Journal on Optimization 22:649–73.
   Web of Science ®Google Scholar
• Facchinei, F. and J. S. Pang. 2003. Finite dimensional variational inequalities and complementarity problems. New York: Springer.
   Google Scholar
• Hans, O. 1961. Random operator equations. In Proceedings of the Fourth Berkeley Symposium on Mathematical Statistics and Probability. Vol. 2 part 1 185–02. University of California Press, California.
   Google Scholar
• Hussain, N., S. Narwal, R. Chugh and V. Kumar. 2016. On convergence of random iterative schemes with errors for strongly pseudo-contractive Lipschitzian map in real Banach space. Journal of Nonlinear Science and Applications 9:3157–68.
   Google Scholar
• Itoh, S. 1979. Random fixed point theorems with an application to random differential equations in Banach spaces. Journal of Mathematical Analysis and Applications 67 (2):261–73.
   Web of Science ®Google Scholar
• Jiang, H. and F. Xu. 2008. Stochastic approximation approaches to the stochastic variational inequality problem. IEEE Transactions on Automatic Control 53:1462–75.
   Web of Science ®Google Scholar
• Jung, J. S. 2012. Iterative methods for mixed equilibrium problems and pseudocontractive mappings. Fixed Point Theory and Applications. 2012:184. doi:10.1186/1687-1812-2012-184
   Google Scholar
• Koshal, J., A. Neddic and U. V. Shanbhag. 2013. Regularized iterative stochastic approximation methods for stochastic variational inequality. IEEE Transactions on Automatic Control 58 (3):594–09.
   Web of Science ®Google Scholar
• Mahmoud, M. A., A. A. Rasha and S. W. Waseem. 2017. Stochastic approximation with series of delayed observations. Journal of The Egyptian Mathematical Society 25:191–96.
   Google Scholar
• Marino, G. and H.-K. Xu. 2007. Weak and strong convergence theorems for strict pseudo-contractions in Hilbert spaces. Journal of Mathematical Analysis and Applications 329 (1):336–46.
   Web of Science ®Google Scholar
• Okeke, G. A. and Abbas M. 2015. Convergence and almost T-stability for a random iterative sequence generated by a generalized random operator. Journal of Inequality and Applications. 2015:146. doi:10.1186/s13660-015-0666-8
   Google Scholar
• Robbins, H. and S. Monro. 1951. A stochastic optimization method. The annals of Mathematical Statistics 22:400–07.
   Google Scholar
• Spacek, A. 1955. Zulfallige gleichungen. Czechoslovian Mathematics Journal 5:462–66.
   Google Scholar
• Tan, K. K. and H. K. Xu. 1993. Approximating fixed point of nonexpansive mappings by the Ishikawa iterative process. Journal of Mathematical Analysis and Applications 178 (2):301–8.
   Web of Science ®Google Scholar
• Tian, M. and L. Huang. 2013. Iterative methods for constrained convex minimization problem in Hilbert spaces. Fixed Point Theory and Applications. 2013:105. doi:10.1186/1687-1812-2013-105
   Google Scholar
• Xu, H. K. 2011. Averaged mappings and the the gradient-projection algorithm. Journal of Optimization: Theory and Application 150 (2):360–78.
   Web of Science ®Google Scholar
• Zhang, S. S., X. R. Wang and M. Liu. 2011. Almost sure T-stability and convergence for iterative algorithms. Applied Mathematics and Mechanics 32 (6):805–10.
   Web of Science ®Google Scholar