Variational inequalities are studied in various models for a large number of mathematical, physical, economics, finance, optimization, game theory, engineering and other problems(see[1],[2],[12], [14],[15], [21]). The fixed point formulation of any variational inequality problem is not only useful for existence of solution of the variational inequality problem, but it also provides the facility to develop algorithms for approximation of solution of variational inequality problem. A lot of research has been carried out to develop various iterative algorithms to find solution of a variational inequality problem. In this paper, we have studied various algorithms or methods used for solving Variational inequality problems and studied the developments of such methods and compared their convergence rate . Our result helps in understanding the development in iterative algorithms for VI. AMS Subject Classification: 49H09; 47H10; 47J20; 49J40, 47J05

Proximal Point Algorithm, KKT based method ,Linear Approximation method, strong convergence, variational inequality, Projection based methods

[1]Censor, Y. and Gibali, A. (2008). Projections onto super-half-spaces for monotone variational inequality problems in finite-dimensional space. Journal of Nonlinear and Convex Analysis, 9:461–475.
[2]Cruz, J. Y. B. and Iusem, A. N. (2010). Convergence of direct methods for paramonotone variational inequalities. Applied Mathematics and Optimization, 46:247 – 263.
[3] Harker, P. T. and Pang, J.-S. (1990). Finite dimensional variational inequality and non linear complementarity problems: A survey of theory, algorithms and applications. Mathematical Programming, 48:161–220.
[4]Robinson, S. M. (1980). Strongly regular generalized equations. Mathematics of Operations Research, 5:43–62.
[5]. Stampacchia,G.:Formes bilineaires coercitives sur les ensembles convexes.C.R.Acad.Bulg. Sci.Paris 258,4413-4416(1964).
[6]Josephy, N. (1979a). Quasi-newton methods for generalized equations.
[7]Josephy, N. H. (1979b). Newton’s method for generalized equations. Technical Summary Report 1965, Mathematics Research Center, University of Wisconsin-Madison, Madison, WI, USA. NTIS Accession No. AD A077 096.
[8]Chan, D. and Pang, J.-S. (1982). Iterative methods for variational and complementarity problems. Mathematical Programming, 24:284–313.
[9] Lawphongpanich, S. and Hearn, D. (1984). Simplicial decomposition of asymmetric traffic assignment problem. Transportation Research, 18B:123–133.
[10]Facchinei, F. and Pang, J.-S. (2003). Finite-Dimensional Variational Inequalities and Complementarity Problems. Springer Series in Operations Research. Springer-Verlag, New York. Published in two volumes, paginated continuously.
[11] Rockafellar, R. (1976). Augmented lagrangian and application of the proximal point algorithm in convex programming. Mathematics of Operations Research, 1:97–116.
[12]Fukushima, M. (1983). An outer appxoximation algorithm for solving general convex programs. Operations Research, 31(1):101–113.
[13]Harker, P. (1984). A variational inequality approch for the determination of oligopolistic market equilibrium. Mathematical Programming, 30:105–111.
[14]He, B. (1997). A class of projection and contraction methods for monotone variational inequalities. Applied Mathematics Optimization, 35:69–76.
[15]In Giannessi, F. and Maugeri, A., editors, Variational Inequalities and Network Equilibrium Problems, pages 257–269, New York. Plenum Press.
[16]Liu, S.,He, H. :Approximating solution of 0T(x) for an H-accretive operator in Banach spaces. J.Math.Anal.Appl. 385,466-476 (2012).
[17]Fang ,Yaping;Huang, Nanjing; H-accretive operators and resolvent operator technique for solving variational inclusions in Banach Spaces,Appl.Math. Lett.17(2004) 647-653.
[18]Aoyama,K.,Iiduka, H.,Takahashi, W.,:Weak convergence of an iterative sequence for accretive operators in Banach spaces.Fixed Point theory and application.doi:10.1155/35390 (2006).
[19] Thuy,NTT.: Regulariztion methods and iterative methods for variational inequality with accretive operator.Acta Math Vietnam, 41:55-68 (2016)
[20] Buong,N., Phuong, N.T.H.: Regularization methods for a class of variational inequalities in Banach spaces. Comput. Math.Phys.52,1487-1496 (2012)
[21]Dafermos, S. (1980). Traffic equilibrium and variation inequalities. Transportation Science, 14:42–54