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A Stochastic Iteration Method for A Class of Monotone Variational Inequalities In Hilbert Space
Pages - 1 - 6     |    Revised - 30-06-2020     |    Published - 01-08-2020
Volume - 8   Issue - 1    |    Publication Date - August 2020  Table of Contents
Linear Monotone Operator, Hilbert Space, Stochastic Approximation.
We examined a general method for obtaining a solution to a class of monotone variational inequalities in Hilbert space. Let H be a real Hilbert space, and Let T : H -> H be a continuous linear monotone operator and K be a non empty closed convex subset of H. From an initial arbitrary point x0 ∈ K. We proposed and obtained iterative method that converges in norm to a solution of the class of monotone variational inequalities. A stochastic scheme {xn} is defined as follows: x(n+1) = xn - anF* (xn), n≥0, F*(xn), n ≥ 0 is a strong stochastic approximation of Txn - b, for all b (possible zero) ∈ H and an ∈ (0,1).
1 Google Scholar 
2 Semantic Scholar 
3 refSeek 
4 Scribd 
5 SlideShare 
A.C. Okoroafor. "A Strong Convergence Stochastic Algorithms for the global solution of Autonomous Differential Equations in Rn." J. Applied Science 6(11), pp 2506-2509, 2006.
C.E. Chidume, "Steepest Descent Approximation for Accretive Operator equations" Nonlinear Anal. Theory and Methods Appl. 26, pp. 299-311.,1996.
H.K.Xu and T.H.Kim. "Convergence of Hybrid Steepest descent methods for Variational inequalities" Journal of Optimization Theory and Application, vol.119, No.1, pp. 185-201., 2003.
H.Walk and L.Zsido. "Convergence of Robbins-Monro Method for Linear Problems in a Banach space.",J.Math. Analysis and Appl., vol. 135, pp. 152-177, 1989.
H.Zhou, Y.Zhou and G. Feng. " Iterative Methods For Solving a Class Of Monotone Variational Inequality Problems with Applications" , Journal of Inequalities and Applications vol 68 (2015), 2015. (https//doi.org/10.1186/s13660-015-0590-y).
L. Ljung, G. Pfug and H.Walk. " Stochastic Approximation and Optimization of Random Systems.",Boston, MA: Birkhauser, 1992.
M.A. Kouritzin. "On The Convergence of Linear Stochastic Convergence Procedures." IEEE trans. Inform. Theory, vol. 42, pp. 1305., 1996.
O. IJIOMA and A.C. Okoroafor. " A Stochastic Approximation Method Applied To A Walrasian Equilibrium Model" IJAM Vol 5, No.1,pp 35-44, 2012.
P.Whitte. " Probability", John Wiley and Sons, London, 1976.
R.T. Rockafellar. "On the Maximal monotonicity of Sub differential mappings", Pacific J. Math., 33, pp. 209-216, 1970.
X. Lud, and J.Yang. " Regularization and Iterative Methods for Monotone Inverse Variational Inequalities" Optim Lett.8, pp 1261-1272, 2014. (https:/doi.org/10.1007/s11590-013-0653-2).
Z.Xu and G.Y.Roach. "A Necessary and Sufficient Condition for Convergence of Steepest Descent Approximation to Accretive Operator Equation." J. Math Anal. Appl.167, pp. 340-354.,1992.