On modified hybrid steepest-descent methods for general variational inequalities

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Abstract

We consider the general variational inequality GVI(F,g,C), where F and g are mappings from a Hilbert space into itself and C is intersection of the fixed point sets of a finite family of nonexpansive mappings. We suggest and analyze an iterative algorithm with variable parameters as follows:un+1=(1αn+1+θn+1)T[n+1]un+αn+1unθn+1g(T[n+1]un)λn+1μn+1F(T[n+1]un),n0. The sequence {un} is shown to converge in norm to the solutions of the general variational inequality GVI(F,g,C) under some mild conditions. Application to constrained generalized pseudo-inverse is included. Since the general variational inequalities include variational inequalities, quasi-variational inequalities and complementarity problems as special cases, results obtained in this paper continue to hold for these problems. Results obtained in this paper may be viewed as a refinement and improvement of the previously known results.

Keywords

Hybrid steepest-descent method with variable parameters
Nonexpansive mappings
Variational inequality
Strong convergence

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Research of Prof. Dr. Muhammad Aslam Noor is supported by the Higher Education Commission, Pakistan, through research grant No. I-29/HEC/HRD/2005/90.