Journal of Optimization Theory and Applications

, Volume 119, Issue 1, pp 185–201 | Cite as

Convergence of Hybrid Steepest-Descent Methods for Variational Inequalities

  • H. K. Xu
  • T. H. Kim


Assume that F is a nonlinear operator on a real Hilbert space H which is η-strongly monotone and κ-Lipschitzian on a nonempty closed convex subset C of H. Assume also that C is the intersection of the fixed point sets of a finite number of nonexpansive mappings on H. We devise an iterative algorithm which generates a sequence (x n ) from an arbitrary initial point x0H. The sequence (x n ) is shown to converge in norm to the unique solution u* of the variational inequality
$$\left\langle {F(u*),\user1{v} - u*} \right\rangle \geqslant 0$$
Applications to constrained pseudoinverse are included.
Iterative algorithms hybrid steepest-descent methods convergence nonexpansive mappings Hilbert space constrained pseudoinverses 


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Copyright information

© Plenum Publishing Corporation 2003

Authors and Affiliations

  • H. K. Xu
    • 1
  • T. H. Kim
    • 2
  1. 1.Department of MathematicsUniversity of Durban-WestvilleDurbanSouth Africa
  2. 2.Division of Mathematical SciencesPukyong National UniversityPusanKorea

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