On Finite Convergence of Iterative Methods for Variational Inequalities in Hilbert Spaces

On Finite Convergence of Iterative Methods for Variational Inequalities in Hilbert Spaces

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Article ID: iaor2014770
Volume: 161
Issue: 3
Start Page Number: 701
End Page Number: 715
Publication Date: Jun 2014
Journal: Journal of Optimization Theory and Applications
Authors: ,
Keywords: iterative methods, proximal point algorithm, Hilbert space
Abstract:

In a Hilbert space, we study the finite termination of iterative methods for solving a monotone variational inequality under a weak sharpness assumption. Most results to date require that the sequence generated by the method converges strongly to a solution. In this paper, we show that the proximal point algorithm for solving the variational inequality terminates at a solution in a finite number of iterations if the solution set is weakly sharp. Consequently, we derive finite convergence results for the gradient projection and extragradient methods. Our results show that the assumption of strong convergence of sequences can be removed in the Hilbert space case.

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