Abstract
This paper is devoted to the study of a new necessary condition in variational inequality problems: approximated gradient projection (AGP). A feasible point satisfies such condition if it is the limit of a sequence of the approximated solutions of approximations of the variational problem. This condition comes from optimization where the error in the approximated solution is measured by the projected gradient onto the approximated feasible set, which is obtained from a linearization of the constraints with slack variables to make the current point feasible.
We state the AGP condition for variational inequality problems and show that it is necessary for a point being a solution even without constraint qualifications (e.g., Abadie’s). Moreover, the AGP condition is sufficient in convex variational inequalities. Sufficiency also holds for variational inequalities involving maximal monotone operators subject to the boundedness of the vectors in the image of the operator (playing the role of the gradients). Since AGP is a condition verified by a sequence, it is particularly interesting for iterative methods.
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Communicated by G. Di Pillo.
Research of R. Gárciga Otero was partially supported by CNPq, FAPERJ/Cientistas do Nosso Estado, and PRONEX Optimization.
Research of B.F. Svaiter was partially supported by CNPq Grants 300755/2005-8 and 475647/2006-8 and by PRONEX Optimization.
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Otero, R.G., Svaiter, B.F. New Condition Characterizing the Solutions of Variational Inequality Problems. J Optim Theory Appl 137, 89–98 (2008). https://doi.org/10.1007/s10957-007-9320-z
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DOI: https://doi.org/10.1007/s10957-007-9320-z