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New Condition Characterizing the Solutions of Variational Inequality Problems

Author

Listed:
  • R. Gárciga Otero

    (Instituto de Economia da Universidade Federal de Rio de Janeiro)

  • B. F. Svaiter

    (Instituto de Matemática Pura e Aplicada)

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.

Suggested Citation

  • R. Gárciga Otero & B. F. Svaiter, 2008. "New Condition Characterizing the Solutions of Variational Inequality Problems," Journal of Optimization Theory and Applications, Springer, vol. 137(1), pages 89-98, April.
    • Handle: RePEc:spr:joptap:v:137:y:2008:i:1:d:10.1007_s10957-007-9320-z
    DOI: 10.1007/s10957-007-9320-z
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    References listed on IDEAS

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    1. J.M. Martínez & B.F. Svaiter, 2003. "A Practical Optimality Condition Without Constraint Qualifications for Nonlinear Programming," Journal of Optimization Theory and Applications, Springer, vol. 118(1), pages 117-133, July.
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    Cited by:

    1. Gabriel Haeser & María Laura Schuverdt, 2011. "On Approximate KKT Condition and its Extension to Continuous Variational Inequalities," Journal of Optimization Theory and Applications, Springer, vol. 149(3), pages 528-539, June.

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