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On Approximate KKT Condition and its Extension to Continuous Variational Inequalities

  • Gabriel Haeser
  • María Laura Schuverdt
Article

Abstract

In this work, we introduce a necessary sequential Approximate-Karush-Kuhn-Tucker (AKKT) condition for a point to be a solution of a continuous variational inequality, and we prove its relation with the Approximate Gradient Projection condition (AGP) of Gárciga-Otero and Svaiter. We also prove that a slight variation of the AKKT condition is sufficient for a convex problem, either for variational inequalities or optimization. Sequential necessary conditions are more suitable to iterative methods than usual punctual conditions relying on constraint qualifications. The AKKT property holds at a solution independently of the fulfillment of a constraint qualification, but when a weak one holds, we can guarantee the validity of the KKT conditions.

Keywords

Optimality conditions Variational inequalities Constraint qualifications Practical algorithms 

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Copyright information

© Springer Science+Business Media, LLC 2011

Authors and Affiliations

  1. 1.Institute of Mathematics and StatisticsUniversity of São PauloSão PauloBrazil
  2. 2.Institute of Science and TechnologyFederal University of São PauloSão José dos Campos SPBrazil
  3. 3.CONICET, Department of Mathematics, FCEUniversity of La Plata, CP 172La PlataArgentina

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