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On Approximate KKT Condition and its Extension to Continuous Variational Inequalities

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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.

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Author notes

  1. Gabriel Haeser

    Present address: Institute of Science and Technology, Federal University of São Paulo, São José dos Campos SP, Brazil

Authors and Affiliations

  1. Institute of Mathematics and Statistics, University of São Paulo, São Paulo, SP, Brazil

    Gabriel Haeser

  2. CONICET, Department of Mathematics, FCE, University of La Plata, CP 172, 1900, La Plata, Argentina

    María Laura Schuverdt

Authors
  1. Gabriel Haeser
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  2. María Laura Schuverdt
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Corresponding author

Correspondence to María Laura Schuverdt.

Additional information

Communicated by G. Di Pillo.

We are indebted to two anonymous referees for insightful comments and suggestions.

G. Haeser was supported by CNPq Grant 503328/2009-0 and FAPESP Grant 09/09414-7.

M.L. Schuverdt was supported by PRONEX-CNPq/FAPERJ Grant E-26/171.164/2003.

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Haeser, G., Schuverdt, M.L. On Approximate KKT Condition and its Extension to Continuous Variational Inequalities. J Optim Theory Appl 149, 528–539 (2011). https://doi.org/10.1007/s10957-011-9802-x

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