On Approximate KKT Condition and its Extension to Continuous Variational Inequalities

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 Garciga-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.

CNPq[503328/2009-0]

Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq)

Fundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP)

FAPESP[09/09414-7]

PRONEX-CNPq/FAPERJ[E-26/171.164/2003]

PRONEX-CNPq/FAPERJ