In this work we introduce a necessary natural sequential Approximate-Karush-Kuhn-Tucker (AKKT) condition for a point to be a solution of a continuous variational inequality problem without constraint quali cations, 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 on a convex problem, and we prove sufficiency results for the AKKT condition on convex optimization problems. Sequential necessary conditions are more suitable to iterative methods than usual punctual conditions relying on constraint quali cations.
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View On approximate KKT condition and its extension to continuous variational inequalities