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The KKT conditions are necessary and sufficient conditions for problems where we maximize over a convex function subject to linear inequality and equality constraints. That is $x^*$ is an optimal point iff it satisfies the KKT conditions.

My question is, are there similar necessary and/or sufficient conditions for points $x$ near $x^*$?

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    $\begingroup$ I've seen Delphine Mico-Umutesi give a talk "Estimating the Violation of the KKT Conditions" that has a similar flavor. You can find an abstract online, but I can't seem to find a full paper. $\endgroup$ Nov 21, 2013 at 3:47

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