On the weakest constraint qualification for sharp local minimizers

The sharp local minimality of feasible points of nonlinear optimization problems is known to possess a characterization by a strengthened version of the Karush–Kuhn–Tucker conditions, as long as the Mangasarian–Fromovitz constraint qualification holds. This strengthened condition is not easy to check algorithmically since it involves the topological interior of some set. In this article, we derive an algorithmically tractable version of this condition, called strong Karush–Kuhn–Tucker condition. We show that the Guignard constraint qualification is the weakest condition under which a feasible point is a strong Karush–Kuhn–Tucker point for every continuously differentiable objective function possessing the point as a sharp local minimizer. As an application, our results yield an algebraic characterization of strict local minimizers of linear programs with cardinality constraints.