Constraint qualifications for convex optimization without convexity of constraints : New connections and applications to best approximation

We study constraint qualifications and necessary and sufficient optimality conditions for a convex optimization problem with inequality constraints where the constraint functions are continuously differentiable but they are not assumed to be convex. We present constraint qualifications under which the Karush-Kuhn-Tucker conditions are necessary and sufficient for optimality without the convexity of the constraint functions and establish new links among various known constraint qualifications that guarantee necessary Karush-Kuhn-Tucker conditions. We also present a new constraint qualification which is the weakest constraint qualification for the Karush-Kuhn-Tucker conditions to be necessary for optimality of the convex optimization problem. Consequently, we present Lagrange multiplier characterizations for the best approximation from a convex set in the face of nonconvex inequality constraints, extending corresponding known results in the literature. We finally give a table summarizing various links among the constraint qualifications.