Abstract

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.