New qualification conditions for convex optimization without convex representation

In this paper, we consider a convex optimization problem with the feasible set is convex and the constraint functions are differentiable, but they are not necessarily convex. We study several known constraint qualifications under which the Karush–Kuhn–Tucker conditions are necessary and sufficient for the optimality. We also establish new connections among various known constraint qualifications that guarantee necessary the Karush–Kuhn–Tucker conditions. Moreover, by using the concept of dual cones, we introduce a new constraint qualification that is the weakest qualification for the Karush–Kuhn–Tucker conditions to be necessary for the optimality. Finally, as an application, we characterize the best approximation to any x ∈ R n from a convex set in the face of non-convex inequality constraints.