The authors introduce a rather large class of differentiable convex functions for which the constraint qualifications is always satisfied. Hence, for programs with the constraints belonging to this class, consistency of the Karush-Kuhn-Tucker condition (as a system) is both necessary and sufficient for optimality. The authors give geometric and algebraic characterizations for this class of functions. In the multiobjective programs the authors proved that, for the same class of functions, the set of Pareto optima coincides with the set of strong Pareto optima.