On programming when the positive cone has an empty interior

In this note, we present a condition which is equivalent to the existence of the Lagrange multiplier for the general convex programming problem. This condition enables one to study a hypothesis distinct from the one of nonempty interior of the positive cone of the space of restrictions that is commonly used. Simple examples of this condition are given. We also explore the relationship of this condition with the subdifferentiability of the primal functional.