On Regularity for Constrained Extremum Problems. Part II: Necessary Optimality Conditions

A particular theorem for linear separation between two sets is applied in the image space associated with a constrained extremum problem. In this space, the two sets are a convex cone, depending on the kind of constraints (equalities and inequalities) of the given problem and the homogenization of its image. It is proved that the particular linear separation is equivalent to the existence of Lagrangian multipliers with a positive multiplier associated with the objective function (i.e., a necessary optimality condition).