A sharp Lagrange multiplier theorem for nonlinear programs

For a nonlinear program with inequalities and under a Slater constraint qualification, it is shown that the duality between optimal solutions and saddle points for the corresponding Lagrangian is equivalent to the infsup-convexity—a not very restrictive generalization of convexity which arises naturally in minimax theory—of a finite family of suitable functions. Even if we dispense with the Slater condition, it is proven that the infsup-convexity is nothing more than an equivalent reformulation of the Fritz John conditions for the nonlinear optimization problem under consideration.