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.