A new result on second-order necessary conditions for nonlinear programming

In this paper, we give a new result on the second-order conditions for nonlinear programming. We show that the Hessian of Lagrangian is positive semi-definite on the critical cone if the Mangasarian–Fromovitz constraint qualification condition holds at a local minimizer, and the rank of the gradients of the active constraints at the local minimizer is one less than that at any point contained in its neighborhood.