On the Hessian of Lagrangian and Second Order Optimality Conditions

Abstract

For a constrained minimization problem, the restriction of a Hessian of Lagrangian to a tangent space of the feasible set can be used to detect whether a Karush–Kuhn–Tucker point is a local minimum, maximum or saddle point of the problem. It is shown in this paper that the restriction of the Hessian to a normal space with respect to the indefinite inner product induced by the Hessian can be used to characterize a Karush–Kuhn–Tucker point for the Wolfe dual. From this result and by an inertia theorem in [S.-P. Han and O. Fujiwara, An inertia theorem and its application to nonlinear programs, manuscript March 1984], we deduce that, under a regularity condition, the Hessian is positive semidefinite if and only if the considered Karush–Kuhn–Tucker point satisfies the second order necessary condition for a local minimum point of the primal problem and, at the same time, satisfies the second order necessary condition for a local maximum point of thee dual. Similar results on the positive definiteness of the Hessian are also discussed, which strengthen some results given in [O. Fujiwara, S.-P. Han and O. L. Mangasarian, Local duality of nonlinear programs, SIAM J. Control Optim., 22 (1984) pp. 162–169], [S.-P. Han and O. L. Mangasarian, Characterization of positive definite and semidefinite matrices via quadratic programming duality, SIAM J. Alg. Disc. Meth., 5 (1984) pp. 26–32].