Local Duality of Nonlinear Programs

It is shown that the second order sufficient (necessary) optimality condition for the dual of a nonlinear program is equivalent to the inverse of the Hessian of the Lagrangian being positive definite (semidefinite) on the normal cone to the local primal constraint surface. This compares with the Hessian itself being positive definite (semidefinite) on the tangent cone on the local primal constraint surface for the corresponding second order condition for the primal problem. We also show that primal second order sufficiency (necessity) and dual second order necessity (sufficiency) is essentially equivalent to the Hessian of the Lagrangian being positive definite. This follows from the following interesting linear algebra result: a necessary and sufficient condition for a nonsingular symmetric n × n matrix to be positive definite is that for some subspace of Rⁿ, the matrix must be positive definite on the subspace and its inverse be positive semidefinite on the orthogonal complement of the subspace.