Local Duality of Nonlinear Programs.
Technical summary rept.,
WISCONSIN UNIV-MADISON MATHEMATICS RESEARCH CENTER
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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 nxn matrix to be positive definite is that for each or 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. Author
- Theoretical Mathematics