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Local Duality of Nonlinear Programs.

Author: O Fujiwara; S -P Han; O L Mangasarian; WISCONSIN UNIV-MADISON MATHEMATICS RESEARCH CENTER.
Publisher: Ft. Belvoir Defense Technical Information Center FEB 1982.
Edition/Format:   Print book : EnglishView all editions and formats
Database:WorldCat
Summary:
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  Read more...
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Document Type: Book
All Authors / Contributors: O Fujiwara; S -P Han; O L Mangasarian; WISCONSIN UNIV-MADISON MATHEMATICS RESEARCH CENTER.
OCLC Number: 227533231
Notes: Sponsored in part by Grant NSF-ENG79-03881.
Description: 23 pages

Abstract:

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 r(n), the matrix must be positive definite on the subspace and its inverse be positive semidefinite on the orthogonal complement of the subspace. (Author).

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