A Linear Approximation Approach to Duality in Nonlinear Programming

Abstract

Linear approximation and linear programming duality theory are used as unifying tools to develop saddlepoint, Fenchel and local duality theory. Among results presented is a new and elementary proof of the necessity and sufficiency of the stability condition for saddlepoint duality, an equivalence between the saddlepoint and Fenchel theories, and nasc for an optimal solution of an optimization problem to be a Kuhn-Tucker point. Several of the classic "constraint qualifications" are discussed with respect to this last condition. In addition, generalized versions of Fenchel and Rockafeller duals are introduced. Finally, a shortened proof is given of a result of Mangasarian and Fromowitz that under fairly general conditions an optimal point is also a Fritz John point.