A New Approach to Lagrange Multipliers

We consider a mathematical programming problem on a Banach space, and we derive necessary conditions for optimality in Lagrange multiplier form. We prove further that "most mathematical programming problems are normal." The novelty of our approach lies on the one hand in the absence of bot...

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Bibliographic Details
Published in:Mathematics of operations research 1976-05, Vol.1 (2), p.165-174
Main Author: Clarke, Frank H
Format: Article
Language:English
Subjects:
Banach space
Banach spaces
Convexity
generalized gradients
Lagrange multipliers
Local minimum
Mathematical functions
Mathematical programming
Mathematical theorems
Necessary conditions for optimality
nonconvex programs
Serenity
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Summary:We consider a mathematical programming problem on a Banach space, and we derive necessary conditions for optimality in Lagrange multiplier form. We prove further that "most mathematical programming problems are normal." The novelty of our approach lies on the one hand in the absence of both differentiability and convexity hypotheses on the functions delimiting the problem, and on the other hand in the method of proof, which is new. The approach unifies the well-known smooth and convex cases besides treating a new general class of problems. The main analytical tool in this paper is an extension to infinite-dimensional spaces of the "generalized gradient" previously introduced by the author. The calculus of the generalized gradient is explored as a preliminary step.
ISSN:0364-765X
1526-5471
DOI:10.1287/moor.1.2.165