A sharp Lagrange multiplier theorem for nonlinear programs

For a nonlinear program with inequalities and under a Slater constraint qualification, it is shown that the duality between optimal solutions and saddle points for the corresponding Lagrangian is equivalent to the infsup-convexity—a not very restrictive generalization of convexity which arises natur...

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Bibliographic Details
Published inJournal of global optimization Vol. 65; no. 3; pp. 513 - 530
Main Author Galan, MRuiz
Format Journal Article
LanguageEnglish
Published New York Springer US 01.07.2016
Springer
Springer Nature B.V
Subjects
Computer Science
Convex analysis
Equivalence
Euclidean space
Inequalities
Lagrange multiplier
Lagrange multipliers
Mathematical analysis
Mathematical functions
Mathematics
Mathematics and Statistics
Minimax technique
Nonlinear programming
Nonlinearity
Operations Research/Decision Theory
Optimization
Real Functions
Saddle points
Studies
Theorems
Fritz John conditions
90C30
90C46
Lagrange multipliers
Karush–Kuhn–Tucker conditions
46A22
Nonlinear programming
26B25
Infsup-convexity
Separation theorem
Online AccessGet full text
ISSN0925-5001
1573-2916
DOI10.1007/s10898-015-0379-z

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Summary:For a nonlinear program with inequalities and under a Slater constraint qualification, it is shown that the duality between optimal solutions and saddle points for the corresponding Lagrangian is equivalent to the infsup-convexity—a not very restrictive generalization of convexity which arises naturally in minimax theory—of a finite family of suitable functions. Even if we dispense with the Slater condition, it is proven that the infsup-convexity is nothing more than an equivalent reformulation of the Fritz John conditions for the nonlinear optimization problem under consideration.
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ISSN:0925-5001
1573-2916
DOI:10.1007/s10898-015-0379-z