Sufficient conditions for the existence of multipliers and Lagrangian duality in abstract optimization problems

Eero Tamminen

Research output: Contribution to journalArticleScientificpeer-review

6 Citations (Scopus)

Abstract

We consider the following optimization problem: in an abstract set X, find and element x that minimizes a real function f subject to the constraints g(x)≤0 and h(x)=0, where g and h are functions from X into normed vector spaces. Assumptions concerning an overall convex structure for the problem in the image space, the existence of interior points in certain sets, and the normality of the constraints are formulated. A theorem of the alternative is proved for systems of equalities and inequalities, and an intrinsic multiplier rule and a Lagrangian saddle-point theorem (strong duality theorem) are obtained as consequences.
Original languageEnglish
Pages (from-to)93-104
Number of pages12
JournalJournal of Optimization Theory and Applications
Volume82
Issue number1
DOIs
Publication statusPublished - 1994
MoE publication typeA1 Journal article-refereed

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Lagrangian Duality
Multiplier
Theorems of the Alternative
Normed vector space
Saddle Point Theorem
Optimization Problem
Strong Duality
Image Space
Sufficient Conditions
Duality Theorems
Strong Theorems
Interior Point
Vector spaces
Normality
Equality
Minimise
Optimization problem
Duality
Intrinsic
Saddlepoint

Cite this

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Sufficient conditions for the existence of multipliers and Lagrangian duality in abstract optimization problems. / Tamminen, Eero.

In: Journal of Optimization Theory and Applications, Vol. 82, No. 1, 1994, p. 93-104.

Research output: Contribution to journalArticleScientificpeer-review

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AB - We consider the following optimization problem: in an abstract set X, find and element x that minimizes a real function f subject to the constraints g(x)≤0 and h(x)=0, where g and h are functions from X into normed vector spaces. Assumptions concerning an overall convex structure for the problem in the image space, the existence of interior points in certain sets, and the normality of the constraints are formulated. A theorem of the alternative is proved for systems of equalities and inequalities, and an intrinsic multiplier rule and a Lagrangian saddle-point theorem (strong duality theorem) are obtained as consequences.

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