The Lagrange approach to infinite linear programs
type: Item
instance of: scholarly article
title: The Lagrange approach to infinite linear programs
zbMATH Open document ID: 0991.90131
DOI: 10.1007/BF02579088
author: J. Rigoberto Gabriel
author: Raquiel R. López-Martínez
author: Onésimo Hernández-Lerma
published in: Top
publication date: 2002-09-03T00:00:00Z
review text: Let \((X, Y)\) and \((Z, W)\) be two dual pairs of vector spaces and let \(X_+\) and \(Z_+\) be positive cones in \(X\) and \(Z\), respectively. Also, let \(Y_+:= X^*_+\subset Y\) and \(W_+:= Z^*_+\subset W\) the dual cones of \(X_+\) and \(Z_+\). The authors study a general, infinite-dimensional, inequality-constrained linear program \[ \text{IP:}\qquad \text{minimize }\{\langle x,c\rangle\mid Ax- b\in Z_+, x\in X_+\}, \] and the corresponding dual program \[ \text{IP}^*:\quad \text{minimize }\langle b,w\rangle\qquad\text{subject to:} -A^*w+ c\in Y_+,\;w\in W_+. \] They use the method of Lagrange multipliers to determine necessary and sufficient conditions for the strong duality of IP, which means that IP and \(\text{IP}^*\) are both solvable and that their optimal values coincide. The authors also show how the results for IP can be applied to programs with equality constraints and illustrate the results with applications to the general capacity and the mass transfer problems.
Mathematics Subject Classification ID: 90C46
Mathematics Subject Classification ID: 90C25
Mathematics Subject Classification ID: 90C08
zbMATH DE Number: 1740557
zbMATH Keywords: infinite-dimensional, inequality-constrained linear program
zbMATH Keywords: Lagrange multipliers
zbMATH Keywords: strong duality
zbMATH Keywords: equality constraints
reviewed by: I. M. Stancu-Minasian
MaRDI profile type: Publication
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cites work: An Infinite Linear Program with a Duality Gap
cites work: Duality for Stochastic Programming Interpreted as L. P. in $L_p $-Space
cites work: Constrained Discounted Dynamic Programming
cites work: Strong duality of the general capacity problem in metric spaces
cites work: Q4357206
cites work: Constrained Average Cost Markov Control Processes in Borel Spaces
cites work: Q4863593
cites work: Approximation Schemes for Infinite Linear Programs
cites work: Q4255598
cites work: Semi-Infinite Programming: Theory, Methods, and Applications
cites work: Q5838558
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cites work: Extremal points and optimal solutions for general capacity problems
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cites work: Q3999495
cites work: Q4050397
cites work: Lagrange Multipliers and Optimality
cites work: Duality in infinite dimensional linear programming
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description: scientific article; zbMATH DE number 1740557