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  • 1
    ISSN: 1432-0606
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract The Farkas-Minkowski systems are characterized through a convex cone associated to the system, and some sufficient conditions are given that guarantee the mentioned property. The role of such systems in semi-infinite programming is studied in the linear case by means of the duality, and, in the nonlinear case, in connection with optimality conditions. In the last case the property appears as a constraint qualification.
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  • 2
    ISSN: 1432-5217
    Keywords: Convex programming ; semi-infinite programming ; linear representation of convex sets ; duality diagrams
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics , Economics
    Description / Table of Contents: Zusammenfassung Die Optimierung einer linearen Funktion auf einer konvexen abgeschlossenen MengeF kann als semi-infinites lineares Programm aufgefaßt werden, daF als Durchschnitt (unendlich) vieler Halbräume dargestellt werden kann. Es werden Dualitätseigenschaften dieser Programme untersucht, wobei von verschiedenen linearen Darstellungen fürF ausgegangen wird. Die Arbeit enthält Sätze über Dualitätsbeziehungen von Farkas-Minkowski, kanonisch abgeschlossene, kompakte und abgeschlossene Systeme. Es werden auch umgekehrte Beziehungen angegeben.
    Notes: Abstract The optimization of a linear function on a closed convex set,F, can be stated as a linear semi-infinite program, sinceF is the solution set of (usually) infinite linear inequality systems, the so-called linear representations ofF. The duality properties of these programs are analyzed when the linear representation ofF ranges in some well known classes of linear inequality systems. This paper provides propositions on the duality diagrams of Farkas-Minkowski, canonically closed, compact and closed systems. Converse statements are also given.
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  • 3
    ISSN: 1573-2878
    Keywords: Linear inequality systems ; convex sets ; semi-infinite programming ; purification algorithms
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract In many interesting semi-infinite programming problems, all the constraints are linear inequalities whose coefficients are analytical functions of a one-dimensional parameter. This paper shows that significant geometrical information on the feasible set of these problems can be obtained directly from the given coefficient functions. One of these geometrical properties gives rise to a general purification scheme for linear semi-infinite programs equipped with so-called analytical constraint systems. It is also shown that the solution sets of such kind of consistent systems form a transition class between polyhedral convex sets and closed convex sets in the Euclidean space of the unknowns.
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  • 4
    ISSN: 1573-2878
    Keywords: Semi-infinite programming ; optimality conditions ; optimal value function ; differentiable convex functions
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract In this paper, we establish different conditions for the uniqueness of the optimal solution of a semi-infinite programming problem. The approach here is based on the differentiability properties of the optimal value function and yields the corresponding extensions to the general linear semi-infinite case of many results provided by Mangasarian and others. In addition, detailed optimality conditions for the most general problem are supplied, and some features of the optimal set mapping are discussed. Finally, we obtain a dimensional characterization of the optimal set, provided that a usual closedness condition (Farkas-Minkowski condition) holds.
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  • 5
    ISSN: 1573-2878
    Keywords: Semi-infinite programming ; duality gap ; discrete approximation ; subdifferential mapping ; convex analysis
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract In this paper, we provide a systematic approach to the main topics in linear semi-infinite programming by means of a new methodology based on the many properties of the sub-differential mapping and the closure of a given convex function. In particular, we deal with the duality gap problem and its relation to the discrete approximation of the semi-infinite program. Moreover, we have made precise the conditions that allow us to eliminate the duality gap by introducing a perturbation in the primal objective function. As a by-product, we supply different extensions of well-known results concerning the subdifferential mapping.
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  • 6
    ISSN: 1573-2878
    Keywords: Semi-infinite optimization ; topological stability ; lower and upper semicontinuity ; linear inequality systems
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract In this note, we analyze the relationship between the lower semicontinuity of the feasible set mapping for linear semi-infinite inequality systems and the so-called topological stability, which is held when the solution sets of all the systems obtained by sufficiently small perturbations of the data are homeomorphic to each other. This topological stability and its relation with the Mangasarian-Fromovitz constraints qualification have been studied deeply by Jongen et al. in Ref. 1. The main difference of our approach is that we are not assuming any kind of structure for the index set and, consequently, any particular property for the functional dependence between the inequalities and the associated indices. In addition, we deal with systems whose solution sets are not necessarily bounded.
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  • 7
    ISSN: 1573-2878
    Keywords: Linear optimization ; semi-infinite programming ; unicity ; strong unicity ; Farkas-Minkowski systems
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract This paper deals with the conditions for the uniqueness of the optimal solution of an optimization problem for which the objective function is linear and the feasible set is a closed convex set in a finite-dimensional space. Some of these conditions, such as strong unicity andw-unicity (a new transition concept), involve only the feasible set. Others are related to the properties of the chosen linear representation. To some extent, the paper surveys the literature about unicity and strong unicity in linear semi-infinite programming.
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  • 8
    Electronic Resource
    Electronic Resource
    Springer
    OR spectrum 10 (1988), S. 145-152 
    ISSN: 1436-6304
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics , Economics
    Description / Table of Contents: Zusammenfassung Die natürliche Verallgemeinerung des Simplexverfahrens zur Behandlung linearer Optimierungsaufgaben mit unendlich vielen Nebenbedingungen gilt für das Duale Problem. Obwohl der zulässige Bereich zwar konvex im Raume der verallgemeinerten endlichen Folgen ist, behält er viele Eigenschaften vom endlichen Fall, die grundlegend um eine geometrische Deutung des Austausch-schritts zu erzeugen sind. Das Problem der Bestimmung von neuen Basismengen wird ebenfalls behandelt.
    Notes: Summary The natural extension of the simplex method to linear optimization problems with infinitely many constraints applies to their dual problems. Although the feasible sets are convex sets in spaces of generalized finite sequences, they preserve many of the properties of polyhedral convex sets in finite dimensional spaces. These properties are fundamental in obtaining a geometrical interpretation of the pivot operation. The problem of finding a basic set is also analyzed.
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  • 9
    Electronic Resource
    Electronic Resource
    Springer
    OR spectrum 18 (1996), S. 209-217 
    ISSN: 1436-6304
    Keywords: Semi-infinite programming ; duality ; Semi-infinite Programmierung ; Dualität
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics , Economics
    Description / Table of Contents: Zusammenfassung Diese Arbeit unterstreicht die Anwendbarkeit des sogenannten Dualproblems von Haar in linearer semi-infiniter Optimierung und analysiert seine Eigenschaften. Dies geschieht im Hinblick auf eine Reduktion in ein gewöhnliches lineares Optimierungsproblem, eine sequentielle Approximation durch endliche Teilprobleme und auch zum Finden einer numerischen Lösung durch Verfahren der zulässigen Richtungen.
    Notes: Abstract This paper emphasizes the great potential applicability of the so-called Haar's dual problem, in linear semi-infinite programming, and analyzes its properties in order to its reduction to an ordinary linear program, its sequential approximation through finite subprograms, as well as to its numerical solution by feasible directions strategies.
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