Farkas' Lemma, Gale's Theorem, and Linear Programming: the Infinite Case in an Algebraic Way

Kód výsledku v IS VaVaI

<a href="https://www.isvavai.cz/riv?ss=detail&h=RIV%2F61988987%3A17310%2F12%3AA130164B" target="_blank" >RIV/61988987:17310/12:A130164B - isvavai.cz</a>

Výsledek na webu

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DOI - Digital Object Identifier

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Jazyk výsledku

angličtina

Název v původním jazyce

Farkas' Lemma, Gale's Theorem, and Linear Programming: the Infinite Case in an Algebraic Way

Popis výsledku v původním jazyce

We study a problem of linear programming in the setting of a vector space over a linearly ordered (possibly skew) field. The dimension of the space may be infinite. The objective function is a linear mapping into another linearly ordered vector space over the same field. In that algebraic setting, we recall known results: Farkas' Lemma, Gale's Theorem of the alternative, and the Duality Theorem for linear programming with finite number of linear constraints. Given that ``semi-infinite'' case, i.e. results for finite systems of linear inequalities in an infinite-dimensional space, we are motivated to consider the infinite case: infinite systems of linear inequalities in an infinite-dimensional space. In the described setting, we formulate an infinite variant of Farkas' Lemma along with an infinite variant of Gale's Theorem of the alternative. Finally, we formulate the problem of an infinite linear programming, its dual problem, and the Duality Theorem for the problems.

Název v anglickém jazyce

Farkas' Lemma, Gale's Theorem, and Linear Programming: the Infinite Case in an Algebraic Way

Popis výsledku anglicky

We study a problem of linear programming in the setting of a vector space over a linearly ordered (possibly skew) field. The dimension of the space may be infinite. The objective function is a linear mapping into another linearly ordered vector space over the same field. In that algebraic setting, we recall known results: Farkas' Lemma, Gale's Theorem of the alternative, and the Duality Theorem for linear programming with finite number of linear constraints. Given that ``semi-infinite'' case, i.e. results for finite systems of linear inequalities in an infinite-dimensional space, we are motivated to consider the infinite case: infinite systems of linear inequalities in an infinite-dimensional space. In the described setting, we formulate an infinite variant of Farkas' Lemma along with an infinite variant of Gale's Theorem of the alternative. Finally, we formulate the problem of an infinite linear programming, its dual problem, and the Duality Theorem for the problems.

Druh

J_x - Nezařazeno - Článek v odborném periodiku (Jimp, Jsc a Jost)

CEP obor

BB - Aplikovaná statistika, operační výzkum

OECD FORD obor

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Projekt

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Návaznosti

S - Specificky vyzkum na vysokych skolach

Rok uplatnění

2012

Kód důvěrnosti údajů

S - Úplné a pravdivé údaje o projektu nepodléhají ochraně podle zvláštních právních předpisů

Název periodika

Global Journal of Mathematical Sciences (GJMS)

ISSN

2164-3709

e-ISSN

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Svazek periodika

1

Číslo periodika v rámci svazku

1

Stát vydavatele periodika

US - Spojené státy americké

Počet stran výsledku

6

Strana od-do

18-23

Kód UT WoS článku

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EID výsledku v databázi Scopus

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