Algebraic duality theorems for infinite LP problems
In this paper, we consider a primal–dual infinite linear programming problem-pair, i.e. LPs on infinite dimensional spaces with infinitely many constraints. We present two duality theorems for the problem-pair: a weak and a strong duality theorem. We do not assume any topology on the vector spaces,...
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| Veröffentlicht in: | Linear algebra and its applications Jg. 434; H. 3; S. 688 - 693 |
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| Format: | Journal Article |
| Sprache: | Englisch |
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01.02.2011
Elsevier |
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| ISSN: | 0024-3795 |
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| Abstract | In this paper, we consider a primal–dual infinite linear programming problem-pair, i.e.
LPs on infinite dimensional spaces with infinitely many constraints. We present two duality theorems for the problem-pair: a weak and a strong duality theorem. We do not assume any topology on the vector spaces, therefore our results are algebraic duality theorems. As an application, we consider transferable utility cooperative games with arbitrarily many players. |
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| AbstractList | In this paper, we consider a primal–dual infinite linear programming problem-pair, i.e.
LPs on infinite dimensional spaces with infinitely many constraints. We present two duality theorems for the problem-pair: a weak and a strong duality theorem. We do not assume any topology on the vector spaces, therefore our results are algebraic duality theorems. As an application, we consider transferable utility cooperative games with arbitrarily many players. |
| Author | Pintér, Miklós |
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| Cites_doi | 10.1016/0022-247X(72)90045-5 10.1002/nav.3800140404 10.1007/BF01753431 10.1016/0022-247X(69)90044-4 10.21236/AD0655106 |
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| Keywords | TU games with infinitely many players Bondareva–Shapley theorem Infinite dimensional duality theorems Exact games Core Vector space Constraint Primal dual method Linear programming Duality Infinite dimension Topology Bondareva-Shapleytheorem |
| Language | English |
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| References | Anderson, Nash (b0010) 1987 Shapley (b0065) 1967; 14 Peleg, Sudhlter (b0050) 2003 Gale, Kuhn, Tucker (b0030) 1951 P. Csóka, P.J.J. Herings, L.Á. Kóczy, Balancedness conditions for exact games, METEOR Research Memorandum RM07/040 (2007) 1–13. Kannai (b0045) 1992; vol. 1 Schmeidler (b0060) 1972; 40 Aliprantis, Border (b0005) 1999 Gillies (b0035) 1959; vol. IV Kannai (b0040) 1969; 27 Fan (b0025) 1956; vol. 38 Bondareva (b0015) 1963; 10 Shapley (b0070) 1971; 1 D. Schmeidler, On balanced games with infinitely many players, Mimmeographed, RM-28, Department of Mathematics, The Hebrew University, Jerusalem, 1967. Aliprantis (10.1016/j.laa.2010.09.007_b0005) 1999 Gillies (10.1016/j.laa.2010.09.007_b0035) 1959; vol. IV Shapley (10.1016/j.laa.2010.09.007_b0070) 1971; 1 Shapley (10.1016/j.laa.2010.09.007_b0065) 1967; 14 Kannai (10.1016/j.laa.2010.09.007_b0040) 1969; 27 Fan (10.1016/j.laa.2010.09.007_b0025) 1956; vol. 38 Peleg (10.1016/j.laa.2010.09.007_b0050) 2003 Schmeidler (10.1016/j.laa.2010.09.007_b0060) 1972; 40 10.1016/j.laa.2010.09.007_b0020 Bondareva (10.1016/j.laa.2010.09.007_b0015) 1963; 10 10.1016/j.laa.2010.09.007_b0055 Anderson (10.1016/j.laa.2010.09.007_b0010) 1987 Gale (10.1016/j.laa.2010.09.007_b0030) 1951 Kannai (10.1016/j.laa.2010.09.007_b0045) 1992; vol. 1 |
| References_xml | – volume: vol. 1 year: 1992 ident: b0045 article-title: The core and balancedness publication-title: Handbook of Game Theory with Economic Applications – year: 1987 ident: b0010 article-title: Linear Programming in Infinite-Dimensional Spaces, Theory and Applications – volume: vol. 38 start-page: 99 year: 1956 end-page: 156 ident: b0025 article-title: On systems of linear inequalities publication-title: Linear Inequalities and Related Systems, Annals of Mathematics Studies – reference: D. Schmeidler, On balanced games with infinitely many players, Mimmeographed, RM-28, Department of Mathematics, The Hebrew University, Jerusalem, 1967. – volume: 1 start-page: 11 year: 1971 end-page: 26 ident: b0070 article-title: Cores of convex games publication-title: Internat. J. Game Theory – volume: vol. IV year: 1959 ident: b0035 article-title: Solutions to general non-zero-sum games publication-title: Contributions to the Theory of Games – year: 1999 ident: b0005 article-title: Infinite Dimensional Analysis – volume: 27 start-page: 227 year: 1969 end-page: 240 ident: b0040 article-title: Countably additive measures in cores of games publication-title: J. Math. Anal. Appl. – volume: 40 start-page: 214 year: 1972 end-page: 225 ident: b0060 article-title: Cores of exact games publication-title: J. Math. Anal. Appl. – volume: 14 start-page: 453 year: 1967 end-page: 460 ident: b0065 article-title: On balanced sets and cores publication-title: Naval Res. Logist. Quart. – volume: 10 start-page: 119 year: 1963 end-page: 139 ident: b0015 article-title: Some applications of linear programming methods to the theory of cooperative games publication-title: Problemy Kybernet. – year: 1951 ident: b0030 article-title: Linear programming and the theory of games publication-title: Activity Analysis of Production and Allocation – reference: P. Csóka, P.J.J. Herings, L.Á. Kóczy, Balancedness conditions for exact games, METEOR Research Memorandum RM07/040 (2007) 1–13. – year: 2003 ident: b0050 article-title: Introduction to the Theory of Cooperative Games – year: 2003 ident: 10.1016/j.laa.2010.09.007_b0050 – volume: 10 start-page: 119 year: 1963 ident: 10.1016/j.laa.2010.09.007_b0015 article-title: Some applications of linear programming methods to the theory of cooperative games publication-title: Problemy Kybernet. – year: 1951 ident: 10.1016/j.laa.2010.09.007_b0030 article-title: Linear programming and the theory of games – volume: 40 start-page: 214 year: 1972 ident: 10.1016/j.laa.2010.09.007_b0060 article-title: Cores of exact games publication-title: J. Math. Anal. Appl. doi: 10.1016/0022-247X(72)90045-5 – ident: 10.1016/j.laa.2010.09.007_b0020 – volume: 14 start-page: 453 year: 1967 ident: 10.1016/j.laa.2010.09.007_b0065 article-title: On balanced sets and cores publication-title: Naval Res. Logist. Quart. doi: 10.1002/nav.3800140404 – volume: 1 start-page: 11 year: 1971 ident: 10.1016/j.laa.2010.09.007_b0070 article-title: Cores of convex games publication-title: Internat. J. Game Theory doi: 10.1007/BF01753431 – year: 1999 ident: 10.1016/j.laa.2010.09.007_b0005 – volume: vol. 38 start-page: 99 year: 1956 ident: 10.1016/j.laa.2010.09.007_b0025 article-title: On systems of linear inequalities – volume: vol. 1 year: 1992 ident: 10.1016/j.laa.2010.09.007_b0045 article-title: The core and balancedness – year: 1987 ident: 10.1016/j.laa.2010.09.007_b0010 – volume: 27 start-page: 227 year: 1969 ident: 10.1016/j.laa.2010.09.007_b0040 article-title: Countably additive measures in cores of games publication-title: J. Math. Anal. Appl. doi: 10.1016/0022-247X(69)90044-4 – ident: 10.1016/j.laa.2010.09.007_b0055 doi: 10.21236/AD0655106 – volume: vol. IV year: 1959 ident: 10.1016/j.laa.2010.09.007_b0035 article-title: Solutions to general non-zero-sum games |
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LPs on infinite dimensional spaces with infinitely many constraints. We... |
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| SubjectTerms | Algebra Bondareva–Shapley theorem Core Exact games Exact sciences and technology Infinite dimensional duality theorems Linear and multilinear algebra, matrix theory Mathematics Sciences and techniques of general use TU games with infinitely many players |
| Title | Algebraic duality theorems for infinite LP problems |
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