Geodesic
Matrix correction of a dual pair of improper linear programming problems
Žurnal vyčislitelʹnoj matematiki i matematičeskoj fiziki, Tome 47 (2007) no. 4, pp. 587-601 
V. I. Erokhin. Matrix correction of a dual pair of improper linear programming problems. Žurnal vyčislitelʹnoj matematiki i matematičeskoj fiziki, Tome 47 (2007) no. 4, pp. 587-601. http://geodesic.mathdoc.fr/item/ZVMMF_2007_47_4_a2/
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     title = {Matrix correction of a~dual pair of improper linear programming problems},
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Voir la notice de l'article provenant de la source Math-Net.Ru

A matrix is sought that solves a given dual pair of systems of linear algebraic equations. Necessary and sufficient conditions for the existence of solutions to this problem are obtained, and the form of the solutions is found. The form of the solution with the minimal Euclidean norm is indicated. Conditions for this solution to be a rank one matrix are examined. On the basis of these results, an analysis is performed for the following two problems: modifying the coefficient matrix for a dual pair of linear programs (which can be improper) to ensure the existence of given solutions for these programs, and modifying the coefficient matrix for a dual pair of improper linear programs to minimize its Euclidean norm. Necessary and sufficient conditions for the solvability of the first problem are given, and the form of its solutions is described. For the second problem, a method for the reduction to a nonlinear constrained minimization problem is indicated, necessary conditions for the existence of solutions are found, and the form of solutions is described. Numerical results are presented.

[1] Voevodin V. V., Kuznetsov Yu. A., Matritsy i vychisleniya, Nauka, M., 1984 | MR | Zbl

[2] Eremin I. I., Mazurov V. D., Astafev H. H., Nesobstvennye zadachi lineinogo i vypuklogo programmirovaniya, Nauka, M., 1983 | MR

[3] Vasilev F. P., Ivanitskii A. Yu., Lineinoe programmirovanie, Faktorial Press, M., 2003 | MR

[4] Ashmanov S. A., Timokhov A. B., Teoriya optimizatsii v zadachakh i uprazhneniyakh, Nauka, M., 1991 | Zbl

[5] Tikhonov A. N., “O normalnykh resheniyakh priblizhennykh sistem lineinykh algebraicheskikh uravnenii”, Dokl. AN SSSR, 254:3 (1980), 549–554 | MR | Zbl

[6] Tikhonov A. N., “O priblizhennykh sistemakh lineinykh algebraicheskikh uravnenii”, Zh. vychisl. matem. i matem. fiz., 20:6 (1980), 1373–1383 | MR | Zbl

[7] Tikhonov A. N., Ryutin A. A., Agayan G. M., “Ob ustoichivom metode resheniya zadachi lineinogo programmirovaniya s priblizhennymi dannymi”, Dokl. AN SSSR, 272:5 (1983), 1058–1063 | MR | Zbl

[8] Vatolin A. A., “Approksimatsiya nesobstvennykh zadach lineinogo programmirovaniya po kriteriyu evklidovoi normy”, Zh. vychisl. matem. i matem. fiz., 24:12 (1984), 1907–1908 | MR | Zbl

[9] Gorelik V. A., “Matrichnaya korrektsiya zadachi lineinogo programmirovaniya s nesovmestnoi sistemoi ogranichenii”, Zh. vychisl. matem. i matem. fiz., 41:11 (2001), 1697–1705 | MR | Zbl

[10] Gorelik V. A., Erokhin V. I., Muraveva O. V., “Nekotorye zadachi approksimatsii matrits koeffitsientov nesovmestnykh sistem lineinykh uravnenii i nesobstvennykh zadach lineinogo programmirovaniya”, Modelirovanie, dekompozitsiya i optimizatsiya slozhnykh dinamich. protsessov, VTs RAN, M., 2001, 57–88 | MR

[11] Erokhin V. I., “Svoistva optimalnoi odnorangovoi korrektsii matrits koeffitsientov nesovmestnykh neodnorodnykh lineinykh modelei”, Diskretnyi analiz i issl. operatsii. Ser. 2, 9:1 (2002), 33–60 | MR

[12] Gorelik V. A., Muraveva O. V., “Matrichnaya korrektsiya dannykh v zadachakh optimizatsii i klassifikatsii”, Modelirovanie, dekompozitsiya i optimizatsiya slozhnykh dinamich. protsessov, VTs RAN, M., 2004, 94–120 | MR

[13] Gorelik V. A., Erokhin V. I., Pechenkin R. V., “Optimalnaya matrichnaya korrektsiya nesovmestnykh sistem lineinykh algebraicheskikh uravnenii s blochnymi matritsami koeffitsientov”, Diskretnyi analiz i issl. operatsii. Ser. 2, 12:2 (2005), 3–23 | MR

[14] Erokhin V. I., “Optimalnaya matrichnaya korrektsiya i regulyarizatsiya nesovmestnykh lineinykh modelei”, Diskretnyi analiz i issl. operatsii. Ser. 2, 9:2 (2002), 41–77 | MR | Zbl

[15] Gorelik V. A., Erokhin V. I., “Optimalnaya (po minimumu poliedralnoi normy) matrichnaya korrektsiya nesovmestnykh sistem lineinykh algebraicheskikh uravnenii i nesobstvennykh zadach lineinogo programmirovaniya”, Modelirovanie, dekompozitsiya i optimizatsiya slozhnykh dinamich. protsessov, VTs RAN, M., 2004, 35–63 | MR

[16] Erokhin V. I. Lemma A. N., “Tikhonova i ee obobscheniya”, Tikhonov i sovremennaya matematika: Obratnye i nekorrektno postavlennye zadachi, Tezisy dokl. Mezhdunar. konf., MGU, Vychisl. matem. i kibernetika, M., 2006, 52–53 | MR

[17] Eremin I. I., Astafev H. H., Vvedenie v teoriyu lineinogo i vypuklogo programmirovaniya, Nauka, M., 1976 | MR

[18] Popov L. D., “Ob odnoetapnom metode resheniya leksikograficheskikh variatsionnykh neravenstv”, Izv. vuzov. Matematika, 12(439) (1998), 71–81 | Zbl