Hostname: page-component-586b7cd67f-2brh9 Total loading time: 0 Render date: 2024-11-22T07:20:08.627Z Has data issue: false hasContentIssue false
  • English
  • Français

POINTWISE RESIDUAL METHOD FOR SOLVING PRIMAL AND DUAL ILL-POSED LINEAR PROGRAMMING PROBLEMS WITH APPROXIMATE DATA

Published online by Cambridge University Press:  12 January 2021

A. Y. IVANITSKIY Open the ORCID record for A. Y. IVANITSKIY [Opens in a new window] ,
V. V. EJOV Open the ORCID record for V. V. EJOV [Opens in a new window]  and
F. P. VASILYEV Open the ORCID record for F. P. VASILYEV [Opens in a new window]
A. Y. IVANITSKIY*
Affiliation:
Faculty of Applied Mathematics, Physics and Information Technology, Chuvash State University, Cheboksary, Russia.
V. V. EJOV
Affiliation:
Mathematical Sciences Laboratory at Flinders University of South Australia, College of Science and Engineering, 1284 South Road, Tonsley, SA5042, Australia; e-mail: [email protected]. Department of Mathematical Analysis, Faculty of Computational Mathematics and Cybernetics, Moscow State University, Leninskie Gory, 1, Moscow119234, Russia; e-mail: [email protected].
F. P. VASILYEV
Affiliation:
Department of Mathematical Analysis, Faculty of Computational Mathematics and Cybernetics, Moscow State University, Leninskie Gory, 1, Moscow119234, Russia; e-mail: [email protected].
*
e-mail: [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

We propose a variation of the pointwise residual method for solving primal and dual ill-posed linear programming with approximate data, sensitive to small perturbations. The method leads to an auxiliary problem, which is also a linear programming problem. Theorems of existence and convergence of approximate solutions are established and optimal estimates of approximation of initial problem solutions are achieved.

Keywords

ill-posed linear programming problemsapproximate datapointwise residual methodprimal and dual linear programming problems

MSC classification

Primary: 90C05: Linear programming
Type
Research Article
Information
The ANZIAM Journal , Volume 62 , Issue 3 , July 2020 , pp. 302 - 317
Check for updates
Copyright
© Australian Mathematical Society 2021

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Abramowitz, M. and Stegun, I. A. (eds), Handbook of mathematical functions with formulas, graphs and mathematical tables, 9th printing, (Dover, New York, 1972) 886; doi:10.1119/1.15378.Google Scholar
Arfken, G. B., Weber, H. J. and Harris, F. E., Mathematical methods for physicists, 7th edn, (Academic Press, Oxford, 2012); doi:10.1016/C2009-0-30629-7.Google Scholar
Avrachenkov, K., Burachik, R. S., Filar, J. A. and Gaitsgory, V., “Constraint augmentation in pseudo-singularity perturbed linear programs”, Math. Program. Ser. A 132 (2012) 179–208; doi:10.1007/s10107-010-0388-0.CrossRefGoogle Scholar
Dantzig, G. B., “Maximization of a linear function of variables subject to linear inequalities”, in: Activity analysis of production and allocation (ed. Koopmans, T. C.), (Wiley, New York, 1951) 339–347.Google Scholar
Hoffman, A., “On approximate solutions of system of linear inequalities”, J. Res. Natl. Bur. Stand. 4 (1952) 263–265; doi:10.6028/jres.049.027.CrossRefGoogle Scholar
Ivanitskiy, A. Y., Stable methods for solving systems of linear equations and inequalities with interval coefficients: candidate dissertation of physical and mathematical sciences (Moscow University Press, Moscow, 1998) (in Russian).Google Scholar
Ivanitskiy, A. Y., Vasilyev and, F. P. Morozov, V. A., “Estimation of speed of convergence of residual method for linear programming problems”, Zh. Vychisl. Mat. Mat. Fiz. 38 (1990) 1257–1262.Google Scholar
Kantorovich, L. V., “Mathematical methods of organizing and planning production” (English translation), Manag. Sci. 6 (1960) 366–422; doi:10.1287/mnsc.6.4.366.CrossRefGoogle Scholar
Karmakar, N., “A new polynomial-time algorithm for linear programming”, Combinatorica 4 (1984) 373–395; doi:10.1007/BF02579150.CrossRefGoogle Scholar
Khachiyan, L., “Polynomial algorithms in linear programming”, Dokl. Akad. Nauk SSSR 244 (1979) 1093–1096; doi:10.1016/0041-5553(80)90061-0.Google Scholar
Leonov, A. S., Solving ill-posed inverse problems, (URSS, Moscow, 2009) (in Russian).Google Scholar
Morozov, V. A., “Stable numerical methods for solving simultaneous systems of linear algebraic equations”, Zh. Vychisl. Mat. Mat. Fiz. 24 (1984) 179–186, available at https://www.sciencedirect.com/sdfe/pdf/download/eid/1-s2.0-0041555384901290/first-page-pdf.Google Scholar
Nesterov, Y. and Nemirovskiy, A., Interior-point polynomial algorithms in convex programming, (John Wiley, New York, 1983); doi:10.1137/1.9781611970791.Google Scholar
Renegar, J., “Some perturbation theory for linear programming”, Math. Program. 65 (1994) 73–91; doi:10.1007/BF01581690.CrossRefGoogle Scholar
Tikhonov, A. N., “On some problems of optimal planning”, Zh. Vychisl. Mat. Mat. Fiz. 6 (1966) 81–89.Google Scholar
Tikhonov, A. N., Karmanov, V. G. and Rudneva, T. L., “On the stability of linear programming”, in: Collection of works of Research Computing Center of Moscow University “Computational methods and programming”, Volume 12, (Moscow University Press, Moscow, 1969) 3–9 (in Russian).Google Scholar
Tikhonov, A. N., Ryutin, A. A. and Agayan, G. M., “A stable method of solving a linear programming problem with approximate data”, Dokl. Akad. Nauk SSSR 272 (1983) 1058–1063, available at http://mi.mathnet.ru/eng/dan9923.Google Scholar
Titarenko, V. N. and Yagola, A., “The problems of linear and quadratic programming for ill-posed problems on some compact sets”, J. Inverse Ill Posed Probl. 11 (2003) 311–328; doi:10.1515/156939403769237079.CrossRefGoogle Scholar
Vanderbei, R. J., Linear programming: foundations and extensions, Volume 196 of Int. Ser. in Oper. Res. & Manag. Sci., (Springer, New York, 2014); doi:10.1007/978-1-4614-7630-6.CrossRefGoogle Scholar
Vasilyev, F. P., Methods for optimizations, (Faktorial Press, Moscow, 2002).Google Scholar
Vasilyev, F. P. and Ivanitskiy, A. Y., In-depth analysis of linear programming, (Springer, Dordrecht, 2001); doi:10.1007/978-94-015-9759-3.CrossRefGoogle Scholar
Vasilyev, F. P., Morozov, V. A. and Yashimovich, M. D., “Estimate of speed of convergence of regularization method for linear programming problems”, Zh. Vychisl. Mat. Mat. Fiz. 29 (1989) 631–635; doi:10.1016/0041-5553(89)90032-3.Google Scholar
von Neumann, J., “Zur Theorie der Gesellschaftsspiele”, Math. Ann. 100 (1928) 295–320; doi:10.1007/BF01448847.CrossRefGoogle Scholar