Abstract
The results of the study of improper linear programming problems are presented, in which the duality theory is essentially used and the approaches of I.I. Eremin (correction of incompatible constraints) and A.N. Tikhonov (creation of compatible systems of constraints equivalent in accuracy to given incompatible constraints). The problem of a stable solution to an approximate (and, possibly, improper) pair of mutually dual linear programming problems with a coefficient matrix of size \(m\times n\) is reduced to a Mathematical Programming problem of dimension \(m + n + 2\). The necessary and sufficient conditions for the existence of a solution and constructive formulas for its calculation are obtained. Computational experiments were carried out on a model Linear Programming problem with an approximate matrix and vectors of the right-hand side and the objective function, demonstrating the convergence of the obtained solutions to the normal solutions to direct and dual problems with a decrease in the level of data error.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Similar content being viewed by others
Notes
- 1.
The author of the theorem is Alexander Krasnikov.
References
Eremin, I.I., Mazurov, V.D., Astaf’ev, N.N.: Improper Problems of Linear and Convex Programming. Nauka, Moscow (1983). (in Russian)
Vasil’ev, F.P., Ivanitskii, A.Yu.: Linear Programming. Faktorial Press, Moscow (2008). (in Russian)
Tikhonov, A.N.: Approximate systems of linear algebraic equations. USSR Comput. Math. Math. Phys. 20(6), 10–22 (1980). (in Russian)
Tikhonov, A.N., Arsenin, V.Ja.: Methods for Solving ill-posed Problems, 3rd edn. Nauka, Moscow (1986). (in Russian)
Volkov, V.V., Erokhin, V.I.: Tikhonov solutions of approximately given systems of linear algebraic equations under finite perturbations of their matrices. Comput. Math. Math. Phys. 50(4), 589–605 (2010). https://doi.org/10.1134/S0965542510040032
Erokhin, V.I., Volkov, V.V.: On A. N. Tikhonov’s regularized least squares method. Vestn. St.-Peterburg Univ. Prikl. Mat. Inform. Protsessy Upr. 13(1), 4–16 (2017). (in Russian). https://doi.org/10.21638/11701/spbu10.2017.101
Erokhin, V.I.: Matrix correction of a dual pair of improper linear programming problems. Comput. Math. Math. Phys. 47(4), 564–578 (2007). https://doi.org/10.1134/S0965542507040033
Erokhin, V.I., Krasnikov, A.S., Khvostov, M.N.: Matrix corrections minimal with respect to the euclidean norm for linear programming problems. Autom. Remote Control 73(2), 219–231 (2012). https://doi.org/10.1134/S0005117912020026
Erokhin, V.I., Krasnikov, A.S., Khvostov, M.N.: On sufficient conditions for the solvability of linear programming problems under matrix correction of their constraints. Tr. Inst. Mat. Mekh. 19(2), 144–156 (2013). (in Russian)
Erokhin, V.I.: On some sufficient conditions for the solvability and unsolvability of matrix correction problems of improper linear programming problems. Tr. Inst. Mat. Mekh. 21(3), 110–116 (2015). (in Russian)
Erokhin, V.I., Krasnikov, A.S., Volkov, V.V., Khvostov, M.N.: Matrix correction minimal with respect to the euclidean norm of a pair of dual linear programming problems. In: CEUR Workshop Proceedings 9th, pp. 196–209 (2016)
Erokhin, V.I.: A stable solution of linear programming problems with the approximate matrix of coefficients. In: 2017 Constructive Nonsmooth Analysis and Related Topics (dedicated to the memory of V.F. Demyanov) (CNSA), St. Petersburg, Russia, pp. 88–90. IEEE (2017). https://doi.org/10.1109/CNSA.2017.7973953
Volkov, V.V., Erokhin, V.I., Krasnikov, A.S., et al.: Minimum-euclidean-norm matrix correction for a pair of dual linear programming problems. Comput. Math. Math. Phys. 57(11), 1757–1770 (2017). https://doi.org/10.1134/S0965542517110148
Erokhin, V.I., Razumov, A.V., Krasnikov, A.S.: Tikhonov’s solution of approximate and improper LP problems. In: IX Moscow International Conference on Operations Research (ORM 2018), Proceedings, vol. I, pp. 131–136. MAKS Press, Moscow (2018). https://doi.org/10.29003/m211.ORM2018_v1
Eremin, I.I., Makarova, D.A., Schultz, L.V.: Questions of stability and regularization of improper linear programming problems. In: Proceedings of the Ural State University, vol. 30, pp. 43–62 (2004)
Gorelik, V., Zolotova, T.: Approximation of the improper linear programming problem with restriction on the norm of the correction matrix of the left-hand side of the constraints. In: 2018 IX International Conference on Optimization and Applications (OPTIMA 2018), pp. 14–24. DEStech Transactions on Computer Science and Engineering (2018). https://doi.org/10.12783/dtcse/optim2018/27918
Shary, S.P.: Interval Regularization for Imprecise Linear Algebraic Equations. arXiv preprint arXiv:1810.01481 (2018)
Artem’eva, L.A., Vasil’ev, F.P., Potapov, M.M.: Extragradient method for correction of inconsistent linear programming problems. Comput. Math. Math. Phys. 58(12), 1919–1925 (2018). https://doi.org/10.1134/S0965542518120163
Popov, L.D.: Methods of interior points, adapted to improper linear programming problems. Tr. Inst. Mat. Mekh. 24(4), 208–216 (2018). (in Russian). https://doi.org/10.21538/0134-4889-2018-24-4-208-210
Skarin, V.D.: Method of penalty functions and regularization in the analysis of improper problems of convex programming. Tr. Inst. Mat. Mekh. 24(3), 187–199 (2018). (in Russian). https://doi.org/10.21538/0134-4889-2018-24-3-187-199
Vatolin, A.A.: Approximation of improper linear programming problems using euclidean norm criterion. USSR Comput. Math. Math. Phys. Fiz. 24(12), 1907–1908 (1984). (in Russian)
Gorelik, V.A.: Matrix correction of a linear programming problem with inconsistent constraints. Comput. Math. Math. Phys. 41(12), 1615–1622 (2001)
Amirkhanova, G.A., Golikov, A.I., Evtushenko, Yu. G.: On an inverse linear programming problem. Proc. Steklov Inst. Math. 295(1), 21–27 (2016)
Agayan, G.M., Ryutin, A.A., Tikhonov, A.N.: The problem of linear programming with approximate data. USSR Comput. Math. Math. Phys. 24(5), 14–19 (1984). (in Russian)
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2019 Springer Nature Switzerland AG
About this paper
Cite this paper
Erokhin, V., Sotnikov, S., Kadochnikov, A., Vaganov, A. (2019). Regularization and Matrix Correction of Improper Linear Programming Problems. In: Bykadorov, I., Strusevich, V., Tchemisova, T. (eds) Mathematical Optimization Theory and Operations Research. MOTOR 2019. Communications in Computer and Information Science, vol 1090. Springer, Cham. https://doi.org/10.1007/978-3-030-33394-2_22
Download citation
DOI: https://doi.org/10.1007/978-3-030-33394-2_22
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-030-33393-5
Online ISBN: 978-3-030-33394-2
eBook Packages: Computer ScienceComputer Science (R0)