Investigation of mathematical models stability and geometry configurations
Authors: Gordeev E.N. | Published: 12.10.2015 |
Published in issue: #5(104)/2015 | |
DOI: 10.18698/0236-3933-2015-5-61-74 | |
Category: Informatics, Computer Engineering and Control | Chapter: Theoretical Computer Science, Cybernetics | |
Keywords: discrete optimization problems, theory of stability, radius of stability, mathematical modelling, computational geometry, parametric programming |
The article discusses application of the theory of stability, previously developed for solving discrete optimization problems. The theory allows considering two types of the applied problems arising during a networks modelling. The modelled process P occurs in time and has several components K_{1},...,K_{s}. Its mathematical models are presented as optimization problems, parametric programming problems or computational geometry problems Z_{1},...,Z_{s}. A practical question arises if there is any relationship between the model and real process. The theory of stability is used in mathematical modelling because it allows linking various components of the process with the help of "uniform" formulae, algorithms and convincingly indicating "bottlenecks" of the model. While analyzing properties of the geometric configurations, the proposed approach allows identifying the "critical" situations. By virtue of parameterization, the input data can be presented as time functions. This allows considering the models of some processes under certain conditions as well as drawing heuristic conclusions about the compatibility of the model with the simulated process. The article describes a general scheme of analyzing the stability analysis problem. It shows the application of this scheme and gives examples illustrating its application.
References
[1] Gordeev E.N., Leont’ev V.K. The General Approach to the Study of the Solution Stability in Discrete Optimization Problems. Zhurnal Vychislit. Mat. i Mat. Fiz. [J. Computational Mathematics and Mathematical Physics], 1996, no. 36, pp. 66-72 (in Russ.).
[2] Leont’ev V.K. Ustoychivost’ v lineynykh diskretnykh zadachakh. Vkn.: Problemy kibernetiki [Stability in Linear Discrete Problems. In the book: Problems of Cybernetics]. Moscow, Nauka Publ., 1979, Iss. 35, pp. 169-185.
[3] Gordeev E.N. Algorithms of Polynomial Complexity for Computing the Radius of Stability in Two Classes of Trajectory Problems. Zhurnal Vychislit. Mat. i Mat. Fiz. [J. Computational Mathematics and Mathematical Physics], 1987, vol. 27, no. 7, pp. 984-992(in Russ.).
[4] Gordeev E.N. Stability of Solutions to the Problem of the Shortest Path on a Graph. Diskretnaya matematika [Discrete Mathematics and Applications], 1989, vol. 1, no. 3, pp. 39-46 (in Russ.).
[5] Leont’ev V.K., Gordeev E.N. Kachestvennoe issledovanie traektornykh zadach. [Qualitative Research of Trajectory Problems]. Kibernetika i sistemnyy analiz [Cybernetics and Systems Analysis], 1986, no. 5, pp. 82-90 (in Russ.).
[6] Gordeev E.N. On the Stability of Solutions in Problems of Computational Geometry. Abstracts of International Scientific Conference "Intelligent Information Processing." Crimean Academy of Sciences, 1996, p. 8 (in Russ.).
[7] Vyalyy M.N., Gordeev E.N., Tarasov S.P. On the Stability of the Voronoi Diagram. Zhurnal Vychislit. Mat. i Mat. Fiz. [J. Computational Mathematics and Mathematical Physics], 1996, vol. 36, no. 3, pp. 405-414 (in Russ.).
[8] Artemenko V.I., Gordeev E.N., Zhuravlev Yu.I. The Method of Constructing the Optimal Program Trajectories for the Robot Manipulator. Kibernetika i sistemnyy analiz [Cybernetics and Systems Analysis], 1986, no. 5, pp. 84-107 (in Russ.).
[9] Gordeev E.N., Leont’ev V.K. Trajectory Parametric Problems. Zhurnal Vychislit. Mat. i Mat. Fiz. [J. Computational Mathematics and Mathematical Physics], 1984, no. 24, pp. 37-46 (in Russ.).
[10] Gordeev E.N. The use of the stability radius of optimization problems to hide and verify correctness of information. Jelektr. Nauchno-Tehn. Izd. "Inzhenernyy zhurnal: nauka i innovacii" MGTU im. Baumana [El. Sc.-Techn. Publ. "Eng. J.: Science and Innovation" of Bauman MSTU], 2013, iss. 11. URL: http://engjournal.ru/catalog/it/hidden/993.html (accessed: 15.09.2014).
[11] Gordeev E.N., Lipkin L.I. O edinstvennosti resheniya nekotorykh kombinatornykh zadach vybora. Metody diskretnogo analiza. Sb. tr. [On the Uniqueness of Solutions of Some Combinatorial Problems of Selection. Methods of Discrete Analysis.]. Proc., 1989, Novosibirsk, iss. 49, pp. 13-31 (in Russ.).
[12] Gordeev E.N. The Adequacy of Modeling Processes in Networks. Elektrosvyaz’ [Telecommunications], 1999, no. 8, pp. 16-21 (in Russ.).