Approaches to finding optimal solutions in high-dimensional problems using duality theory

M.N. Belozyorov; E.A. Kalashnikov

doi:10.18127/j19998554-202601-03

Approaches to finding optimal solutions in high-dimensional problems using duality theory

M.N. Belozyorov¹, E.A. Kalashnikov²

research article UDC 519.852, 004.42 Pages: 32-37 doi.org/10.18127/j19998554-202601-03

¹ Sberbank of Russia PJSC (Moscow, Russia)
² National Research Technological University «MISIS» (Moscow, Russia)

Keywords: Linear programming optimal solution search high-dimensional problems duality theory

Abstract

Optimization problems based on numerical linear programming methods are used in various sectors of the economy. At the same time, there are a number of difficulties associated with this method. For example, an increase in the number of variables and constraints complicates the analysis; contradictory constraints require adjustment, etc.

The purpose of the research is to develop approaches to finding optimal solutions in high-dimensional linear programming problems based on the theory of duality.

The approaches have been proposed taking into account technological options and their subsets with the solution of a dual problem for an optimized matrix. The duality method is a powerful tool for solving optimization problems of high-dimensional linear programming. Solving a dual problem reduces computational complexity and simplifies analysis, as well as accelerates algorithm convergence.

Received: 12.05.2025
Approved after review: 21.07.2025
Accepted: 14.01.2026

How to cite

Белозеров М.Н., Калашников Е.А. Подходы к поиску оптимальных решений в задачах большой размерности с использованием теории двойственности // Нейрокомпьютеры: разработка, применение. 2026. Т. 28. № 1. С. 32–37. DOI: https://doi.org/10.18127/ j19998554-202601-03

References

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[1] Kibzun A.I., Rasskazova V.A. Model' tselochislennogo linejnogo programmirovaniya kak matematicheskoe obespechenie sistemy optimal'nogo planirovaniya potokovogo proizvodstva na etape operativnogo grafikovaniya. Avtomatika i telemekhanika. 2023. № 5. S. 113–132. (in Russian)

[2] Arkhipova V.I. Linejnoe programmirovanie v ekonomike. Innovatsionnye nauchnye issledovaniya v sovremennom mire: teoriya, metodologiya, praktika. 2022. S. 8–14. (in Russian)

[3] Sivakova T.V., Sudakov V.A., Shimko V.S. Issledovanie metodov resheniya zadach smeshannogo tselochislennogo linejnogo programmirovaniya. Preprinty Instituta prikladnoj matematiki im. M.V. Keldysha RAN. 2024. № 24. S. 1–18. (in Russian)

[4] Yakubova U.Sh., Mirkhodzhaeva N.Sh., Parpieva N.T. Nekotorye primeneniya teorii dvojstvennosti pri reshenii zadach linejnogo programmirovaniya. Byulleten' nauki i praktiki. 2022. T. 8. № 5. S. 621–628. (in Russian)

[5] Gurbanberdieva A., Atadzhanova B.E. Ispol'zovanie linejnogo programmirovaniya dlya resheniya zadach raspredeleniya resursov. Vsemirnyj uchenyj. 2024. T. 1. № 18. S. 82–87. (in Russian)

[6] Fedorova G.V., Makarycheva I.V., Shishkanova G.A. Prinyatie upravlencheskikh reshenij na osnove metodov linejnogo programmirovaniya. Aktual'nye problemy upravleniya. 2022. S. 125–129. (in Russian)

[7] Cong X. et al. Decentralized diagnosis by Petri nets and integer linear programming. IEEE Transactions on Systems, Man, and Cybernetics Systems. 2017. V. 48. № 10. P. 1689–1700.

[8] Dorfman R. Application of linear programming to the theory of the firm: including an analysis of monopolistic firms by non-linear programming. University of California Press. 2022.

[9] Cohen M.B., Lee Y.T., Song Z. Solving linear programs in the current matrix multiplication time. Journal of the ACM (JACM). 2021. V. 68. № 1. P. 1–39.

[10] Jdid M., Salama A.A., Khalid H.E. Neutrosophic handling of the simplex direct algorithm to define the optimal solution in linear programming. International Journal of Neutrosophic Science (IJNS). 2022. V. 18. № 1. P. 30–41.

[11] Singaravelan A. et al. Application of two-phase simplex method (TPSM) for an efficient home energy management system to reduce peak demand and consumer consumption cost. IEEE Access. 2021. V. 9. P. 63591–63601.

[12] Antoniou A. et al. Linear programming. Part I: The simplex method. Practical Optimization: Algorithms and Engineering Applications. 2021. P. 339–392.

[13] Sigalo M.B. et al. An energy management system for the control of battery storage in a grid-connected microgrid using mixed integer linear programming. Energies. 2021. V. 14. № 19. P. 6212.

[14] Liu S. et al. An enhanced branch-and-bound algorithm for bilevel integer linear programming. European Journal of Operational Research. 2021. V. 291. № 2. P. 661–679.

[15] Putz D. et al. A comparison between mixed-integer linear programming and dynamic programming with state prediction as novelty for solving unit commitment. International Journal of Electrical Power & Energy Systems. 2021. V. 125. P. 106426.

Abstract

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