Development of iterative algorithms for solving the inverse problem using inverse calculations
DOI:
https://doi.org/10.15587/1729-4061.2020.205048Keywords:
inverse calculations, function optimization, nonlinear programming, gradient method, inverse problem.Abstract
Iterative algorithms for solving the inverse problem, presented as a quadratic programming problem, developed by modifying algorithms based on the inverse calculation mechanism are proposed. Iterative algorithms consist in a sequential change of the argument values using iterative formulas until the function reaches the value that most corresponds to the constraint. Two solutions are considered: by determining the shortest distance to the line of the given level determined by the constraint, and by moving along the gradient. This approach was also adapted to solve more general nonlinear programming optimization problems. The solution of four problems is considered: formation of production output and storage costs, optimization of the securities portfolio and storage costs for the given volume of purchases. It is shown that the solutions obtained using iterative algorithms are consistent with the result of using classical methods (Lagrange multiplier, penalty), standard function of the MathCad package. In this case, the greatest degree of compliance was obtained using the method based on constructing the level line; the method based on moving along the gradient is more universal.
The advantage of the algorithms is a simpler computer implementation of iterative formulas, the ability to get a solution in less time than known methods (for example, the penalty method, which requires multiple optimizations of a modified function with a change in the penalty parameter). The algorithms can also be used to solve other nonlinear programming problems of the presented kind.
The paper can be useful for specialists when solving problems in the field of economics, as well as developing decision support systems.
References
- Barmina, E. A., Kvyatkovskaya, I. Yu. (2010). Monitoring of quality of work of a commercial organization. Indicators structuring. Application of cognitive maps. Vestnik Astrakhanskogo gosudarstvennogo tekhnicheskogo universiteta, 2, 15–20.
- Odintsov, B. E. (2004). Obratnye vychisleniya v formirovanii ekonomicheskikh resheniy. Moscow: Finansy i statistika, 256.
- Zheng, G.-H., Zhang, Q.-G. (2018). Solving the backward problem for space-fractional diffusion equation by a fractional Tikhonov regularization method. Mathematics and Computers in Simulation, 148, 37–47. doi: https://doi.org/10.1016/j.matcom.2017.12.005
- Park, Y., Reichel, L., Rodriguez, G., Yu, X. (2018). Parameter determination for Tikhonov regularization problems in general form. Journal of Computational and Applied Mathematics, 343, 12–25. doi: https://doi.org/10.1016/j.cam.2018.04.049
- Bai, Z.-Z., Buccini, A., Hayami, K., Reichel, L., Yin, J.-F., Zheng, N. (2017). Modulus-based iterative methods for constrained Tikhonov regularization. Journal of Computational and Applied Mathematics, 319, 1–13. doi: https://doi.org/10.1016/j.cam.2016.12.023
- Wang, H., Yang, W., Guan, N. (2019). Cauchy sparse NMF with manifold regularization: A robust method for hyperspectral unmixing. Knowledge-Based Systems, 184, 104898. doi: https://doi.org/10.1016/j.knosys.2019.104898
- Scardapane, S., Comminiello, D., Hussain, A., Uncini, A. (2017). Group sparse regularization for deep neural networks. Neurocomputing, 241, 81–89. doi: https://doi.org/10.1016/j.neucom.2017.02.029
- Xu, J., Schreier, F., Doicu, A., Trautmann, T. (2016). Assessment of Tikhonov-type regularization methods for solving atmospheric inverse problems. Journal of Quantitative Spectroscopy and Radiative Transfer, 184, 274–286. doi: https://doi.org/10.1016/j.jqsrt.2016.08.003
- Gribanova, E. (2020). Algorithm for solving the inverse problems of economic analysis in the presence of limitations. EUREKA: Physics and Engineering, 1, 70–78. doi: https://doi.org/10.21303/2461-4262.2020.001102
- Qi, Y., Liu, D., Li, X., Lei, J., Xu, X., Miao, Q. (2020). An adaptive penalty-based boundary intersection method for many-objective optimization problem. Information Sciences, 509, 356–375. doi: https://doi.org/10.1016/j.ins.2019.03.040
- El-Sobky, B., Abo-Elnaga, Y. (2018). A penalty method with trust-region mechanism for nonlinear bilevel optimization problem. Journal of Computational and Applied Mathematics, 340, 360–374. doi: https://doi.org/10.1016/j.cam.2018.03.004
- Trunov, A. N. (2015). Modernization of means for analysis and solution of nonlinear programming problems. Quantitative Methods in Economics, 16 (2), 133–141.
- Li, J., Yang, Z. (2018). A QP-free algorithm without a penalty function or a filter for nonlinear general-constrained optimization. Applied Mathematics and Computation, 316, 52–72. doi: https://doi.org/10.1016/j.amc.2017.08.013
- Mitsel', A. A., Khvaschevskiy, A. N. (1999). Noviy algoritm resheniya zadachi kvadratichnogo programmirovaniya. Avtometriya, 3, 93–98.
- Morovati, V., Pourkarimi, L. (2019). Extension of Zoutendijk method for solving constrained multiobjective optimization problems. European Journal of Operational Research, 273 (1), 44–57. doi: https://doi.org/10.1016/j.ejor.2018.08.018
- Tsai, J.-T. (2015). Improved differential evolution algorithm for nonlinear programming and engineering design problems. Neurocomputing, 148, 628–640. doi: https://doi.org/10.1016/j.neucom.2014.07.001
- Hosseini, A. (2016). A non-penalty recurrent neural network for solving a class of constrained optimization problems. Neural Networks, 73, 10–25. doi: https://doi.org/10.1016/j.neunet.2015.09.013
- Darabi, A., Bagheri, M., Gharehpetian, G. B. (2020). Dual feasible direction-finding nonlinear programming combined with metaheuristic approaches for exact overcurrent relay coordination. International Journal of Electrical Power & Energy Systems, 114, 105420. doi: https://doi.org/10.1016/j.ijepes.2019.105420
- Gribanova, E. (2019). Development of a price optimization algorithm using inverse calculations. Eastern-European Journal of Enterprise Technologies, 5 (4 (101)), 18–25. doi: https://doi.org/10.15587/1729-4061.2019.180993
- Demin, D. (2017). Synthesis of optimal control of technological processes based on a multialternative parametric description of the final state. Eastern-European Journal of Enterprise Technologies, 3 (4 (87)), 51–63. doi: https://doi.org/10.15587/1729-4061.2017.105294
- Zhang, Q., Dong, W., Wen, C., Li, T. (2020). Study on factors affecting corn yield based on the Cobb-Douglas production function. Agricultural Water Management, 228, 105869. doi: https://doi.org/10.1016/j.agwat.2019.105869
- Sarmah, S. P., Acharya, D., Goyal, S. K. (2008). Coordination of a single-manufacturer/multi-buyer supply chain with credit option. International Journal of Production Economics, 111 (2), 676–685. doi: https://doi.org/10.1016/j.ijpe.2007.04.003
- Kalayci, C. B., Ertenlice, O., Akbay, M. A. (2019). A comprehensive review of deterministic models and applications for mean-variance portfolio optimization. Expert Systems with Applications, 125, 345–368. doi: https://doi.org/10.1016/j.eswa.2019.02.011
Downloads
Published
How to Cite
Issue
Section
License
Copyright (c) 2020 Ekaterina Gribanova
This work is licensed under a Creative Commons Attribution 4.0 International License.
The consolidation and conditions for the transfer of copyright (identification of authorship) is carried out in the License Agreement. In particular, the authors reserve the right to the authorship of their manuscript and transfer the first publication of this work to the journal under the terms of the Creative Commons CC BY license. At the same time, they have the right to conclude on their own additional agreements concerning the non-exclusive distribution of the work in the form in which it was published by this journal, but provided that the link to the first publication of the article in this journal is preserved.
A license agreement is a document in which the author warrants that he/she owns all copyright for the work (manuscript, article, etc.).
The authors, signing the License Agreement with TECHNOLOGY CENTER PC, have all rights to the further use of their work, provided that they link to our edition in which the work was published.
According to the terms of the License Agreement, the Publisher TECHNOLOGY CENTER PC does not take away your copyrights and receives permission from the authors to use and dissemination of the publication through the world's scientific resources (own electronic resources, scientometric databases, repositories, libraries, etc.).
In the absence of a signed License Agreement or in the absence of this agreement of identifiers allowing to identify the identity of the author, the editors have no right to work with the manuscript.
It is important to remember that there is another type of agreement between authors and publishers – when copyright is transferred from the authors to the publisher. In this case, the authors lose ownership of their work and may not use it in any way.