519.85 Method of finding non-dominant solutions in decomposition problems
Kiselev V. V. (Bauman Moscow State Technical University)
MATHEMATICAL PROGRAMMING, LARGE DIMENSION, PARETO–OPTIMALITY, DECOMPOSITION, MONOTONICITY
doi: 10.18698/2309-3684-2022-1-129140
The article discusses the method of finding optimal solutions in the presence of a model of a complex technical system in the optimal design problem. The method is based on the use of nondominable, lambda optimal solutions and is a generalization of the method of Krasnoshchekov P.S., Morozov V.V., Fedorov V.V. [1]. The method allows in many cases (for lambda monotone objective functions) to reduce the number of calculations and reduce the dimension of the original problem. A numerical method for constructing lambda optimal solutions has been developed. A numerical example is given in which it is shown that the number of lambda optimal solutions consists of a single point, and the set of Pareto–optimal solutions is a curve on which it is necessary to build an ε–network to find the optimal solution.
Krasnoshchekov P.S., Morozov V.V., Fedorov V.V. Dekompoziciya v zadachah proektirovaniya [Decomposition in design problems]. Izvestiya Akademii nauk SSSR. Tekhnicheskaya kibernetika [Proceedings of the USSR Academy of Sciences. Technical Cybernetics], 1979, No. 2, pp. 7-9.
Krasnoshchekov P.S. Morozov V.V., Popov N.M. Optimizaziya v avtomatizirovannom proektirovanii [Optimization in computer-aided design]. Moscow, MAKS Press, 2018, 323 p.
Mesarovich M., Takahara Ya. Obschay teoriya system: matematicheskie osnovi[General theory of systems: mathematical foundations]. Moscow, Mir Publ., 1978, 316 p.
Morozov V.V., Sukharev A.G., Fedorov V.V. Issledovanie operatiy v zadachax i uprazhneniyax [Investigation of operations in tasks and exercises]. Moscow, URSS Publ., 2009, 288 p.
Podinovsky V.V., Nogin V.D. Pareto-optimalnie rescheniya mnogokriterialnix zadachax [Pareto-optimal solutions of multicriteria problems]. Moscow, Fizmatlit Publ., 2007, 256 p.
Tsurkov V.I. Decompozitiya v zadachaxbolschoy razmernosti [Decomposition in problems of large dimension]. Moscow, Nauka Publ., 1981, 352 p.
Vasin A.A., Krasnoshchekov P.S., Morozov V.V. Issledovanie operatziy[Operations research]. Moscow, Publishing Center "Academy", 2008, 464 p.
Kalman R., Falb P., Arbib M. Ocherki po matematicheskoy teorii system [Essays on the mathematical theory of systems]. Moscow, Editorial URSS Publ., 2004, 400 p.
Pavlovsky Yu.N., Smirnova T.G. Problema dekompozizii v matematicheskom modelirovanii [The problem of decomposition in mathematical modeling]. Moscow, Phasis Publ., 1998 266 p.
Yu P.L. Cone convexity, cone extreme points, and nondominated solutions in decision problems with multiobjectives. Journal of Optimization Theory and Applications, 1974, vol. 4, no. 3, pp. 319–377.
Tzeng G.H., Wang H.F., Wen U.P., Yu P.L. Multiple Criteria Decision Making. New York, Springer, 2011, 398 p.
Kiselev V.V. Application of the Λ-monotonicity to the search for optimal solutions in higher-dimensional problems, Journal of Mashematical Science, 2016, vol. 216, no. 5, pp. 667–673.
Kiselev V.V. Pareto-optimality and L–optimal for solving some classes of optimal control problem. Bulletin of the Financial University, 2016, no. 4, pp.72–77.
Sulimov V.D., Shkapov P.M., Goncharov D.A. Use of hybrid algorithms in extremum eigenproblems of Lagrangian dynamical systems. Маthematical Modeling and Coтputational Methods, 2016, no. 4, pp. 84–102.
Zoitendijk G. Metodi vozmozhnix napravleniy [Methods of possible directions]. Moscow, Foreign Literature Publ., 1963, 176 p.
Karmanov V.G. Matematicheskoe programmirovanie [Mathematical programming]. Moscow, Fizmatlit Publ., 2004, 264 p.
Киселев В.В. Метод нахождения недоминируемых решений в задачах декомпозиции моделей сложных систем. Математическое моделирование и численные методы, 2022, № 1, с. 129–140.
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