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Optimization problem solving by means of generalized gradient method and symmetry principle

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Category: Economy
Last Updated on Wednesday, 25 December 2013 11:43
Published on Wednesday, 19 December 2012 16:24
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Authors:

K.V. Litvinenko, State Higher Educational Institution “National Mining University”, doctoral student, Dnipropetrovsk, Ukraine

Abstract:

Existing mathematical methods of optimisation of practical problems are developed enough forsolving many problems of designing, forecasting, planning and management. However the majority of real problems cannot be adequately described within the limits of linear models, as reflects modest successes of linear programming in technical appendices. In natural statement, nonlinear problems aredescribed by convex, concave functions and convex areas of admissible operating parameters. The active search of effective strategy with a wide scope for the problems with nonlinear criterion functions and restrictions proceeds.

Purpose. Creationof the synthesised algorithm of criterion function extreme determining at nonlinear restrictions with properties of global convergence.

Methodology. Theoretical researches are based on substantive provisions of the theory of nonlinear programming and the optimisation theory.

Findings. The problem of construction of the method of the nonlinear function extreme determining has been investigated, at nonlinear restrictions, as synthesised procedureon the basis of the symmetry principle and the generalised gradient. Initial nonlinear function and restriction will be transformed consistently to auxiliary symmetric function by methods of updating of algorithm of Woolf and K-transformation for the decision of a problem of minimisation. The constructed auxiliary function can be unimodal or multimodal. Further to auxiliary function the two-step method of ‘heavy ball’ is applied for the purpose of the best use of the information received during the previous iteration. The given procedure allows us not to exclude variables from the initial problem that is important for problems with essentially nonlinear restrictions. The convergence of the method and the operability of the synthesised algorithm has been tested on a problem inMathCAD environment.

Originality. The effective procedure of nonlinearcriterion function extreme determining at nonlinear restrictions for the decision of the optimisation problem, possessing properties of global convergence has been offered.

Practical value. The presented decision algorithm of the global optimizationproblem can be used for the decision of a wide range of practical problems of optimisation.

 

References: 
 
1. Корсун В.И. Использование симметрии для распараллеливания процесса поиска экстремума целевой функции в задачах оптимального проектирования и адаптивной идентификации / Корсун В.И. // Мат. модели и современные инф. технологии. Сб. науч. Тр. НАН Украины. Ин-т математики. – К., 1998. – С. 66–68.
Korsun, V.I. (1998), “Use of symmetry for paralleling of the criterion function extremes determining process in problems of optimum designing and adaptive identification”, Math. Models and modern inf. technologies. Collect. of sci. papers NАS Ukraine. Mathematics Institution, pp. 66–68.
 
2. Химмельблау Д. Прикладное нелинейное программирование [Электронный ресурс] / Химмельблау Д. – Режим доступа: www/http//twirpx.com/file/6966/
Himmelblau, D.M. Prikladnoe nelinejnoe programmirovanie [Applied nonlinear programming] available at: www/http//twirpx.com/file/6966/. 
 
3. Корсун В.И. Исследование алгоритма поиска экстремума целевой функции, основанного на применении концепции симметрии и параллельного пространства / Корсун В.И., Демиденко М.А. – Днепропетровск: Науковий вісник НГУ, 2000. – №2. – С. 101–104.
Korsun, V.I. and Demidenko, M.A. (2000), “Research of algorithm of the criterion function extremes determining on the base of application of the concept of symmetry and parallel space”, Naukovyi visnyk Natsionalnoho Hirnychoho Universytetu, no.2, pp. 101–104.
 
4. Корсун В.И. Параллельное пространство сопряженных направлений и экстремальные свойства функций / Корсун В.И., Литвиненко К.В. // Прикладна геометрія та інженерна графіка. Праці – Мелітополь: ТДАТУ, 2011. – Вип. 4. – т.50. – С. 90–97.
Korsun, V.I. and Litvinenko, K.V. (2011), “Parallel space of the conjugated directions and extreme properties of functions”, Collection of sci. papers of MSATU, no.50, pp. 90–97.
 
5. Литвиненко К.В. Метод двойного штрафа в задачах оптимизации / Литвиненко К.В. // Наукові вісті. Сучасні проблеми металургії. – Дніпропетровськ: НМетАУ, 2011. – Вип.14. – С. 32–38.
Litvinenko, K.V. “Method of the double penalty in optimisation problems”, Naukovi visti. Suchasni problemi metalurgii, no.14, pp. 32–38.
 
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