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

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


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.


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Date 2013-12-24 Filesize 374.51 KB Download 676


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ISSN (print) 2071-2227,
ISSN (online) 2223-2362.
Journal was registered by Ministry of Justice of Ukraine.
Registration number КВ No.17742-6592PR dated April 27, 2011.


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