Approximate method of optimization of nonlinear programming problems

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Currently, the problem of choosing the optimal solution is one of the most important and urgent problems in industry, economy, agriculture and the military sphere. Methods and approaches of the theory of nonlinear programming are used to solve many applied optimization problems. The main difficulty of nonlinear optimization is the lack of a universal method for solving this class of problems. To solve this problem, special methods are being developed for solving particular nonlinear programming problems, for example, for positive or limited initial data. The paper investigates the problem of analytical optimization of nonlinear programming problems. The purpose of this work is to develop a new approximate method for solving optimization problems of a nonlinear function under nonlinear constraints in the form of equalities. To do this, an approximation (expansion in a series) of the objective function and constraints is performed. All variables are considered bounded at the top and bottom. The objective function and constraints are considered infinitely differentiable by the set of arguments, and all their derivatives are assumed to be limited in absolute value by a given number. In this article, the theorem on the conditional maximum of the objective function under given constraints is proved. the results of which are the justification of the developed method. Since the developed optimization method is approximate, the error of the proposed representation of the objective function and constraint functions is estimated. In problems of an applied nature, the boundaries of variable changes are often set approximately and they can be adjusted. In addition, it is possible to adjust the point relative to which the functions are decomposed into series. Therefore, the article analyzes the sensitivity of the optimal solution of the problem when changing the decomposition point into a series of functions for different values of the coordinates of the left boundaries when searching for the maximum of the objective function. To explain the operation of the method, a specific numerical example is analyzed in detail. Modeling in the MS Excel environment was used to solve it. Graphs of the sensitivity study of the solution of the problem when changing the initial data are constructed. Nonlinear programming models are used, for example, to solve the following practically important issues: minimizing costs in the sale of products, optimizing consumer choice, maximizing production volume, determining the rational behavior of an individual in a given situation, rational use of resources, forming an optimal portfolio of securities.