NECESSARY AND SUFFICIENT CONDITIONS FOR THE GLOBAL EXTREMUM OF OBJECTIVE FUNCTIONS

Global optimization of multi-extremal, multi-variable functions is an important problem for the development of various areas of science. Its relevance is that the need to search for global extremum of functions constantly arises both in theoretical research and in practice. In this work, a new method for determining the global minimum of a multi-extremal, multi-variable function is proposed. In a previously published work by one of the authors of this article, a special function called the "auxiliary function" was constructed by transforming the objective function, and its important properties (non-negativity, uniform discontinuity, differentiability, monotonicity, etc.) were studied. In the presented article, necessary and sufficient conditions for the global minimum of the objective function are rigorously formulated and proven. As a result, the problem of finding the global minimum of a multi-extremal and multi-variable function was reduced to the problem of determining the "greatest zero" of a convex function of one variable: it was proved that the global minimum of the objective function is equal to the exact upper bound of the zeros of the auxiliary function. And the problem of rational application of known numerical methods to determine the "greatest zero" of the auxiliary function with high accuracy was considered.

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