2022 , Volume 27, № 4, p.77-83

The article addresses the problem of two-sided estimation of the minimum of a function of one real variable using interval methods. These methods are widely used for finding the ranges of functions, but their order of accuracy often does not exceed 3 for the functions of one variable and 2 for the case of many variables. The paper proposes a method for finding an interval estimate for the minimum of a function of one variable with any odd order greater than 2.

The proposed technique uses so-called functional intervals, the boundaries of which are described by polynomials of even degrees obtained from the Taylor formula for the function under consideration. The article provides a derivation of interval estimates for the minimum, and also proves an estimate for the order of convergence of the width of the resulting interval with the decrease in the size of the domain of the function. An example is given in which estimating the minimum of a function reaches the theoretical order of convergence.

The approach developed in this paper allows, in principle, to achieve any odd accuracy order in the interval estimation of the minimum of a function. In practice, this method is convenient to use for the third order of accuracy, since, on the one hand, we need to find an interval estimate for derivatives of high degrees, and, on the other hand, we need to find roots of polynomials of even degrees.

The new approach is shown to be useful and practical, as it uses the local Taylor expansion and therefore behaves stably as the width of the function domain decreases. The new approach can be used to improve the performance of interval global optimization algorithms based on adaptive domain partitioning. An increase in the order of estimation accuracy entails the acceleration of these algorithms, as well as a decrease in the number of subintervals generated in the process of partitioning the domain