2026 , Volume 31, № 1, p.57-65

This research addresses a new method for reducing the two-dimensional problem of finding a global minimum of a function to a sequence of simpler one-dimensional problems. This approach helps to overcome the computational difficulties inherent in multidimensional optimization and enhances the efficiency of existing interval computation algorithms.

The method is based on an interval representation of a function of two variables using a Taylor expansion with a Lagrange remainder term. A key aspect of the new technique is the estimation of the second-order mixed term using quadratic forms, which allows for the transformation of the original expression into a sum of two one-dimensional functional intervals. The natural interval extensions of the derivatives are used to estimate their ranges of values. In practice, the method is integrated into a branch-and-bound scheme, where a technique based on solving inequalities with functional interval bounds is applied to narrow the search domain.

The proposed method was successfully tested on the Booth function minimization problem. As a result of its application, the original two-dimensional search box was significantly reduced. Furthermore, a two-sided estimate for the global minimum of the function was obtained, which proved to be significantly more accurate than the estimates computed using classical methods — namely, the natural interval extension and the centered differential form.

The developed method demonstrates high practical value. It not only improves the estimates of the global minimum but also reduces the search domain by reducing the multidimensional problem to one-dimensional ones. The use of functional intervals opens up new possibilities for transforming and simplifying complex expressions. The proposed approach is promising for practical application in interval global optimization algorithms and can be generalized to functions of a larger number of variables.