Every year the interest of theorists and practitioners in optimisation problems is growing, and extreme problems are found in all branches of science. Local optimisation problems are well studied and there are constructive methods for their solution. However, global optimisation problems do not meet the requirements in practice; therefore, the search for the global minimum remains one of the major challenges for computational and applied mathematics. This study discusses the search for the global minimum of multidimensional and multiextremal problems with high precision. Mechanical quadrature formulas, that is, the formulas for approximate integration were applied to calculate the integrals. Of all the approximate integration formulas, the Sobolev lattice cubature formulas with a regular boundary layer were chosen. In multidimensional examples, the Sobolev formulas are optimal. Computational experiments were carried out in the most popular C++ programming language. Based on the computational experiments, a new algorithm was proposed. In three-dimensional space, the calculations of the global minimum have been described using specific examples. Computational experiments show that the proposed algorithm works for multiextremal problems with the same amount of time as for convex ones.
Some approximate integration formulas of statistical interest