Algorithm for Calculating the Global Minimum of a Smooth Function of Several Variables

Abstract

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.

Көмекші Функция Негізінде Глобальды Оңтайландыруға Кесіндіні Қақ Бөлу Әдісін Қолдану

“…In [12], a new algorithm is described, which is very different from the methods described above. The search begins with the value of the global minimum, and not with the definition of coordinates.…”

“…The calculation of the auxiliary function, a complete description of the algorithm and computational experiments are published in previous articles [12], [15]. Now we will prove the convergence of the new global optimization algorithm.…”

“…A full description of the method using the auxiliary function is described in the author's articles [12], [15] Similarly, the following midpoints of the sub-sections are defined:…”

Application of the Auxiliary Function Method to the Search for the Global Minimum of Functions of Many Variables

“…We use test functions based on hyperbolic and exponential abilities [15,16], which, in our opinion, are more representative for solving optimization problems compared to the well-known test functions (power, trigonometric and others) considered in our previous works [17,18]. In addition, in this article, unlike previous ones, independent functions with three functions with at least several structures are used.…”