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

Abstract

In early works, we presented a new economical and effective method for finding the global optimum of a function of many variables, which was conditionally called the auxiliary function method. The essence of the method is that a multi-extremal and multivariable objective function is transformed into a convex function 𝑔 𝑚 (𝐹, 𝛼) of one variable, which is the Lebesgue integral over a compact where the objective function is considered:, 𝑚 ∈ 𝑁 . The function 𝑔 𝑚 (𝐹, 𝛼) was called the auxiliary function. In early works, the properties of the auxiliary function and the algorithm of the new method were studied, the convergence of the method was proven, and computational experiments were carried out with multiextremal functions in three-dimensional space. Based on these results and in order to demonstrate the advantages of using the auxiliary function method, this paper considers the problem of finding global minima of objective functions in a fourdimensional space constructed on the basis of hyperbolic and exponential potentials and conducts a comparative analysis of the results obtained. In this work, as a result of completed computational experiments on test functions in three-dimensional and fourdimensional space, where auxiliary functions with different values of the degree 𝑚 𝜖 𝑁 were expanded, important conclusions were obtained and proven. As a result, the change in the auxiliary function depending on its degree m is clearly shown. This result provides even more opportunities to improve the efficiency of the constructed method. Next, you can set up first-and second-order methods to find the "oldest " zero auxiliary function.