Wegstein's Method for Calculating the Global Extremum

Abstract

This study discusses an economical and efficient method for calculating the global optimum of a function of many variables. The proposed algorithm can be attributed to methods based on auxiliary functions. The auxiliary function itself is obtained by converting the objective function using the Lebesgue integral and is a function of one variable. In a previously published paper by one of the authors of this article, this auxiliary function was used to calculate the global minimum of smooth multiextremal functions on convex closed sets. In the same article, an algorithm was proposed for dividing a segment into half to find a global minimum. And in this paper we consider the problem of finding the global minimum of continuous functions defined on bounded closed subsets of an n-dimensional Euclidean space. In addition, curious properties of the auxiliary function are established that are valid for any continuous objective function. For example, its non-negativity, positive homogeneity of some order, uniform continuity, differentiability and strict convexity are proved, and higher-order derivatives are calculated. The optimality criterion is established. The essence of this optimality criterion is that the value of a variable at which the auxiliary function and its derivatives are equal to zero up to a certain order turns out to be equal to the global minimum of the objective function. It follows from this optimality criterion that to calculate the global minimum of the objective function, it is sufficient to find the zero of the auxiliary function or its derivative up to the m-th order. Therefore, Wegstein's algorithm was used as a way to find the root of an equation with one unknown. In addition, the advantage of the Wegstein’s method is that it always converges. And in this situation, it turned out to be more efficient, despite its slow convergence, since it requires almost half the number of calculations of the values of the auxiliary function and that halves the need for numerical calculations of multiple integrals with a large number of variables.