Towards the Analysis of the simulated annealing method in the multiextremal case

There are many applied problems in which it is necessary to calculate global extrema whose number is large or even infinite. These problems include, for example, some experimental design problems, and the problem of solving large systems of equations. For a single extremum of a function of several variables, one of the commonly used numerical algorithms is the simulated annealing, which is also successfully used in high volume discrete problems (travelling salesman problem). In discrete problems, it is known that the simulated annealing method searches equal global extrema with an equal probability. The continuous case has not been investigated yet. It was assumed that equal extrema are to be found consistently, sharing their neighborhood during the computation. This method is not always effective, especially in the case when multiple extrema fill up a certain region in R ⁿ . The results obtained in this study outline a general approach to the problem. We give computational examples showing the effectiveness of the approach. It can be used to create programs, algorithms indicating the localization of the roots of large equation systems. It can also be noted that many problems of design for regression experiments have an infinite number of solutions.