Monotonous random search on a torus: Integral upper bounds for the complexity

The paper consists of three parts. The first part is dedicated to a Markov monotonous random search on a general optimization space. Under certain restrictions, an upper bound for the complexity of search is presented in an integral form, suitable for further analysis. This estimate is applied to the case of a torus, where several specific results on the rate of convergence are obtained with the help of a supplementary optimization problem, discussed in Appendix.