Monte Carlo method of batch iterations: Probabilistic characteristics

A method was proposed for approximate solution of the system of nonlinear algebraic equations and inequalities by computer‐aided generation of the sequence of residues of this system that were calculated through the collections of random vectors generated at each algorithmic step. It is based on the batch iterations using the simple Monte Carlo trials. The almost sure convergence with an exponential rate of this sequence to the global minimum of residue was proved. For the finite number of iterations, the probabilistic estimates of the deviation of the residue value from its global minimum were established. The method can be used for approximate solution of systems of equations and inequalities with algorithmically defined functions satisfying the Hölder condition.