Abstract
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.
Similar content being viewed by others
References
Strongin, R.G. and Sergeyev, Ya.D., Global Optimization with Non-Convex Constraints. Sequential and Parallel Algorithms, Dordrecht: Kluwer, 2000.
Sobol’, I.M., On the Distribution of Points in Cube and Approximate Calculation of Integrals, Zh. Vychisl. Mat. Mat. Fiz., 1967, vol. 7, no. 4, pp. 784–802.
Bertsimas, D. and Vempala, S., Solving Convex Programs by Random Walks, J. ACM, 2004, vol. 51, no. 4, pp. 540–556.
Rubinstein, R.Y. and Kroese, D.P., Simulation and the Monte Carlo Methods, Probability and Statistics, New Jersey: Wiley, 2007.
Strongin, R.G., Gergel’, V.P., Grishagin, V.A., et al., Parallel’nye vychisleniya v zadachakh global’noi optimizatsii (Parallel Computations in the Global Optimization Problems), Moscow: Mosk. Gos. Univ., 2012.
Polyak, B. and Gryasina, E., Hit-and-Run: New Design Technique for Stabilization, Robustness and Optimization of Linear Systems, in Proc. IFAC World Congr., 2008, pp. 376–380.
Nesterov, Yu.E., Vvedenie v vypukluyu optimizatsiyu (Introduction to Convex Optimization), Moscow: MTsNMO, 2010.
Statnikov, R. and Statnikov, A., The Parameter Space Investigation Method Toolkit, Boston: Artech House, 2011.
Author information
Authors and Affiliations
Corresponding author
Additional information
Original Russian Text © B.S. Darkhovskii, Yu.S. Popkov, A.Yu. Popkov, 2015, published in Avtomatika i Telemekhanika, 2015, No. 5, pp. 60–71.
Rights and permissions
About this article
Cite this article
Darkhovskii, B.S., Popkov, Y.S. & Popkov, A.Y. Monte Carlo method of batch iterations: Probabilistic characteristics. Autom Remote Control 76, 776–785 (2015). https://doi.org/10.1134/S0005117915050045
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1134/S0005117915050045