Abstract
The paper presents a quantitative explanation of failure of generic Monte Carlo techniques as applied to optimization problems of high dimensions. Deterministic grids are also discussed.
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Kroese, D.P., Taimre, T., Botev, Z.I.: Handbook of Monte Carlo Methods. Wiley, New York (2011)
Tempo, R., Calafiore, G., Dabbene, F.: Randomized Algorithms for Analysis and Control of Uncertain Systems, with Applications. Springer, London (2013)
Zhigljavsky, A., Z̆hilinskas, A.: Stochastic Global Optimization. Springer, New York (2008)
Horst, R., Pardalos, Panos M. (eds.): Handbook of Global Optimization, vol. 1, Kluwer, Dordrecht (1995)
Pardalos, Panos M., Romeijn, H. Edwin (eds.): Handbook of Global Optimization, vol. 2, Kluwer, Dordrecht (2002)
Simon, D.: Evolutionary Optimization Algorithms. Wiley, New York (2013)
Goldberg, D.: Genetic Algorithms in Search, Optimization and Machine Learning. Addison-Wesley, Reading, MA (1989)
Solis, F.J., Wets, R.J.-B.: Minimization by random search techniques. Math. Oper. Res. 6(1), 19–30 (1981)
Nemirovski, A., Yudin, D.B.: Problem Complexity and Method Efficiency in Optimization. Wiley, New York (1983)
Nesterov, Y.U.: Introductory Lectures on Convex Optimization: a Basic Course. Springer, New York (2004)
Deb, K.: Multiobjective optimization. In: Burke, E.K., Kendall, G. (eds.) Search Methodologies: Introductory Tutorials in Optimization and Decision Support Techniques, pp. 444–463. Springer, New York (2014)
Statnikov, R., Matusov, J.: Multicriteria Analysis in Engineering Using the PSI Method with MOVI 1.0. Kluwer Academic Publishers, Dordrect (2002)
Statnikov, R., Matusov, J., Statnikov, A.: Multicriteria engineering optimization problems: statement, solution and applications. J. Optim. Theory Appl. 155(2), 355–375 (2012)
Hopcroft, J., Kannan, R.: Foundations of Data Science. Available at http://research.microsoft.com/en-US/people/kannan/book-no-solutions-aug-21-2014
Polyak, B.T., Shcherbakov, P.S.: Random spherical uncertainty in estimation and robustness. IEEE Trans. Autom. Control 45(11), 2145–2150 (2000)
Milman, V.D., Schechtman, G.: Asymptotic Theory of Finite Dimensional Normed Spaces. Springer-Verlag, Berlin (2001)
Wilks, S.S.: Mathematical Statistics. Wiley, New York (1962)
Polyak, B., Shcherbakov, P., Khlebnikov, M.: Quadratic image of a ball: towards efficient description of the boundary. In: International conference on system theory, control, and computing (ICSTCC 2014), pp. 105–110. IEEE CSS (2014)
Dabbene, F., Shcherbakov, P., Polyak, B.T.: A randomized cutting plane method with probabilistic geometric convergence. SIAM J. Optim. 20(6), 3185–3207 (2010)
Kuipers, L., Niederreiter, H.: Uniform Distribution of Sequences. Wiley, New York (2005)
Sobol, I.M.: On the distribution of points in a cube and the approximate evaluation of integrals. USSR Comput. Math. Math. Phys. 7(4), 86–112 (1967)
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Financial support for this work was provided by the Russian Science Foundation through project no. 16-11-10015.
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Polyak, B., Shcherbakov, P. Why Does Monte Carlo Fail to Work Properly in High-Dimensional Optimization Problems?. J Optim Theory Appl 173, 612–627 (2017). https://doi.org/10.1007/s10957-016-1045-4
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DOI: https://doi.org/10.1007/s10957-016-1045-4