Abstract
The concept of implementable nonlinear stochastic programming by finite series of Monte-Carlo samples is surveyed addressing the topics related with stochastic differentiation, stopping rules, conditions of convergence, rational setting of the parameters of algorithms, etc. Our approach distinguishes itself by treatment of the accuracy of solution in a statistical manner, testing the hypothese of optimality according to statistical criteria, and estimating confidence intervals of the objective and constraint functions. The rule for adjusting the Monte-Carlo sample size is introduced which ensures the convergence with the linear rate and enables us to solve the stochastic optimization problem using a reasonable number of Monte-Carlo trials. The issues of implementation of the developed approach in optimal decision making, portfolio optimization, engineering are considered, too.
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References
Arrow, K.J., Hurwicz, L., and Uzawa, H. eds. (1958). Studies in Linear and Nonlinear Programming. Stanford University Press, Stanford, California.
Bartkute V. and Sakalauskas L. (2006). Simultaneous Perturbation Stochastic Approximation for Nonsmooth Functions. European Journal on Operational Research (accepted).
Bentkus V. and Goetze F. (1999) Optimal bounds in non-Gaussian limit theorems for U-statistics. Annals of Probability, 27(1), 454–521.
Bertsekas, D.I. (1982). Constrained Optimization and Lagrange Multiplier Methods. Academic Press, Paris-Toronto.
Bhattacharya, R.N., and Ranga Rao, R. (1976). Normal Approximation and Asymptotic Expansions. John Wiley, New York, London, Toronto.
Ermolyev, Ju.M. (1976). Methods of Stochastic Programming. Nauka, Moscow. 240 pp. (in Russian).
Ermolyev Yu., and Wets R. (1988). Numerical Techniques for Stochastic Optimization. Springer-Verlag, Berlin.
Ermolyev Yu. and Norkin I. (1995). On nonsmooth problems of stochastic systems optimization. WP-95-96, IIASA, A-2361, Laxenburg, Austria.
Jun Shao (1989) Monte-Carlo approximations in Bayessian decision theory. JASA, 84(407), 727–732.
Katkovnik V.J. (1976). Linear Estimators and Problems of Stochastic Optimization. Nauka, Moscow (in Russian).
King J. (1988). Stochastic Programming problems: Examples from the Literature. In: Ermolyev Ju and Wets R. (eds.), Numerical Techniques for Stochastic Optimization, Springer-Verlag, Berlin.
Kushner H, and Jin G.G. (2003). Stochastic Approximation and Recursive Algorithms and Applications. Springer, NY.
Stochastic optimization methods. Springer, NY. Marti K. (2005).
Mikhalevitch V.S., Gupal A.M. and Norkin V.I. (1987). Methods of Nonconvex Optimization. Nauka, Moscow (in Russian).
Pflug G.Ch. (1988). Step size rules, stopping times and their implementation in stochastic optimization algorithms. Numerical Techniques for Stochastic Optimization, Ermolyev, Ju., and Wets, R. (eds.), Springer-Verlag, Berlin, 353–372.
Polyak B.T. (1987). Introduction to Optimization. Translation Series in Mathematics and Engineering. Optimization Software, Inc, Publication Division, New York.
Prekopa A. (1980) Logarithmic concave metric and related topics. In: M.A.H. Dempster (ed.) Stochastic Programming. Academic Press, London, 63–82.
Rockafellar, R.T., and Wets, R. J.-B. (1994). A Lagrangian finite generation technique for solving linear-quadratic problems in stochastic programming. Mathematical Programming, 28, 63–93.
Rubinstein R. (1983). Smoothed functionals in stochastic optimization. Mathematical Operations Research, 8, 26–33.
Rubinstein R. and Shapiro A. (1993). Discrete Events Systems: Sensitivity Analysis and Stochastic Optimization by the score function method. Wiley, New York, NY.
Sakalauskas L. (1997). A centering by the Monte-Carlo method. Stochastic Analysis and Applications, 15(4), 615–627.
Sakalauskas L. (1998). Portfolio management by the Monte-Carlo method. Proceedings of the 23rd Meeting of the European Working Group on Financial Modelling, Crakow, Progress & Business Publ., 179–188.
Sakalauskas L. (2000). Nonlinear Optimization by Monte-Carlo estimators. Informatica, 11(4), 455–468.
Sakalauskas L. (2002) Nonlinear stochastic programming by Monte-Carlo estimators. European Journal on Operational Research, 137, 558–573.
Sakalauskas L. (2004). Application of the Monte-Carlo method to nonlinear stochastic optimization with linear constraionts. Informatica, 15(2), 271–282.
Shapiro A. (1989). Asymptotic properties of statistical estimators in stochastic programming. The Annals of Statistics, 17(2), 841–858.
Shapiro A. and Homem-de-Mello T. (1998). A simulation-based approach to two-stage stochastic programming with recourse. Mathematical Programming, 81, 301–325.
Yudin D.B. (1965). Qualitative methods for analysis of complex systems. Izv. AN SSSR, ser. Technicheskaya. Kibernetika, 1, 3–13 (in Russian).
Ziemba W. (1972) Solving nonlinear programming problems with stochastic objective functions. Journal of Financial and Quantitative Analysis, 7, 1995–2001.
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Sakalauskas, L. (2006). Towards Implementable Nonlinear Stochastic Programming. In: Coping with Uncertainty. Lecture Notes in Economics and Mathematical Systems, vol 581. Springer, Berlin, Heidelberg . https://doi.org/10.1007/3-540-35262-7_15
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DOI: https://doi.org/10.1007/3-540-35262-7_15
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-35258-7
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