Abstract
The statistical method of empirical means is applied to solve the general stochastic programming problem with compound risk functions. The convergence of the method is established in probabilistic terms.
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Literature Cited
Yu. M. Ermoliev and R. J.-B. Wets, eds., Numerical Techniques for Stochastic Optimization, Springer, Berlin (1988).
P. J. Huber, Robust Statistics, Wiley, New York (1961).
A. S. Korkhin, "On some properties of estimators of regression parameters subject to prior inequality constraints," Kibernetika, No. 6, 104–114 (1985).
E. Yubi, "Statistical analysis and solution method for stochastic programming problems," Izv. Akad. Nauk ÉstSSR, Fiz.-Mat.,26, No. 4, 369–375 (1977).
E. Tamm, "Chebyshev inequality for E-models in stochastic programming," Izv. Akad. Nauk ÉstSSR, Fiz.-Mat.,27, No. 4, 448–450 (1978).
V. Kankova, "An approximate solution of a stochastic optimization problem," Trans. 8th Prague Conf., 1978, Academia, Prague (1978), pp. 349–353.
J. Dupačová, "Experience in stochastic programming models," Survey Math. Program., Proc. 9th Int. Math. Program. Symp., Akademia Kiado, Budapest (1979), pp. 99–105.
R. West, "A statistical approach to the solution of stochastic programs with (convex) simple resource," Working Paper, Univ. of Kentucky, Lexington (1979).
G. Salinetti and R. J.-B. Wets, "On the convergence in distribution of measurable multifunctions (random sets), normal integrands, stochastic processes and stochastic infima," Math. Oper. Res.,11, No. 3, 385–419 (1986).
J. Dupačová and R. J.-B. Wets, "Asymptotic behavior of statistical estimators and optimal solutions for stochastic optimization problems," Ann. Stat.,16, 1517–1549 (1988).
A. J. King and R. J.-B. Wets, "Epi-consistency of convex stochastic programs," WP-88-57, IIASA, Laxenburg, Austria (1988).
P. S. Knopov, "One approach to solving stochastic optimization problems," Kibernetika, No. 4, 126–127 (1988).
V. I. Norkin, "Stability of stochastic optimization models and statistical methods of stochastic programming," Preprint 89-53, Inst. Kiber. im. V. M. Glushkova, Akad. Nauk UkrSSR, Kiev (1989).
V. N. Vapnik, Estimation of Dependences from Experimental Data [in Russian], Nauka, Moscow (1979).
A. J. King, "Asymptotic distributions for solutions in stochastic optimization and generalizedM-estimation," WP-88-58, IIASA, Laxenburg, Austria (1988).
A. Shapiro, "Asymptotic properties of statistical estimators in stochastic programming," Ann. Stat.,17, No. 2, 841–858 (1989).
A. Shapiro, "Asymptotic analysis of stochastic programs," Res. Rep. Navorsingsverslag 90/89(12), Univ. of South Africa, Pretoria, South Africa (1989).
Yu. M. Ermoliev and V. I. Norkin, "Normalized convergence in stochastic optimization," WP-89-91, IIASA, Laxenburg, Austria (1989).
J. A. Reeds, On the Definition of von Mises Functionals, PhD Thesis, Harvard Univ. (1976).
L. T. Fernholz, Von Mises Calculus for Statistical Functionals, Lect. Notes Stat., Vol. 19, Springer, New York (1983).
R. Grubel, "The length of the short," Ann. Stat.,16, 619–628 (1988).
R. D. Gill, "Non- and semi-parametric maximum likelihood estimators and von Mises method (Part I)," Scand. J. Stat.,16, 97–124 (1989).
Yu. M. Ermol'ev and V. I. Norkin, "Normalized convergence of random variables and its application," Kibernetika, No. 6, 85–89 (1990).
Yu. M. Ermol'ev, Stochastic Programming Methods [in Russian], Nauka, Moscow (1976).
A. J. King and R. T. Rockafellar, "Non-normal asymptotic behaviour of solution estimates in linear-quadratic stochastic optimization," Unpublished manuscript, Univ. of Washington, Seattle (1986).
S. M. Robinson, "Local epi-continuity and local optimization," Math. Program.,37, 208–222 (1987).
J.-P. Aubin and I. Ekeland, Applied Nonlinear Analysis [Russian translation], Mir, Moscow (1988).
N. Z. Shor, Methods of Minimization of Nondifferentiable Functions and Their Applications [in Russian], Naukova Dumka, Kiev (1979).
V. G. Karmanov, Mathematical Programming [in Russian], Nauka, Moscow (1975).
P.-J. Laurent, Approximation and Optimization [Russian translation], Mir, Moscow (1975).
A. A. Gaivoronskii, "Some methods of solution of the nonstationary stochastic programming problem with constraints," in: Methods of Operations Research and Reliability Theory in System Analysis [in Russian], Inst. Kiber. im. V. M. Glushkova Akad. Nauk UkrSSR, Kiev (1977), pp. 70–79.
A. A. Gaivoronskii, "On nonstationary stochastic programming problems," Kibernetika, No. 4, 89–92 (1978).
V. M. Gorbachuk, "Search for Nash equilibria under conditions of nonstationarity of nonsmooth payoff functions," in: Mathematical Methods of Analysis and Optimization of Complex Systems Under Uncertainty [in Russian], Inst. Kiber. im. V. M. Glushkova Akad. Nauk UkrSSR, Kiev (1986), pp. 8–12.
R. T. Rockafellar, Convex Analysis, Princeton Univ. Press (1970).
L. LeCam, "On some asymptotic properties of maximum likelihood estimates and related Bayes' estimates," Univ. Calif. Publ. Stat.,1, 277–330 (1953).
R. I. Jenrich, "Asymptotic properties of non-linear least squares estimators," Ann. Math. Stat.,40, 633–643 (1969).
P. Billingsley, Convergence of Probability Measures, Wiley, New York (1968).
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Translated from Kibernetika i Sistemnyi Analiz, No. 2, pp. 107–120, March–April, 1992.
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Norkin, V.I. Convergence of the empirical mean method in statistics and stochastic programming. Cybern Syst Anal 28, 253–264 (1992). https://doi.org/10.1007/BF01126212
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DOI: https://doi.org/10.1007/BF01126212