Abstract
To justify the use of sampling to solve stochastic programming problems one usually relies on a law of large numbers for random lsc (lower semicontinuous) functions when the samples come from independent, identical experiments. If the samples come from a stationary process, one can appeal to the ergodic theorem proved here. The proof relies on the ‘scalarization’ of random lsc functions.
Research supported in part by a grant of the National Science Foundation
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References
Wets, R. J-B (1989) Stochastic Programming. Handbook for Operations Research and Management sciences # 1, Elsevier Science, North Holland
Birge, J.R., Louveaux, F. (1997) Introduction to Stochastic Programming. Springer, Heidelberg
Korf, L.A., Wets, R. J-B (1999) Ergodic limit laws for stochastic optimization problems. Manuscript, University of California, Davis
King, A.J., Wets, R. J-B (1990) Epi-consistency of convex stochastic programs. Stochastics and Stochastics Reports 34 83–92
Attouch, H., Wets, R. J-B (1990) Epigraphical processes: laws of large numbers for random lsc functions. Séminaire d’Analyse Convexe, Montpellier
Artstein, Z., Wets, R. J-B (1995) Consistency of minimizers and the SLLN for stochastic programs. J. of Convex Analysis 2 1–17
Hess, C. (1991) Epi-convergence of sequences of normal integrands and strong consistency of the maximum likelihood estimator. Cahiers no. 9121, Cérémade, Paris
Castaing, C., Ezzaki, F. (1995) SLLN for convex random sets and random lower semicontinuous integrands. Manuscript, Université de Montpellier
Attouch, H. (1984) Variational Convergence for Functions and Operators. Applicable Mathematics Series, Pitman
Aubin, J.-P, Frankowska, H. (1990) Set-Valued Analysis. Birkhäuser, Basel
Rockafellar, R.T., Wets, R. J-B (1997) Variational Analysis. Grundlehren der mathematischen Wissenschaften 317, Springer-Verlag, Heidelberg
King, A.J., Rockafellar R.T., Somlyody, L., Wets, R. J-B (1988) Lake eutrophication management: the Lake Balaton project. In: Ermoliev Y., Wets R. (Eds.) Numerical Techniques for Stochastic Optimization, Springer-Verlag, Heidelberg, 435–444
Salinger, D. (1997) A Splitting Algorithm for Multistage Stochastic Programming with Application to Hydropower Scheduling. Ph. D. Thesis, University of Washington, Seattle
Kall, P., Wallace, S. (1994) Stochastic Programming. J. Wiley&Sons, New York
Kibzun, A.I., Kan, Y.S. (1996) Stochastic Programming Problems with Probability and Quantile Functions. J. Wiley&Sons, New York
Prékopa, A. (1995) Stochastic Programming. Kluwer, Dordrecht
Wets, R. J-B (1998) Stochastic programs with chance constraints: generalized convexity and approximation issues. In: Crouzeix J.-P., Martinez-Legaz J.E., Volle M. (Eds.) Generalized Convexity, Generalized Monotonicity: Recent Results. Kluwer, Dordrecht, 61–74
Korf, L.A., Wets, R. J-B (1999) Random lsc functions: scalarization. Manuscript, University of California, Davis
Durrett, R. (1991) Probability: Theory and Examples, Wadsworth and Brooks
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Korf, L.A., Wets, R.JB. (2000). An Ergodic Theorem for Stochastic Programming Problems. In: Nguyen, V.H., Strodiot, JJ., Tossings, P. (eds) Optimization. Lecture Notes in Economics and Mathematical Systems, vol 481. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-57014-8_14
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DOI: https://doi.org/10.1007/978-3-642-57014-8_14
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