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Probabilistic bounds (via large deviations) for the solutions of stochastic programming problems

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Abstract

Several exponential bounds are derived by means of the theory of large deviations for the convergence of approximate solutions of stochastic optimization problems. The basic results show that the solutions obtained by replacing the original distribution by an empirical distribution provides an effective tool for solving stochastic programming problems.

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References

  1. H. Attouch, R. Lucchetti and R.J-B Wets, The topology of the ρ-Hausdorff distance, Ann. Mat. pura ed appl. 160 (1991) 303–320.

    Article  Google Scholar 

  2. H. Attouch and R.J-B Wets, Quantitative stability of variational systems: I. The epigraphical distance, Trans. Amer. Math. Soc. 328 (1991) 695–729.

    Google Scholar 

  3. H. Attouch and R.J-B Wets, Epigraphical processes: laws of large numbers for random lsc functions, manuscript, University of California, Davis (1991), in: Séminaire d'Analyse Convexe, Montepellier (1990) pp. 13.1–13.29.

  4. H. Attouch and R.J-B Wets, Quantitative stability of variational systems: II. A framework for nonlinear conditioning, SIAM J. Optim. (1992), to appear; IIASA Working Paper 88–9, Laxenburg, Austria.

  5. G. Beer, Conjugate convex functions and the epi-distance topology, Proc. Amer. Math. Soc. 108 (1990) 117–126.

    Google Scholar 

  6. P. Billingsley,Convergence of Probability Measures (Wiley, New York, 1968).

    Google Scholar 

  7. J.-D. Deuschel and D.W. Stroock,Large Deviations (Academic Press, Boston, 1989).

    Google Scholar 

  8. N. Dunford and J. Schwartz,Linear Operators, Part I:General Theory (Interscience, New York, 1957).

    Google Scholar 

  9. J. Dupačová, Stochastic programming with incomplete information, Optimization 18 (1987) 507–532.

    Google Scholar 

  10. J. Dupačová and R.J-B Wets, Asymptotic behavior of statistical estimators and of optimal solutions for stochastic optimization problems, Ann. Statist. 16 (1988) 1517–1549.

    Google Scholar 

  11. Yu.M. Ermoliev, A.A. Gaivoronski and C. Nedeva, Stochastic optimization problems with incomplete information of distribution functions, SIAM J. Contr. Optim. 23 (1985) 696–716.

    Google Scholar 

  12. W. Hoeffding, Probability inequalities for sums of bounded random variables, J. Amer. Statist. Assoc. 58 (1963) 13–30.

    Google Scholar 

  13. P. Huber, Robust estimation of a location parameter, Ann. Math. Statist. 35 (1964) 73–101.

    Google Scholar 

  14. P. Kall, On approximations and stability in stochastic programming, in:Parametric Optimization and Related Topics, eds. J. Guddat, H. Jonge, B. Kummer and F. Nozicka (Akademie-Verlag, Berlin, 1987).

    Google Scholar 

  15. V. Kaňková, An approximate solution of a stochastic optimization problem, in:Trans. 8th Conf. on Information Theory, Statistical Decision Functions and Random Processes (Academia, Prague, 1978) pp. 349–353.

    Google Scholar 

  16. A.J. King and R.T. Rockafellar, Asymptotic theory for solutions in generalizedM-estimation and stochastic programming, Math. Oper. Res. 18 (1993) 148–162.

    Article  Google Scholar 

  17. A.J. King and R.J-B Wets, Epi-consistency of convex stochastic programs, Stoch. and Stoch. Reports 34 (1990) 83–92.

    Google Scholar 

  18. L. Le Cam, On some asymptotic properties of maximum likelihood estimates and related Bayes estimates, University of California Publications in Statistics 1 (1953) 277–330.

    Google Scholar 

  19. R. Lucchetti and R.J-B Wets, Convergence of minima of integral functionals, with applications to optimal control and stochastic optimization, Statist. Dec. 11 (1993); IIASA WP-90-81.

  20. S.M. Robinson and R.J-B Wets, Stability in two-stage stochastic programming, SIAM J. Control Optim. 25 (1987) 1409–1416.

    Article  Google Scholar 

  21. R.T. Rockafellar and R.J-B Wets, Variational systems, an introduction, in:Multifunctions and Integrands: Stochastic Analysis, Approximation and Optimization, ed. G. Salinetti, Lecture Notes in Mathematics 1091 (Springer, Berlin, 1984) pp. 1–54.

    Google Scholar 

  22. R. Schultz, Strong convexity in stochastic programs with recourse, Techical Report no. 407, Humboldt University, Berlin (1992).

    Google Scholar 

  23. A. Shapiro, Asymptotic analysis of stochastic programs, Ann. Oper. Res. 30 (1991) 169–186.

    Google Scholar 

  24. A.B. Tsybakov, Error bounds for the methods of minimization of empiral risk, Probl. Inf. Transmission 17 (1981) 50–61 (in Russian).

    Google Scholar 

  25. S. Vogel, Stability results for stochastic programming problems, Optimization 19 (1988) 269–288.

    Google Scholar 

  26. A. Wald, Note on the consistency of the maximum likelihood estimate, Ann. Math. Statist. 20 (1949) 595–601.

    Google Scholar 

  27. R.J-B Wets, Convergence of convex functions, variational inequalities and convex optimization problems, in:Variational Inequalities and Complementarity Problems, eds. R. Cottle, F. Giannessi and J.L. Lions (Wiley, Chichester, 1980) pp. 405–419.

    Google Scholar 

  28. R.J-B Wets, Constrained estimation: consistency and asymptotic, Appl. Stoch. Models Data Anal. 7 (1991) 17–32.

    Google Scholar 

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Authors and Affiliations

  1. International Institute of Applied Systems Analysis, Laxenburg, Austria

    Yuri M. Kaniovski

  2. Department of Mathematical Sciences, IBM T.J. Watson Research Center, 10598, Yorktown Heights, NY, USA

    Alan J. King

  3. Department of Mathematics, University of California, 95616, Davis, CA, USA

    Roger J-B Wets

Authors
  1. Yuri M. Kaniovski
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  2. Alan J. King
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  3. Roger J-B Wets
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Supported in part by a grant from the US-Israel Science Foundation.

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Kaniovski, Y.M., King, A.J. & Wets, R.JB. Probabilistic bounds (via large deviations) for the solutions of stochastic programming problems. Ann Oper Res 56, 189–208 (1995). https://doi.org/10.1007/BF02031707

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