Location via proxy:   [ UP ]  
[Report a bug]   [Manage cookies]                
Skip to main content
Springer Nature Link
Log in

Asymptotic analysis of stochastic programs

  • Theory
  • Published:
Annals of Operations Research Aims and scope Submit manuscript

Abstract

In this paper we discuss a general approach to studying asymptotic properties of statistical estimators in stochastic programming. The approach is based on an extended delta method and appears to be particularly suitable for deriving asymptotics of the optimal value of stochastic programs. Asymptotic analysis of the optimal value will be presented in detail. Asymptotic properties of the corresponding optimal solutions are briefly discussed.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Subscribe and save

Springer+ Basic
$34.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. A. Araujo and E. Giné,The Central Limit Theorem for Real and Banach Valued Random Variables (Wiley, 1980).

  2. S. Asmussen and R. Y. Rubinstein, The efficiency and heavy traffic properties of the score function method in sensitivity analysis of queueing models, Preprint, Aalborg University, Denmark, and Technion, Haifa, Israel (1989).

    Google Scholar 

  3. V. I. Averbukh and O. G. Smolyanov, The theory of differentiation in linear topological spaces. Russian Math. Surveys 22 (1967) 201–258.

    Google Scholar 

  4. P. Billingsley,Convergence of Probability Measures (Wiley, 1968).

  5. R. Cominetti, Metric regularity tangent sets and second order optimality conditions, Appl. Math. Optim. 21 (1990) 265–287.

    Google Scholar 

  6. J.M. Danskin,The theory of Max-min and Its Applications to Weapons Allocation Problems (Springer, 1967).

  7. G.B. Dantzig and W. Glynn, Parallel processors for planning under uncertainty, Preprint, Stanford University (1989).

  8. J. Dupačová, Stability in stochastic programming with recourse. Contaminated distribution, Math. Progr. Study 27 (1986) 133–144.

    Google Scholar 

  9. J. Dupačová, Stochastic programming with incomplete information: A survey of results on post optimization and sensitivity analysis, Optimization 18 (1987) 507–532.

    Google Scholar 

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

    Google Scholar 

  11. Yu. Ermoliev and R. J.-B. Wets (eds.),Numerical Techniques for Stochastic Optimization (Springer, Berlin, 1988).

    Google Scholar 

  12. L. T. Fernholz, Von Mises calculus for statistical functionals,Lecture Notes in Statistics (Springer, New York, 1983).

    Google Scholar 

  13. A. V. Fiacco,Introduction to Sensitivity and Stability Analysis in Nonlinear Programming (Academic Press, 1983).

  14. J. Gauvin and F. Dubeau, Differential properties of the marginal function in mathematical programming. Math. Progr. Study 19 (1982) 101–119.

    Google Scholar 

  15. J. Gauvin and R. Janin, Directional behaviour of optimal solutions in nonlinear mathematical programming, Math. Oper. Res. 13 (1988) 629–649.

    Google Scholar 

  16. R. D. Gill, Non- and semiparametric maximum likelihood estimators and the von Mises method (Part I), Scand. J. Statist. 16 (1989) 97–124.

    Google Scholar 

  17. E. G. Golshtein,Theory of Convex Programming, Trans. Math. Monographs 26 (American Mathematical Society, Providence, RI, 1972).

    Google Scholar 

  18. R. Grübel, The length of the shorth, Ann. Statist. 16 (1988) 619–628.

    Google Scholar 

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

    Google Scholar 

  20. P.J. Huber, The behaviour of maximum likelihood estimates under nonstandard condition, in:Proc. 5th Berkeley Symp. on Mathematical Statistics and Probability, vol. 1 (University of California Press, Berkeley, 1967) pp. 221–233.

    Google Scholar 

  21. P. J. Huber,Robust Statistics (Wiley, 1981).

  22. A. D. Ioffe and V. M. Tihomirov,Theory of Extremal Problems (North-Holland, 1979).

  23. A. J. King, Asymptotic behavior of solutions in stochastic optimization: nonsmooth analysis and the derivation of non-normal limit distributions, Dissertation, University of Washington (1986).

  24. A. J. King, Asymptotic distributions for solutions in stochastic optimization and generalizedM-estimation, Working paper, International Institute for Applied Systems Analysis, Laxenburg, Austria (1988).

    Google Scholar 

  25. A. J. King. Generalized delta theorems for multivalued mappings and measurable selections. Math. Oper. Res. 14 (1989) 720–736.

    Google Scholar 

  26. A. J. King and R. T. Rockafellar, Sensitivity analysis for nonsmooth generalized equations. Working paper, IBM Research Division, T.J. Watson Research Center (1989).

  27. J. Kyparisis, On uniqueness of Kuhn-Tucker multipliers in nonlinear programming, Math. Progr. 32 (1985) 242–246.

    Google Scholar 

  28. F. Lempio and H. Maurer. Differential stability in infinite-dimensional nonlinear programming, Appl. Math. Optim. 6 (1980) 139–152.

    Google Scholar 

  29. E. S. Levitin, On differential properties of the optimum value of parametric problems of mathematical programming, Sov. Math. Dokl. 15 (1974) 603–608.

    Google Scholar 

  30. M. Z. Nashed, Differentiability and related properties of nonlinear operators: some aspects of the role of differentials in nonlinear functional analysis, in:Nonlinear Functional Analysis and Applications, ed. L. B. Rall (Academic Press, New York, 1971) pp. 103–309.

    Google Scholar 

  31. D. Pollard, Asymptotics via empirical processes, Statist. Sci. 4 (1989) 341–366.

    Google Scholar 

  32. C. R. Rao,Linear Statistical Inference and Its Applications (Wiley, 1973).

  33. J. A. Reeds, On the definition of von Mises functionals, Ph.D. thesis, Harvard University (1976).

  34. S. M. Robinson, Stability theory for systems of inequalities, part II: Differentiable nonlinear systems, SIAM J. Numer. Anal. 13 (1976) 497–513.

    Google Scholar 

  35. S. M. Robinson. Generalized equations and their solutions, part 1: basic theory, Math. Progr. Study 10 (1979) 128–141.

    Google Scholar 

  36. S. M. Robinson, Strongly regular generalized equations, Math. Oper. Res. 5 (1980) 43–62.

    Google Scholar 

  37. S. M. Robinson, An implicit function theorem for B-differentiable functions, Working paper. International Institute for Applied Systems Analysis, Laxenburg, Austria (1988).

    Google Scholar 

  38. R. T. Rockafellar,Convex Analysis (Princeton University Press, 1970).

  39. R. Y. Rubinstein, Monte Carlo methods for performance evaluation, sensitivity analysis and optimization of stochastic systems, to appear in:Encyclopedia of Computer Science and Technology, eds. Kent and Williams (1990).

  40. A. Shapiro, Second-order sensitivity analysis and asymptotic theory of parametrized nonlinear programs, Math. Progr. 33 (1985) 280–299.

    Google Scholar 

  41. A. Shapiro, Gâteaux, Frechet and Hadamard directional differentiability of functional solutions in stochastic programming, Operations Research and Statistics Mimeograph Series No. 395. Technion, Haifa. Israel (1988).

    Google Scholar 

  42. A. Shapiro, Asymptotic properties of statistical estimators in stochastic programming. Ann. Statist. 17 (1989) 841–858.

    Google Scholar 

  43. A. Shapiro, On differential stability in stochastic programming, Math. Progr. 47 (1990) 107–116.

    Google Scholar 

  44. A. Shapiro, On concepts of directional differentiability, J. Optim. Theory Appl. 66 (1990) 477–487.

    Google Scholar 

  45. C. Ursescu, Multifunctions with convex closed graph, Czech. Math. J. 25 (1975) 438–441.

    Google Scholar 

  46. J. A. Wellner, Discussion of Gill's paper “Non- and semiprametric maximum likelihood estimators and the von Mises method (Part I)”, Scand. J. Statist. 16 (1989) 124–127.

    Google Scholar 

  47. R. J.-B. Wets, Stochastic programming: Solution techniques and approximation schemes, in:Mathematical Programming. The State of the Art, eds. A. Bachem, M. Grötschel and B. Korte (Springer, Berlin, 1983).

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics and Applied Mathematics, University of South Africa, 0001, Pretoria, South Africa

    Alexander Shapiro

Authors
  1. Alexander Shapiro
    View author publications

    You can also search for this author in PubMed Google Scholar

Rights and permissions

Reprints and permissions

About this article

Cite this article

Shapiro, A. Asymptotic analysis of stochastic programs. Ann Oper Res 30, 169–186 (1991). https://doi.org/10.1007/BF02204815

Download citation

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF02204815

Keywords

Search

Navigation