Skip to main content

Part of the book series: The Kluwer International Series in Engineering and Computer Science ((SECS,volume 373))

  • 324 Accesses

Abstract

In this chapter we approach the main problem of finding the minimizer of

$$F(x) = \int {H(x,\omega )d{{\mu }_{x}}(\omega )}$$

by discussing various notions of differentiability of parameter integrals.

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

Access this chapter

Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 169.00
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Hardcover Book
USD 219.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

Bibliography

  1. Aubin J.P. and Frankowska H. (1990). Set valued analysis. Birkhäuser Verlag, Basel.

    Google Scholar 

  2. Bickel P.J. and Freedman, D.A. (1981). Some asymptotic theory for the bootstrap. Ann. Statist. 9 (6), 1196–1217.

    Article  Google Scholar 

  3. Billingsley P. (1968). Convergence of Probability Measures. J. Wiley & Sons, New York.

    Google Scholar 

  4. Devroye L. (1986). Non-Uniform Random Variate Generation. Springer Verlag, New York, Berlin, Heidelberg.

    Google Scholar 

  5. Dunford N. and Schwartz, J.T. (1957). Linear operators, Part I. Inter-science publishers, New York.

    Google Scholar 

  6. Donoghue W.F. (1969). Distributions and Fourier Transforms. Pure and Applied Mathematics 32, Academic Press, New York.

    Google Scholar 

  7. Evtushenko Y. (1990). Automatic differentiation vieved from optimal control theory. In: Automatic differentiation of algorithms. (A. Griewank, G. Corliss eds.), SIAM, Philadelphia.

    Google Scholar 

  8. Ermoliev Yu. (1976). Methods of stochastic programming. Monographs in Optimization and Operations Research, Nauka, Moskwa.

    Google Scholar 

  9. Ermoliev Yu., Gaivoronski A. (1991). On Optimization of Discontinuous Systems. IIASA Working paper WP-91–49, Laxenburg, Austria.

    Google Scholar 

  10. Ermoliev Yu., Norkin V. and Wets R. (1995). The minimization of semi-continuous functions: mollifier subgradients. Siam J. on Control and Optimization, 33 (1), 149–167.

    Article  Google Scholar 

  11. Glynn P.W. (1987). Construction of process-differentiable representations for parametric families of distributions. University of Wisconsin-Madison, Mathematics Research Center.

    Google Scholar 

  12. Gong W.B. and Bremaud P. (1993). Derivatives of likelihood ratios and smoothed perturbation analysis for the routing problem. ACM Trans, on Modeling and Comp. Simul. 3 (2), 134–161.

    Article  Google Scholar 

  13. Gong W.B. and Ho Y.C. (1987). Smoothed (Conditional) Perturbation Analysis of Discrete Event Systems. IEEE Transactions on Automatic Control 32, 858–866.

    Google Scholar 

  14. Ho Y.C. and Cao X. (1983). Perturbation Analysis and Optimization of Queueing Networks. J. Optim. Theory Applic. 40 (4), 559–582.

    Article  Google Scholar 

  15. Ho Y.C. and Cao X. (1991). Perturbation Analysis of Discrete Event Systems. Kluwer, Boston.

    Book  Google Scholar 

  16. Lang, S. (1973). Calculus of several variables. Addison Wesley, Reading, Massachusetts.

    Google Scholar 

  17. Major P. (1978). On the invariance principle for sums of independent, identically distributed random variables. J. Multivariate Analysis 8, 487–501.

    Article  Google Scholar 

  18. Pflug G. Ch. (1988). Derivatives of probability measures — concepts and applications to the optimization of stochastic systems. Lecture Notes in Control, Vol. 103 (P. Varaiya and A. Kurzhanski eds.), Springer Verlag, 252–274.

    Google Scholar 

  19. Pflug, G. Ch. (1989). Sampling derivatives of probability measures. Computing 42, 315–328.

    Google Scholar 

  20. Raik E. (1975). The differentiability in the parameter of the probability function and optimization via the stochastic pseudogradient method. Izvestiya Akad. Nauk Est. SSR, Phis. Math. 24 (1), 3–6 (in Russian).

    Google Scholar 

  21. Rosenblatt M. (1956). Remarks on some non-parametric estimates of a density function. Ann. Math. Statist. 27, 3, 832–837.

    Article  Google Scholar 

  22. Rubinstein R.Y. (1981). Simulation and the Monte Carlo Method. John Wiley & Sons, New York.

    Book  Google Scholar 

  23. Rubinstein R.Y. and Shapiro A. (1992). Discrete Event Systems: Sensitivity Analysis and Stochastic Optimization by the Score Function Method. John Wiley & Sons, New York.

    Google Scholar 

  24. Rubinstein R.Y. (1992). Sensitivity Analysis of Discrete Event Systems by the “Push Out” Method. Ann. Oper. Res. 39, 229–250.

    Article  Google Scholar 

  25. Rudin W. (1962). Fourier Analysis on Groups. Interscience Publishers, (J. Wiley & Sons) New York.

    Google Scholar 

  26. Sahin I. (1990). Regenerative Inventory Systems. Bilkent University Lecture Series, Springer Verlag, New York.

    Google Scholar 

  27. Suri R. (1987). Infinitesimal perturbation analysis for general discrete-event systems. Journal of the ACM 34 (3), 686–717.

    Article  Google Scholar 

  28. Uryasev S. (1989). A differentiation formula for integrals over sets given by inclusion. Numer. Funct. Anal, and Optimiz. 10, 827–841.

    Article  Google Scholar 

  29. Uryasev S. (1995). Derivatives of probability functions and integrals over sets given by inequalities. J. Computational and Applied Mathematics 56, 197–223.

    Article  Google Scholar 

  30. Vallander S.S. (1973). Calculation of the Wasserstein distance between probability distributions on the line. Theor. Prob. Appi 18, 784–786.

    Article  Google Scholar 

Download references

Authors

Rights and permissions

Reprints and permissions

Copyright information

© 1996 Kluwer Academic Publishers

About this chapter

Cite this chapter

Pflug, G.C. (1996). Derivatives. In: Optimization of Stochastic Models. The Kluwer International Series in Engineering and Computer Science, vol 373. Springer, Boston, MA. https://doi.org/10.1007/978-1-4613-1449-3_3

Download citation

  • DOI: https://doi.org/10.1007/978-1-4613-1449-3_3

  • Publisher Name: Springer, Boston, MA

  • Print ISBN: 978-1-4612-8631-8

  • Online ISBN: 978-1-4613-1449-3

  • eBook Packages: Springer Book Archive

Publish with us

Policies and ethics