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Stochastic optimization problems with nondifferentiable cost functionals

  • D. P. Bertsekas
Article

Abstract

In this paper, we examine a class of stochastic optimization problems characterized by nondifferentiability of the objective function. It is shown that, in many cases, the expected value of the objective function is differentiable and, thus, the resulting optimization problem can be solved by using classical analytical or numerical methods. The results are subsequently applied to the solution of a problem of economic resource allocation.

Keywords

Probability Measure Convex Function Optimal Control Problem Directional Derivative Differentiability Property 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

  1. 1.
    Bazaraa, M. S., Goode, J. J., andShetty, C. M.,Optimality Criteria in Nonlinear Programming without Differentiability, Operations Research, Vol. 19, No. 1, 1971.Google Scholar
  2. 2.
    Bazaraa, M. S.,Nonlinear Programming: Nondifferentiable Functions, Georgia Institute of Technology, Ph.D. Thesis, 1971.Google Scholar
  3. 3.
    Bertsekas, D. P., andMitter, S. K.,Steepest Descent for Optimization Problems with Nondifferentiable Cost Functionals, Paper Presented at the 5th Annual Princeton Conference on Information Sciences and Systems, Princeton, New Jersey, 1971.Google Scholar
  4. 4.
    Bertsekas, D. P., andMitter, S. K.,Descent Numerical Methods for Optimization Problems with Nondifferentiable Cost Functionals, SIAM Journal on Control, Vol. 11, No. 4, 1973.Google Scholar
  5. 5.
    Dem'yanov, V. F., andRubinov, A. M.,Minimization of Functionals in Normed Spaces, SIAM Journal on Control, Vol. 6, No. 1, 1968.Google Scholar
  6. 6.
    Heins, W., andMitter, S. K.,Conjugate Convex Functions, Duality, and Optimal Control Problems, I. Systems Governed by Ordinary Differential Equations, Information Sciences, Vol. 2, No. 2, 1970.Google Scholar
  7. 7.
    Chanem, M. Z. E.,Optimal Control Problems with Nondifferentiable Cost Functionals, Stanford University, Department of Engineering-Economic Systems, Ph.D. Thesis, 1970.Google Scholar
  8. 8.
    Luenberger, D. G.,Control Problems with Kinks, IEEE Transactions on Automatic Control, Vol. AC-15, No. 5, 1970.Google Scholar
  9. 9.
    Neustadt, L. W.,A General Theory of Extremals, Journal of Computational and System Science, Vol. 3, No. 1, 1969.Google Scholar
  10. 10.
    Rockafellar, R. T.,Conjugate Convex Functions in Optimal Control and the Calculus of Variations, Journal of Mathematical Analysis and Applications, Vol. 32, No. 1, 1970.Google Scholar
  11. 11.
    Rockafellar, R. T.,Convex Analysis, Princeton University Press, Princeton, New Jersey, 1970.MATHGoogle Scholar
  12. 12.
    Halmos, P. R.,Measure Theory, Van Nostrand Reinhold Company, New York, New York, 1950.MATHGoogle Scholar
  13. 13.
    Dunford, N., andSchwartz, J. T.,Linear Operators, Part I, John Wiley and Sons (Interscience Publishers), New York, New York, 1957.Google Scholar
  14. 14.
    Royden, H. L.,Real Analysis, The Macmillan Company, New York, New York, 1968.Google Scholar
  15. 15.
    Rockafellar, R. T.,Measurable Dependence of Convex Sets and Functions on Parameters, Journal of Mathematical Analysis and Applications, Vol. 28, No. 1, 1969.Google Scholar
  16. 16.
    Aumann, R. J.,Integrals of Set Valued Functions, Journal of Mathematical Analysis and Applications, Vol. 12, No. 1, 1965.Google Scholar
  17. 17.
    Bringland, T. F.,Trajectory Integrals of Set Valued Functions, Pacific Journal of Mathematics, Vol. 33, No. 1, 1970.Google Scholar

Copyright information

© Plenum Publishing Corporation 1973

Authors and Affiliations

  • D. P. Bertsekas
    • 1
  1. 1.Department of Engineering-Economic SystemsStanford UniversityStanford

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