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Inequalities for Stochastic Nonlinear Programming Problems

Author

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  • O. L. Mangasarian

    (Shell Development Company, Emeryville, California)

  • J. B. Rosen

    (Shell Development Company, Emeryville, California)

Abstract

Many actual situations can be represented in a realistic manner by the two-stage stochastic nonlinear programming problem Min x E min y [(phi)( x ) + (psi)( y )] subject to g ( x ) + h ( y ) ≧ b , where b is a random vector with a known distribution, and E denotes expectation taken with respect to the distribution of b . Madansky has obtained upper and lower bounds on the optimum solution to this two-stage problem for the completely linear case. In the present paper these results are extended, under appropriate convexity, concavity, and continuity conditions, to the two-stage nonlinear problem. In many cases of practical interest the calculation of these bounds will require only slightly more effort than two solutions of a deterministic problem of the same size, that is, a problem with a known constant value for the vector b . A small nonlinear numerical example illustrates the calculation of these bounds. For this example the bounds closely bracket the optimum solution to the two-stage problem.

Suggested Citation

  • O. L. Mangasarian & J. B. Rosen, 1964. "Inequalities for Stochastic Nonlinear Programming Problems," Operations Research, INFORMS, vol. 12(1), pages 143-154, February.
    • Handle: RePEc:inm:oropre:v:12:y:1964:i:1:p:143-154
    DOI: 10.1287/opre.12.1.143
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    Cited by:

    1. Li, Changmin & Yang, Hai & Zhu, Daoli & Meng, Qiang, 2012. "A global optimization method for continuous network design problems," Transportation Research Part B: Methodological, Elsevier, vol. 46(9), pages 1144-1158.
    2. Dokumacı, Emin & Sandholm, William H., 2011. "Large deviations and multinomial probit choice," Journal of Economic Theory, Elsevier, vol. 146(5), pages 2151-2158.

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