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Stability Analysis of One Stage Stochastic Mathematical Programs with Complementarity Constraints

Author

Listed:
  • Yongchao Liu

    (Dalian University of Technology)

  • Huifu Xu

    (University of Southampton)

  • Gui-Hua Lin

    (Dalian University of Technology)

Abstract

We study the quantitative stability of the solution sets, optimal value and M-stationary points of one stage stochastic mathematical programs with complementarity constraints when the underlying probability measure varies in some metric probability space. We show under moderate conditions that the optimal solution set mapping is upper semi-continuous and the optimal value function is Lipschitz continuous with respect to probability measure. We also show that the set of M-stationary points as a mapping is upper semi-continuous with respect to the variation of the probability measure. A particular focus is given to empirical probability measure approximation which is also known as sample average approximation (SAA). It is shown that optimal value and M-stationary points of SAA programs converge to their true counterparts with probability one (w.p.1.) at exponential rate as the sample size increases.

Suggested Citation

  • Yongchao Liu & Huifu Xu & Gui-Hua Lin, 2012. "Stability Analysis of One Stage Stochastic Mathematical Programs with Complementarity Constraints," Journal of Optimization Theory and Applications, Springer, vol. 152(2), pages 537-555, February.
    • Handle: RePEc:spr:joptap:v:152:y:2012:i:2:d:10.1007_s10957-011-9903-6
    DOI: 10.1007/s10957-011-9903-6
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    References listed on IDEAS

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