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Stability analysis of stochastic programs with second order dominance constraints

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Abstract

In this paper we present a stability analysis of a stochastic optimization problem with stochastic second order dominance constraints. We consider a perturbation of the underlying probability measure in the space of regular measures equipped with pseudometric discrepancy distance (Römisch in Stochastic Programming. Elsevier, Amsterdam, pp 483–554, 2003). By exploiting a result on error bounds in semi-infinite programming due to Gugat (Math Program Ser B 88:255–275, 2000), we show under the Slater constraint qualification that the optimal value function is Lipschitz continuous and the optimal solution set mapping is upper semicontinuous with respect to the perturbation of the probability measure. In particular, we consider the case when the probability measure is approximated by an empirical probability measure and show an exponential rate of convergence of the sequence of optimal solutions obtained from solving the approximation problem. The analysis is extended to the stationary points.

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Notes

  1. The property is also known as calmness of \({\vartheta }\) at \(P\), see Section F in [34, Chapter 8] for general discussions on calmness.

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Acknowledgments

The authors are grateful to the two anonymous referees, the associate editor, Tito Homem-de-Mello and Werner Römisch for their valuable suggestions and comments which have significantly helped improve the presentation of the paper.

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Authors and Affiliations

  1. Department of Mathematics, Dalian Maritime University, Dalian, 116023, China

    Yongchao Liu

  2. School of Mathematics, University of Southampton, Southampton, SO17 1BJ, UK

    Huifu Xu

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  1. Yongchao Liu
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Correspondence to Huifu Xu.

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The work of Yongchao Liu is carried out while he is visiting Huifu Xu in the School of Mathematics, University of Southampton sponsored by China Scholarship Council. It is also supported by the Fundamental Research Funds for the Central Universities (N0.2012TD032) and National Natural Science Foundation of China.

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Liu, Y., Xu, H. Stability analysis of stochastic programs with second order dominance constraints. Math. Program. 142, 435–460 (2013). https://doi.org/10.1007/s10107-012-0585-0

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