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Expected residual minimization formulation for a class of stochastic linear second-order cone complementarity problems

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  • Wang, Guoxin
  • Zhang, Jin
  • Zeng, Bo
  • Lin, Gui-Hua

Abstract

This paper considers a class of stochastic linear second-order cone complementarity problems (SLSOCCP). Noticing that the SLSOCCP does not have a solution suitable to all realizations in general, we present a deterministic formulation, called the expected residual minimization (ERM) formulation, for it. The coercive property of the ERM problem and the robustness of its solutions are discussed. Due to the existence of expectation in the ERM problem, we employ the Monte Carlo approximation techniques to approximate the ERM problem and show that, under mild conditions, this approximation approach possesses exponential convergence rate. Then, we extend the above results to a general mixed SLSOCCP. Furthermore, we apply the theoretical results to a stochastic optimal power flow model in radial network and report some numerical dispatching experiments for real-world Southern California Edison 47-bus network.

Suggested Citation

  • Wang, Guoxin & Zhang, Jin & Zeng, Bo & Lin, Gui-Hua, 2018. "Expected residual minimization formulation for a class of stochastic linear second-order cone complementarity problems," European Journal of Operational Research, Elsevier, vol. 265(2), pages 437-447.
    • Handle: RePEc:eee:ejores:v:265:y:2018:i:2:p:437-447
    DOI: 10.1016/j.ejor.2017.09.008
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    References listed on IDEAS

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    1. Jein-Shan Chen & Shaohua Pan, 2010. "A one-parametric class of merit functions for the second-order cone complementarity problem," Computational Optimization and Applications, Springer, vol. 45(3), pages 581-606, April.
    2. Egging, Ruud, 2013. "Benders Decomposition for multi-stage stochastic mixed complementarity problems – Applied to a global natural gas market model," European Journal of Operational Research, Elsevier, vol. 226(2), pages 341-353.
    3. Jein-Shan Chen, 2006. "Two classes of merit functions for the second-order cone complementarity problem," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 64(3), pages 495-519, December.
    4. Lopez, Marco & Still, Georg, 2007. "Semi-infinite programming," European Journal of Operational Research, Elsevier, vol. 180(2), pages 491-518, July.
    5. Gabriel, Steven A. & Zhuang, Jifang & Egging, Ruud, 2009. "Solving stochastic complementarity problems in energy market modeling using scenario reduction," European Journal of Operational Research, Elsevier, vol. 197(3), pages 1028-1040, September.
    6. Xiaojun Chen & Masao Fukushima, 2005. "Expected Residual Minimization Method for Stochastic Linear Complementarity Problems," Mathematics of Operations Research, INFORMS, vol. 30(4), pages 1022-1038, November.
    7. G. L. Zhou & L. Caccetta, 2008. "Feasible Semismooth Newton Method for a Class of Stochastic Linear Complementarity Problems," Journal of Optimization Theory and Applications, Springer, vol. 139(2), pages 379-392, November.
    8. Shaohua Pan & Jein-Shan Chen, 2010. "A semismooth Newton method for SOCCPs based on a one-parametric class of SOC complementarity functions," Computational Optimization and Applications, Springer, vol. 45(1), pages 59-88, January.
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