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The Decomposition Method for Two-Stage Stochastic Linear Programming Problems with Quantile Criterion

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Abstract

We consider the two-stage stochastic linear programming problem with quantile criterion in case when the vector of random parameters has a discrete distribution with a finite number of realizations. Based on the confidence method and duality theorems, we construct a decompositional algorithm for finding guaranteeing solutions.

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References

  1. Birge, J.R., and Louveaux, F., Introduction to Stochastic Programming, New York: Springer Verlag, 1997.

    MATH  Google Scholar 

  2. Yudin, D.B., Zadachi i metody stokhasticheskogo programmirovaniya (Problems and Methods in Stochastic Programming), Moscow: Sovetskoe Radio, 1979.

    MATH  Google Scholar 

  3. Kibzun, A.I. and Kan, Yu.S., Zadachi stokhasticheskogo programmirovaniya s veroyatnostnymi kriteriyami (Stochastic Programming Problems with Probabilistic Criteria), Moscow: Fizmatlit, 2009.

    MATH  Google Scholar 

  4. Bunto, T.V. and Kan, Yu.S., Quantile Criterion-Based Control of the Securities Portfolio with a Nonzero Ruin Probability, Autom. Remote Control, 2013, vol. 74, no. 5, pp. 811–828.

    Article  MathSciNet  MATH  Google Scholar 

  5. Vishnyakov, B.V. and Kibzun, A.I., A Two-Step Capital Variation Model: Optimization by Different Statistical Criteria, Autom. Remote Control, 2005, vol. 66, no. 7, pp. 1137–1152.

    Article  MathSciNet  MATH  Google Scholar 

  6. Pankov, A.R., Platonov, E.N., and Semenikhin, K.V., Minimax Optimization of Investment Portfolio by Quantile Criterion, Autom. Remote Control, 2003, vol. 64, no. 7, pp. 1122–1137.

    Article  MathSciNet  MATH  Google Scholar 

  7. Naumov, A.V. and Ulanov, S.V., Risk in Two-Stage Optimal Resource Allocation, Autom. Remote Control, 2003, vol. 64, no. 7, pp. 1115–1121.

    Article  MATH  Google Scholar 

  8. Naumov, A.V. and Bogdanov, A.B., Solution to a Two-Step Logistics Problem in a Quintile Statement, Autom. Remote Control, 2006, vol. 67, no. 12, pp. 1893–1899.

    Article  MathSciNet  MATH  Google Scholar 

  9. Norkin, V.I., Kibzun, A.I., and Naumov, A.V., Reducing Two-Stage Stochastic Optimization Problems with a Discrete Distribution of Random Data to Partially Integer Programming Problems, Kibern. Sist. Anal., 2014, vol. 50, no. 5, pp. 34–48.

    MATH  Google Scholar 

  10. Artstein, Z. and Wets, R.J-B., Consistency of Minimizers and the SLLN for Stochastic Programs, J. Convex Anal., 1995, vol. 2, no. 1/2, pp. 1–17.

    MathSciNet  MATH  Google Scholar 

  11. Naumov, A.V. and Bobylev, I.M., On the Two-Stage Problem of Linear Stochastic Programming with Quantile Criterion and Discrete Distribution of the Random Parameters, Autom. Remote Control, 2012, vol. 73, no. 2, pp. 265–275.

    Article  MathSciNet  MATH  Google Scholar 

  12. Ivanov, S.V. and Naumov, A.V., Algorithm to Optimize the Quantile Criterion for the Polyhedral Loss Function and Discrete Distribution of Random Parameters, Autom. Remote Control, 2012, vol. 73, no. 1, pp. 105–117.

    Article  MathSciNet  MATH  Google Scholar 

  13. Van Slyke, R. and Wets, R.J., L-Shaped Linear Program with Application to Optimal Control and Stochastic Linear Programming, SIAM. J. Appl. Math., 1969, vol. 17, pp. 638–663.

    Article  MathSciNet  MATH  Google Scholar 

  14. Eremin, I.I., Lineinaya optimizatsiya i sistemy lineinykh neravenstv (Linear Optimization and Systems of Linear Inequalities), Moscow: Akademiya, 2007.

    MATH  Google Scholar 

  15. Kibzun, A.I. and Naumov, A.V., Two-Stage Problems of Quantile Linear Programming, Autom. Remote Control, 1994, vol. 56, no. 1, pp. 68–76.

    MathSciNet  MATH  Google Scholar 

  16. Kibzun, A.I., Naumov, A.V, and Ivanov, S.V., The Optimization Problem for the Operation of a Railroad Transportation Node, Upravlen. Bol’shimi Sist., 2012, no. 38, pp. 140–160.

    Google Scholar 

  17. Naumov, A.V. and Ivanov, C.V., The Investment Distribution Problem in the Development of Ground Space Industries, Tr. MAI (Electronic Journal), 2011, no. 50. http://trudymai.ru/published. php?ID=28677

  18. Ashmanov, A.S., Lineinoe programmirovanie (Linear Programming), Moscow: Nauka, 1981.

    MATH  Google Scholar 

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Authors and Affiliations

  1. Moscow Aviation Institute (National State University), Moscow, Russia

    I. D. Zhenevskaya & A. V. Naumov

Authors
  1. I. D. Zhenevskaya
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  2. A. V. Naumov
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Correspondence to I. D. Zhenevskaya.

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Original Russian Text © I.D. Zhenevskaya, A.V. Naumov, 2018, published in Avtomatika i Telemekhanika, 2018, No. 2, pp. 36–50.

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Zhenevskaya, I.D., Naumov, A.V. The Decomposition Method for Two-Stage Stochastic Linear Programming Problems with Quantile Criterion. Autom Remote Control 79, 229–240 (2018). https://doi.org/10.1134/S0005117918020030

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  • DOI: https://doi.org/10.1134/S0005117918020030

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