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Variable Neighborhood Search for a Two-Stage Stochastic Programming Problem with a Quantile Criterion

  • Stochastic Systems
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Abstract

We consider a two-stage stochastic programming problem with a bilinear loss function and a quantile criterion. The problem is reduced to a single-stage stochastic programming problem with a quantile criterion. We use the method of sample approximations. The resulting approximating problem is considered as a stochastic programming problem with a discrete distribution of random parameters. We check convergence conditions for the sequence of solutions of approximating problems. Using the confidence method, the problem is reduced to a combinatorial optimization problem where the confidence set represents an optimization strategy. To search for the optimal confidence set, we adapt the variable neighborhood search method. To solve the problem, we develop a hybrid algorithm based on the method of sample approximations, the confidence method, variable neighborhood search.

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References

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

    MATH  Google Scholar 

  2. Shapiro, A., Dentcheva, D., and Ruszczyński, A., Lectures on Stochastic Programming, Philadelphia: Society for Industrial and Applied Mathematics (SIAM), 2009.

    Book  Google Scholar 

  3. Beraldi, P. and Ruszczyński, A., A Branch and Bound Method for Stochastic Integer Problems under Probabilistic Constraints, Optim. Method. Software, 2002, vol. 17, no. 3, pp. 359–382.

    Article  MathSciNet  MATH  Google Scholar 

  4. Kibzun, A.I. and Kan, Y.S., Stochastic Programming Problems with Probability and Quantile Functions, Chichester: Wiley, 1996.

    MATH  Google Scholar 

  5. Kibzun, A.I., Comparison of Two Algorithms for Solving a Two-Stage Bilinear Stochastic Programming Problem with Quantile Criterion, Appl. Stoch. Model. Business Industry, 2015, vol. 31, no. 6, pp. 862–874.

    Article  MathSciNet  MATH  Google Scholar 

  6. Artstein, Z. and Wets, R.J.-B., Consistency of Minimizers and the SLLN for Stochastic Programs, J. Convex Anal., 1996, vol. 2, pp. 1–17.

    MathSciNet  MATH  Google Scholar 

  7. Pagnoncelli, B.K., Ahmed, S., and Shapiro, A., Sample Average ApproximationMethod for Chance Constrained Programming: Theory and Applications, J. Optim. Theory Appl., 2009, vol. 142, pp. 399–416.

    Article  MathSciNet  MATH  Google Scholar 

  8. Ivanov, S.V. and Kibzun, A.I., On the Convergence of Sample Approximations for Stochastic Programming Problems with Probabilistic Criteria, Autom. Remote Control, 2018, vol. 79, no. 2, pp. 216–228.

    Article  MathSciNet  MATH  Google Scholar 

  9. Ivanov, S.V. and Kibzun, A.I., Sample Average Approximation in the Two-Stage Stochastic Linear Programming Problem with Quantile Criterion, Tr. Inst. Mat. Mekh. UrO RAN, 2017, vol. 23, no. 3, pp. 134–143.

    Google Scholar 

  10. Guigues, V., Juditsky, A., and Nemirovski, A., Non-Asymptotic Confidence Bounds for the Optimal Value of a Stochastic Program, Optim. Method. Software, 2017, vol. 32, no. 5, pp. 1033–1058.

    Article  MathSciNet  MATH  Google Scholar 

  11. Lepp, R., Approximate Solution of Stochastic Programming Problems with Recourse, Kybernetika, 1987, vol. 23, no. 6, pp. 476–482.

    MathSciNet  MATH  Google Scholar 

  12. Lepp, R., Approximation of the Quantile Minimization Problem with Decision Rules, Optim. Method. Software, 2002, vol. 17, no. 3, pp. 505–522.

    Article  MathSciNet  MATH  Google Scholar 

  13. Benati, S. and Rizzi, R., A Mixed Integer Linear Programming Formulation of the Optimal Mean/Valueat-Risk Portfolio Problem, Eur. J. Oper. Res., 2007, vol. 176, no. 1, pp. 423–434.

    Article  MATH  Google Scholar 

  14. Norkin, V.I., Kibzun, A.I., and Naumov, A.V., Reducing Two-Stage Probabilistic Optimization Problems with Discrete Distribution of Random Data to Mixed-Integer Programming Problems, Cybernet. Syst. Anal., 2014, vol. 50, no. 5, pp. 679–692.

    Article  MathSciNet  MATH  Google Scholar 

  15. Ruszczyński, A., Probabilistic Programming with Discrete Distributions and Precedence Constrained Knapsack Polyhedra, Math. Program., 2002, vol. 93, pp. 195–215.

    Article  MathSciNet  MATH  Google Scholar 

  16. Mladenović, N. and Hansen, P., Variable Neighborhood Search, Comput. Oper. Res., 1997, vol. 24, no. 11, pp. 1097–1100.

    Article  MathSciNet  MATH  Google Scholar 

  17. Hansen, P., Mladenović, N., and Pérez, J.A.M., Variable Neighbourhood Search: Methods and Applications, Ann. Oper. Res., 2010, vol. 175, no. 1, pp. 367–407.

    Article  MathSciNet  MATH  Google Scholar 

  18. Hansen, P. and Mladenović, N., Variable Neighborhood Search, in Handbook of Heuristics, Mart´ı, R., Pardalos, P., and Resende, M., Eds., Cham: Springer, 2016.

    Google Scholar 

  19. Hansen, P., Mladenović, N., Todosijević, R., and Hanafi, S., Variable Neighborhood Search: Basics and Variants, EURO J. Comput. Optim., 2017, vol. 5, no. 5, pp. 423–454.

    Article  MathSciNet  MATH  Google Scholar 

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Authors and Affiliations

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

    S. V. Ivanov & A. I. Kibzun

  2. Emirates College of Technologies, Abu Dhabi, UAE

    N. Mladenović

  3. Ural Federal University, Yekaterinburg, Russia

    N. Mladenović

Authors
  1. S. V. Ivanov
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  2. A. I. Kibzun
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  3. N. Mladenović
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Corresponding author

Correspondence to S. V. Ivanov.

Additional information

Russian Text © S.V. Ivanov, A.I. Kibzun, N. Mladenovi´c, 2019, published in Avtomatika i Telemekhanika, 2019, No. 1, pp. 54–66.

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Ivanov, S.V., Kibzun, A.I. & Mladenović, N. Variable Neighborhood Search for a Two-Stage Stochastic Programming Problem with a Quantile Criterion. Autom Remote Control 80, 43–52 (2019). https://doi.org/10.1134/S0005117919010041

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  • DOI: https://doi.org/10.1134/S0005117919010041

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