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Applying the Minimum Risk Criterion in Stochastic Recourse Programs

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Abstract

In the setting of stochastic recourse programs, we consider the problem of minimizing the probability of total costs exceeding a certain threshold value. The problem is referred to as the minimum risk problem and is posed in order to obtain a more adequate description of risk aversion than that of the accustomed expected value problem. We establish continuity properties of the recourse function as a function of the first-stage decision, as well as of the underlying probability distribution of random parameters. This leads to stability results for the optimal solution of the minimum risk problem when the underlying probability distribution is subjected to perturbations. Furthermore, an algorithm for the minimum risk problem is elaborated and we present results of some preliminary computational experiments.

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References

  1. S. Ahmed, M. Tawarmalani, and N.V. Sahinidis, “A finite branch and bound algorithm for two-stage stochastic integer programs,” Submitted for publication. Available at http://www.isye.gatech.edu/∼sahmed, 2000.

  2. Z. Artstein and R.J.-B. Wets, “Stability results for stochastic programs and sensors, allowing for discontinuous objective functions,” SIAM Journal on Optimization, vol. 4, pp. 537-550, 1994.

    Google Scholar 

  3. B. Bank, J. Guddat, D. Klatte, B. Kummer, and K. Tammer, Non-Linear Parametric Optimization, Akademie-Verlag: Berlin, 1982.

    Google Scholar 

  4. B. Bereanu, “Minimum risk criterion in stochastic optimization,” Economic Computation and Economic Cybernetics Studies and Research, vol. 2, pp. 31-39, 1981.

    Google Scholar 

  5. R.N. Bhattacharya and R. Ranga Rao, “Normal Approximation and Assymptotic Expansions,” John Wiley: New York, 1976.

    Google Scholar 

  6. P. Billingsley, Convergence of Probability Measures, Wiley: New York, 1968.

    Google Scholar 

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

    Google Scholar 

  8. C.C. Carøe and R. Schultz, “Dual decomposition in stochastic integer programming,” Operations Research Letters, vol. 24, pp. 37-45, 1999.

    Google Scholar 

  9. J. Dupačová, “Stochastic programming with incomplete information: A survey of results on postoptimization and sensitivity analysis,” Optimization, vol. 18, pp. 507-532, 1987.

    Google Scholar 

  10. W.K. Klein Haneveld and M.H. van der Vlerk, “On the expected value function of a simple integer recourse problem with random technology matrix,” Journal of Computational and Applied Mathematics, vol. 56, pp. 45-53, 1994.

    Google Scholar 

  11. R. Hemmecke and R. Schultz, “Decomposition of test sets in stochastic integer programming,” Mathematical Programming (to appear).

  12. J.L. Higle and S. Sen, “Stochastic decomposition: An algorithm for two-stage linear programs with recourse,” Mathematics of Operations Research, vol. 16, pp. 650-669, 1991.

    Google Scholar 

  13. J. Hoffmann-Jørgensen, Probability with a View Toward Statistics, vol. 1, Chapman &; Hall: New York, 1994.

    Google Scholar 

  14. P. Kall, “On approximations and stability in stochastic programming,” in Parametric Optimization and Related Topics, J. Guddat, H.Th. Jongen, B. Kummer, and F. Nožiøka (Eds.), Akademie Verlag: Berlin, 1987, pp. 387-407.

    Google Scholar 

  15. P. Kall and S.W. Wallace, Stochastic Programming, John Wiley: Chichester, UK, 1994.

    Google Scholar 

  16. A.H.G. Rinooy Kan and L. Stougie, “Stochastic integer programming,” in NumericalTechniques for Stochastic Optimization, Y. Ermoliev and R.J.-B. Wets (Eds.), Springer-Verlag: Berlin, 1988, pp. 201-213.

    Google Scholar 

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

    Google Scholar 

  18. D. Klatte, “A note on quantitative stability results in nonlinear optimization,” in Proceedings of the 19. Jahrestagung Mathematische Optimierung, K. Lommatzsch (Ed.), Seminarbericht Nr. 90, Sektion Mathematik, Humboldt-Universität Berlin, 1987, pp. 77-86.

  19. A. Løkketangen and D.L. Woodruff, “Progressive hedging and tabu search applied to mixed integer (0,1) multistage stochastic programming,” Journal of Heuristics, vol. 2, pp. 111-128, 1996.

    Google Scholar 

  20. F.V. Louveaux and M.H. van der Vlerk, “Stochastic programming with simple integer recourse,” Mathematical Programming, vol. 61, pp. 301-325, 1993.

    Google Scholar 

  21. A. Prékopa, Stochastic Programming, Kluwer Academic Publishers: Dordrecht, Netherlands, 1995.

    Google Scholar 

  22. S.T. Rachev and W. Römisch, “Quantitative stability in stochastic programming: The method of probability metrics,” Preprint 2000–22, Humboldt-Universität zu Berlin, Institut für Mathematik, 2000.

  23. E. Raik, “Qualitative research into the stochastic nonlinear programming problems,” Eesti NSV Teaduste Akademia Toimetised (News of the Estonian Academy of Sciences) Füüs Mat., vol. 20, pp. 8-14, 1971 (In Russian).

    Google Scholar 

  24. E. Raik, “On the stochastic programming problem with the probability and quantile functionals,” Eesti NSV Teaduste Akademia Toimetised (Newsof the Estonian Academy of Sciences) Füüs Mat., vol. 21, pp. 142-148, 1972 (In Russian).

    Google Scholar 

  25. S.M. Robinson, “Local epi-continuity and local optimization,” Mathematical Programming, vol. 37, pp. 208-222, 1987.

    Google Scholar 

  26. S.M. Robinson and R.J.-B. Wets, “Stability in two-stage stochastic programming,” SIAM Journal on Control and Optimization, vol. 25, pp. 1409-1416, 1987.

    Google Scholar 

  27. W. Römisch and R. Schultz, “Distribution sensitivity in stochastic programming,” Mathematical Programming, vol. 50, pp. 197-226, 1991.

    Google Scholar 

  28. W. Römisch and R. Schultz, “Stability analysis for stochastic programs,” Annals of Operations Research, vol. 30, pp. 241-266, 1991.

    Google Scholar 

  29. W. Römisch and R. Schultz, “Stability of solutions for stochastic programs with complete recourse, Mathematics of Operations Research, vol. 18, pp. 590-609, 1993.

    Google Scholar 

  30. A. Ruszczyński, “A regularized decomposition method for minimizing a sum of polyhedral functions,” Mathematical Programming, vol. 35, pp. 309-333, 1986.

    Google Scholar 

  31. R. Schultz, “Continuity and stability in two-stage stochastic integer programming,” in Stochastic Optimization, Numerical Methods and Technical Applications, volume 379 of Lecture Notes in Economics and Mathematical Systems, K. Marti (Ed.), Springer-Verlag: Berlin, 1992, pp. 81-92.

    Google Scholar 

  32. R. Schultz, “Continuity properties of expectation functions in stochastic integer programming,” Mathematics of Operations Research, vol. 18, pp. 578-589, 1993.

    Google Scholar 

  33. R. Schultz, “On structure and stability in stochastic programs with random technology matrix and complete integer recourse,” Mathematical Programming, vol. 70, pp. 73-89, 1995.

    Google Scholar 

  34. R. Schultz, “Rates of convergence in stochastic programs with complete integer recourse,” SIAM Journal on Optimization, vol. 6, pp. 1138-1152, 1996.

    Google Scholar 

  35. R. Schultz, L. Stougie, and M.H. van der Vlerk, “Solving stochastic programs with integer recourse by enumeration: A framework using Grübner basis reductions,” Mathematical Programming, vol. 83, pp. 229-252, 1998.

    Google Scholar 

  36. A. Shapiro, “Quantitative stability instochastic programming,” Mathematical Programming, vol. 67, pp. 99-108, 1994.

    Google Scholar 

  37. L. Stougie, Design and Analysis of Algorithms for Stochastic Integer Programming, CWI Tract, 37. Center for Mathematics and Computer Science: Amsterdam, 1987.

    Google Scholar 

  38. R.M. Van Slyke and R.J.-B. Wets, “L-shaped linear programs with applications to optimal control and stochastic linear programming,” SIAM Journal of Applied Mathematics, vol. 17, pp. 638-663, 1969.

    Google Scholar 

  39. R.J.-B. Wets, “Stochastic programs with fixed recourse: The equivalent deterministic program,” SIAM Review, vol. 16, pp. 309-9339, 1974.

    Google Scholar 

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Authors and Affiliations

  1. Department of Operations Research, University of Aarhus, Building 530, Ny Munkegade, DK—8000, Århus C, Denmark

    Morten Riis

  2. Institute of Mathematics, Gerhard-Mercator University Duisburg, Lotharstraße 65, D—47048, Duisburg, Germany

    Rüdiger Schultz

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  1. Morten Riis
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  2. Rüdiger Schultz
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Correspondence to Morten Riis.

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Riis, M., Schultz, R. Applying the Minimum Risk Criterion in Stochastic Recourse Programs. Computational Optimization and Applications 24, 267–287 (2003). https://doi.org/10.1023/A:1021862109131

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