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Risk Aversion in Two-Stage Stochastic Integer Programming

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Stochastic Programming

Part of the book series: International Series in Operations Research & Management Science ((ISOR,volume 150))

Abstract

Some recent developments in the area of risk aversion in stochastic integer programming are surveyed. After a discussion of modeling guidelines and resulting mean–risk stochastic integer programs emphasis is placed on structural properties of these optimization problems and on algorithms for their solution. Bibliographical notes conclude the Chapter.

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References

  • Ahmed, S.: Convexity and decomposition of mean-risk stochastic programs. Math. Program. 106(3), 433–446 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  • Alonso-Ayuso, A., Escudero, L. F., Garín, A., Ortuño, M. T., Pérez, G.: An approach for strategic supply chain planning under uncertainty based on stochastic 0-1 programming. J. Global Optim. 26(1), 97–124 (2003a)

    Article  MathSciNet  MATH  Google Scholar 

  • Alonso-Ayuso, A., Escudero, L.F., Ortuño, M.T.: BFC: A branch-and-fix coordination algorithmic framework for solving some types of stochastic pure and mixed 0-1 programs. Eur. J. Oper. Res. 151(3), 503–519 (2003b)

    Article  MATH  Google Scholar 

  • Artzner, P., Delbaen, F., Eber, J.-M., Heath, D., Ku, H.: Coherent multiperiod risk adjusted values and bellman’s principle. Ann. Oper. Res. 152(1), 5–22 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  • Bank, B., Mandel, R.: Parametric Integer Optimization. Akademie-Verlag, Berlin (1988)

    MATH  Google Scholar 

  • Bank, B., Guddat, J., Klatte, D., Kummer, B., Tammer, K.: Non-linear Parametric Optimization. Akademie-Verlag, Berlin (1982)

    Google Scholar 

  • Bereanu, B.: Programme de risque minimal en programmation linéaire stochastique. Comptes Rendus de l’ Académie des Sciences Paris 259(5), 981–983 (1964)

    MathSciNet  MATH  Google Scholar 

  • Bereanu, B.: Minimum risk criterion in stochastic optimization. Econ. Comput. Econ. Cybern. Stud. Res. 2, 31–39 (1981)

    MathSciNet  Google Scholar 

  • Billingsley, P.: Convergence of Probability Measures. Wiley, New York, NY (1968)

    MATH  Google Scholar 

  • Birge, J.R., Louveaux, F.: Introduction to Stochastic Programming. Springer, New York, NY (1997)

    MATH  Google Scholar 

  • Blair, C.E., Jeroslow, R.G.: The value function of a mixed integer program: I. Discr. Math. 19, 121–138 (1977)

    Article  MathSciNet  MATH  Google Scholar 

  • Bonnans, J.F., Shapiro, A.: Perturbation Analysis of Optimization Problems. Springer, New York, NY (2000)

    MATH  Google Scholar 

  • Carøe, C.C., Schultz, R.: Dual decomposition in stochastic integer programming. Oper. Res. Lett. 24(1–2), 37–45 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  • Charnes, A., Cooper, W.W.: Deterministic equivalents for optimizing and satisficing under chance constraints. Oper. Res. 11(1), 18–38 (1963)

    Article  MathSciNet  MATH  Google Scholar 

  • Dentcheva, D., Römisch, W.: Duality gaps in nonconvex stochastic optimization. Math. Program. 101(3), 515–535 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  • Dentcheva, D., Ruszczyński, A.: Stochastic optimization with dominance constraints. SIAM J. Optim. 14(2), 548–566 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  • Dentcheva, D., Ruszczyński, A.: Optimality and duality theory for stochastic optimization with nonlinear dominance constraints. Math. Program. 99(2), 329–350 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  • Eichhorn, A., Römisch, W.: Polyhedral risk measures in stochastic programming. SIAM J. Optim. 16(1), 69–95 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  • Helmberg, C., Kiwiel, K.C.: A spectral bundle method with bounds. Math. Program. 93(7), 173–194 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  • Kall, P., Wallace, S.W.: Stochastic Programming. Wiley, Chichester (1994)

    MATH  Google Scholar 

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

    MATH  Google Scholar 

  • Kiwiel, K.C.: Proximity control in bundle methods for convex nondifferentiable optimization. Math. Program. 46(1–2), 105–122 (1990)

    Article  MathSciNet  MATH  Google Scholar 

  • Kristoffersen, T.: Deviation measures in linear two-stage stochastic programming. Math. Methods Oper. Res. 62(2), 255–274 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  • Levy, H.: Stochastic dominance and expected utility: survey and analysis. Manage. Sci. 38(4), 555–593 (1992)

    Article  MATH  Google Scholar 

  • Louveaux, F.V., Schultz, R.: Stochastic integer programming. In: Ruszczyński, A., Shapiro, A., (eds.) Stochastic Programming, Handbooks in Operations Research and Management Science, pp. 213–266. Elsevier, Amsterdam (2003)

    Chapter  Google Scholar 

  • Märkert, A., Schultz, R.: On deviation measures in stochastic integer programming. Oper. Res. Lett. 33(5), 441–449 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  • Markowitz, H.M.: Portfolio selection. J. Finance 7(1), 77–91 (1952)

    Google Scholar 

  • Müller, A., Stoyan, D.: Comparison Methods for Stochastic Models and Risks. Wiley, Chichester (2002)

    MATH  Google Scholar 

  • Mulvey, J.M., Vanderbei, R.J., Zenios, S.A.: Robust optimization of large-scale systems. Oper. Res. 43(2), 264–281 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  • Ogryczak, W., Ruszczyński, A.: From stochastic dominance to mean-risk models: Semideviations as risk measures. Eur. J. Oper. Res. 116(1), 33–50 (1999)

    Article  MATH  Google Scholar 

  • Ogryczak, W., Ruszczyński, A.: Dual stochastic dominance and related mean-risk models. SIAM J. Optim. 13(1), 60–78 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  • Pflug, G.C.: Some remarks on the value-at-risk and the conditional value-at-risk. In: Uryasev, S., (ed.) Probabilistic Constrained Optimization: Methodology and Applications, pp. 272–281. Kluwer, Dordrecht (2000)

    Google Scholar 

  • Pflug, G.C.: A value-of-information approach to measuring risk in multiperiod economic activity. J. Bank. Finance. 30(2), 695–715 (2006)

    Article  MathSciNet  Google Scholar 

  • Pflug, G.C., Ruszczyński, A.: Risk measures for income streams. In: Szegö, G., (ed.) Risk Measures for the 21st. Century, pp. 249–269. Wiley, Chichester (2004)

    Google Scholar 

  • Prékopa, A.: Stochastic Programming. Kluwer, Dordrecht (1995)

    Google Scholar 

  • Raik, E.: Qualitative research into the stochastic nonlinear programming problems. Eesti NSV Teaduste Akademia Toimetised / Füüsika, Matemaatica (News of the Estonian Academy of Sciences / Physics, Mathematics), 20, 8–14 (1971) In Russian

    MathSciNet  MATH  Google Scholar 

  • Raik, E.: On the stochastic programming problem with the probability and quantile functionals. Eesti NSV Teaduste Akademia Toimetised / Füüsika, Matemaatica (News of the Estonian Academy of Sciences / Physics, Mathematics), 21, 142–148 (1972) In Russian

    MathSciNet  MATH  Google Scholar 

  • Riis, M., Schultz, R.: Applying the minimum risk criterion in stochastic recourse programs. Comput. Optim. Appl. 24(2–3), 267–287 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  • Robinson, S.M.: Local epi-continuity and local optimization. Math. Program. 37(2), 208–222 (1987)

    Article  MATH  Google Scholar 

  • Rockafellar, R.T., Uryasev, S.: Conditional value-at-risk for general loss distributions. J. Bank. Finance. 26(7), 1443–1471 (2002)

    Article  Google Scholar 

  • Römisch, W.: Stability of stochastic programming problems. In: Ruszczyński, A.A., Shapiro, A., (eds.) Stochastic Programming, Handbooks in Operations Research and Management Science, pp. 483–554. Elsevier, Amsterdam (2003)

    Chapter  Google Scholar 

  • Ruszczyński, A., Shapiro, A.: Stochastic Programming. Handbooks in Operations Research and Management Science. Elsevier, Amsterdam (2003)

    Google Scholar 

  • Ruszczyński, A., Vanderbei, R.J.: Frontiers of stochastically nondominated portfolios. Econometrica 71(4), 1287–1297 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  • Schultz, R.: On structure and stability in stochastic programs with random technology matrix and complete integer recourse. Math. Program. 70(1–3), 73–89 (1995)

    MATH  Google Scholar 

  • Schultz, R.: Some aspects of stability in stochastic programming. Ann. Oper. Res. 100(1–4), 55–84 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  • Schultz, R.: Mixed-integer value functions in stochastic programming. In: Jünger, M., Reinelt, G., Rinaldi, G., (eds.) Combinatorial Optimization – Eureka, You Shrink!, Papers Dedicated to Jack Edmonds, LNCS 2570, pp. 171–184. Springer, Berlin (2003)

    Chapter  Google Scholar 

  • Schultz, R., Tiedemann, S.: Risk aversion via excess probabilities in stochastic programs with mixed-integer recourse. SIAM J. Optim. 14(1), 115–138 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  • Schultz, R., Tiedemann, S.: Conditional value-at-risk in stochastic programs with mixed-integer recourse. Math. Program. 105(2–3), 365–386 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  • Takriti, S., Ahmed, S.: On robust optimization of two-stage systems. Math. Program. 99(1), 109–126 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  • Tiedemann, S.: Risk measures with preselected tolerance levels in two-stage stochastic mixed-integer programming. PhD thesis, University of Duisburg-Essen, Cuvillier Verlag, Göttingen (2005)

    Google Scholar 

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Authors and Affiliations

  1. Department of Mathematics, University of Duisburg-Essen, Campus Duisburg, Duisburg, Germany

    Rüdiger Schultz

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  1. Rüdiger Schultz
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Correspondence to Rüdiger Schultz .

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  1. Dept. Management Science & Engineering, Stanford University, Stanford, 94305, California, USA

    Gerd Infanger

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Schultz, R. (2010). Risk Aversion in Two-Stage Stochastic Integer Programming. In: Infanger, G. (eds) Stochastic Programming. International Series in Operations Research & Management Science, vol 150. Springer, New York, NY. https://doi.org/10.1007/978-1-4419-1642-6_8

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