Multilevel Optimization Modeling for Risk-Averse Stochastic Programming

Published Online:https://doi.org/10.1287/ijoc.2015.0665

References

  • Acerbi C (2002) Spectral measures of risk: A coherent representation of subjective risk aversion. J. Banking Finance 26:1505–1518.CrossrefGoogle Scholar
  • Artzner P, Delbaen F, Eber JM, Heath D (1999) Coherent measures of risk. Math. Finance 9:203–228.CrossrefGoogle Scholar
  • Asamov T, Ruszczyński R (2015) Time-consistent approximations of risk-averse multistage stochastic optimization problems. Math. Programming 153:459–493.CrossrefGoogle Scholar
  • Bard JF (1991) Some properties of the bilevel programming problem. J. Optim. Theory Appl. 68:371–378.CrossrefGoogle Scholar
  • Beale EML, Tomlin JA (1970) Special facilities in a general mathematical programming system for non-convex problems using ordered sets of variables. Lawrence J, ed. Proc. Fifth Internat. Conf. Oper. Res. (Tavistock Publications, London), 447–454.Google Scholar
  • Ben-Ayed O, Blair CE (1990) Computational difficulties of bilevel linear programming. Oper. Res. 38:556–560.LinkGoogle Scholar
  • Benson HY, Sen A, Shanno DF, Vanderbei RJ (2006) Interior-point algorithms, penalty methods and equilibrium problems. Comput. Optim. Appl. 34:155–182.CrossrefGoogle Scholar
  • Boda K, Filar JA (2006) Time consistent dynamic risk measures. Math. Methods Oper. Res. 63:169–186.CrossrefGoogle Scholar
  • Cheridito P, Delbaen F, Kupper M (2006) Dynamic monetary risk measures for bounded discrete-time processes. Electron. J. Probab. 11:57–106.CrossrefGoogle Scholar
  • Collado RA, Papp D, Ruszczyński A (2012) Scenario decomposition of risk-averse multistage stochastic programming problems. Ann. Oper. Res. 200:147–170.CrossrefGoogle Scholar
  • Colombo M, Grothey A, Hogg J, Woodsend K, Gondzio J (2009) A structure-conveying modelling language for mathematical and stochastic programming. Math. Programming Comput. 1:223–247.CrossrefGoogle Scholar
  • Delbaen F (2002) Coherent risk measures on general probability spaces. Sandmann K, Schönbucher PJ, eds. Advances in Finance and Stochastics (Springer, Berlin), 1–37.CrossrefGoogle Scholar
  • Föllmer H, Schied A (2004) Stochastic Finance: An Introduction in Discrete Time, 2nd ed. (DeGruyter, Berlin).CrossrefGoogle Scholar
  • Garey MR, Johnson DS (1979) Computers and Intractability: A Guide to the Theory of 𝒩𝒫-Completeness (W. H. Freeman, San Francisco).Google Scholar
  • Gurobi Optimization, Inc. (2014) Gurobi optimizer reference manual. Accessed November 2014, http://www.gurobi.com/documentation/6.0/refman/index.html.Google Scholar
  • Hansen P, Jaumard B, Savard G (1992) New branch-and-bound rules for linear bilevel programming. SIAM J. Sci. Statist. Comput. 13:1194–1217.CrossrefGoogle Scholar
  • Jeroslow RG (1985) The polynomial hierarchy and a simple model for competitive analysis. Math. Programming 32:146–164.CrossrefGoogle Scholar
  • Kupper M, Schachermayer W (2009) Representation results for law invariant time consistent functions. Math. Financial Econom. 2:189–210.CrossrefGoogle Scholar
  • Luo ZQ, Pang JS, Ralph D (1996) Mathematical Programs with Equilibrium Constraints (Cambridge University Press, Cambridge, UK).CrossrefGoogle Scholar
  • Marcotte P, Savard G (1991) A note on the Pareto optimality of solutions to the linear bilevel programming problem. Comput. Oper. Res. 18:355–359.CrossrefGoogle Scholar
  • Riedel F (2004) Dynamic coherent risk measures. Stochastic Processes Appl. 112:185–200.CrossrefGoogle Scholar
  • Rockafellar RT, Uryasev S (2000) Optimization of conditional value-at-risk. J. Risk 2:21–42.CrossrefGoogle Scholar
  • Rockafellar RT, Uryasev S (2002) Conditional value-at-risk for general loss distributions. J. Banking Finance 26:1442–1471.CrossrefGoogle Scholar
  • Rockafellar RT, Uryasev S, Zabarankin M (2006) Generalized deviations in risk analysis. Finance Stochastics 10:51–74.CrossrefGoogle Scholar
  • Ruszczyński A (2010) Risk-averse dynamic programming for Markov decision processes. Math. Programming 125:235–261.CrossrefGoogle Scholar
  • Ruszczyński A, Shapiro A (2006a) Conditional risk mappings. Math. Oper. Res. 31:544–561.LinkGoogle Scholar
  • Ruszczyński A, Shapiro A (2006b) Optimization of convex risk functions. Math. Oper. Res. 31:433–452.LinkGoogle Scholar
  • Scandolo G (2003) Risk measures in a dynamic setting. Ph.D. thesis, University of Milan, Milan.Google Scholar
  • Shapiro A (2012) Time consistency of dynamic risk measures. Oper. Res. Lett. 40:436–439.CrossrefGoogle Scholar
  • Uryasev S, Rockafellar RT (2001) Conditional value-at-risk: Optimization approach. Uryasev S, Pardalos PM, eds. Stochastic Optimization: Algorithms and Applications, Applied Optimization, Vol. 54 (Kluwer, Dordrecht, Netherlands), 411–435.CrossrefGoogle Scholar
  • Valente C, Mitra G, Sadki M, Fourer R (2009) Extending algebraic modelling languages for stochastic programming. INFORMS J. Comput. 21:107–122.LinkGoogle Scholar
  • Watson J-P, Woodruff DL, Hart WE (2012) PySP: Modeling and solving stochastic programs in Python. Math. Programming Comput. 4:109–149.CrossrefGoogle Scholar
  • Zhang G, Lu J, Gao Y (2015) Multi-Level Decision Making: Models, Methods and Applications, Intelligent Systems Reference Library, Vol. 82 (Springer, Berlin).CrossrefGoogle Scholar
INFORMS site uses cookies to store information on your computer. Some are essential to make our site work; Others help us improve the user experience. By using this site, you consent to the placement of these cookies. Please read our Privacy Statement to learn more.