Multilevel Optimization Modeling for Risk-Averse Stochastic Programming
Published Online:27 Jan 2016https://doi.org/10.1287/ijoc.2015.0665
References
- (2002) Spectral measures of risk: A coherent representation of subjective risk aversion. J. Banking Finance 26:1505–1518.Crossref, Google Scholar
- (1999) Coherent measures of risk. Math. Finance 9:203–228.Crossref, Google Scholar
- (2015) Time-consistent approximations of risk-averse multistage stochastic optimization problems. Math. Programming 153:459–493.Crossref, Google Scholar
- (1991) Some properties of the bilevel programming problem. J. Optim. Theory Appl. 68:371–378.Crossref, Google Scholar
- (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
- (1990) Computational difficulties of bilevel linear programming. Oper. Res. 38:556–560.Link, Google Scholar
- (2006) Interior-point algorithms, penalty methods and equilibrium problems. Comput. Optim. Appl. 34:155–182.Crossref, Google Scholar
- (2006) Time consistent dynamic risk measures. Math. Methods Oper. Res. 63:169–186.Crossref, Google Scholar
- (2006) Dynamic monetary risk measures for bounded discrete-time processes. Electron. J. Probab. 11:57–106.Crossref, Google Scholar
- (2012) Scenario decomposition of risk-averse multistage stochastic programming problems. Ann. Oper. Res. 200:147–170.Crossref, Google Scholar
- (2009) A structure-conveying modelling language for mathematical and stochastic programming. Math. Programming Comput. 1:223–247.Crossref, Google Scholar
- (2002) Coherent risk measures on general probability spaces. Sandmann K, Schönbucher PJ, eds. Advances in Finance and Stochastics (Springer, Berlin), 1–37.Crossref, Google Scholar
- (2004) Stochastic Finance: An Introduction in Discrete Time, 2nd ed. (DeGruyter, Berlin).Crossref, Google Scholar
- (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
- (1992) New branch-and-bound rules for linear bilevel programming. SIAM J. Sci. Statist. Comput. 13:1194–1217.Crossref, Google Scholar
- (1985) The polynomial hierarchy and a simple model for competitive analysis. Math. Programming 32:146–164.Crossref, Google Scholar
- (2009) Representation results for law invariant time consistent functions. Math. Financial Econom. 2:189–210.Crossref, Google Scholar
- (1996) Mathematical Programs with Equilibrium Constraints (Cambridge University Press, Cambridge, UK).Crossref, Google Scholar
- (1991) A note on the Pareto optimality of solutions to the linear bilevel programming problem. Comput. Oper. Res. 18:355–359.Crossref, Google Scholar
- (2004) Dynamic coherent risk measures. Stochastic Processes Appl. 112:185–200.Crossref, Google Scholar
- (2000) Optimization of conditional value-at-risk. J. Risk 2:21–42.Crossref, Google Scholar
- (2002) Conditional value-at-risk for general loss distributions. J. Banking Finance 26:1442–1471.Crossref, Google Scholar
- (2006) Generalized deviations in risk analysis. Finance Stochastics 10:51–74.Crossref, Google Scholar
- (2010) Risk-averse dynamic programming for Markov decision processes. Math. Programming 125:235–261.Crossref, Google Scholar
- (2006a) Conditional risk mappings. Math. Oper. Res. 31:544–561.Link, Google Scholar
- (2006b) Optimization of convex risk functions. Math. Oper. Res. 31:433–452.Link, Google Scholar
- (2003) Risk measures in a dynamic setting. Ph.D. thesis, University of Milan, Milan.Google Scholar
- (2012) Time consistency of dynamic risk measures. Oper. Res. Lett. 40:436–439.Crossref, Google Scholar
- (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.Crossref, Google Scholar
- (2009) Extending algebraic modelling languages for stochastic programming. INFORMS J. Comput. 21:107–122.Link, Google Scholar
- (2012) PySP: Modeling and solving stochastic programs in Python. Math. Programming Comput. 4:109–149.Crossref, Google Scholar
- (2015) Multi-Level Decision Making: Models, Methods and Applications, Intelligent Systems Reference Library, Vol. 82 (Springer, Berlin).Crossref, Google Scholar

