Polyhedral Risk Measures in Stochastic Programming

We consider stochastic programs with risk measures in the objective and study stability properties as well as decomposition structures. Thereby we place emphasis on dynamic models, i.e., multistage stochastic programs with multiperiod risk measures. In this context, we define the class of polyhedral risk measures such that stochastic programs with risk measures taken from this class have favorable properties. Polyhedral risk measures are defined as optimal values of certain linear stochastic programs where the arguments of the risk measure appear on the right-hand side of the dynamic constraints. Dual representations for polyhedral risk measures are derived and used to deduce criteria for convexity and coherence. As examples of polyhedral risk measures we propose multiperiod extensions of the Conditional-Value-at-Risk.

[1]  Peter Kall,et al.  Stochastic Programming , 1995 .

[2]  S. Weber.  Distribution-Invariant Dynamic Risk Measures , 2003 .

[3]  Thorsten Rheinländer.  Risk Management: Value at Risk and Beyond , 2003 .

[4]  Werner Römisch,et al.  Polyhedral risk measures in electricity portfolio optimization , 2004 .

[5]  David Heath,et al.  Coherent multiperiod risk adjusted values and Bellman’s principle , 2007, Ann. Oper. Res..

[6]  Philippe Artzner,et al.  Coherent Measures of Risk , 1999 .

[7]  Wlodzimierz Ogryczak,et al.  On consistency of stochastic dominance and mean–semideviation models , 2001, Math. Program..

[8]  S. Kusuoka.  On law invariant coherent risk measures , 2001 .

[9]  Andrzej Ruszczynski,et al.  Decomposition methods in stochastic programming , 1997, Math. Program..

[10]  R. Schultz,et al.  Multistage Stochastic Integer Programs: An Introduction , 2001 .

[11]  R. Wets,et al.  STABILITY OF ε-APPROXIMATE SOLUTIONS TO CONVEX STOCHASTIC PROGRAMS∗ , 2007 .

[12]  B. WETSt,et al.  STOCHASTIC PROGRAMS WITH FIXED RECOURSE : THE EQUIVALENT DETERMINISTIC PROGRAM , 2022 .

[13]  Wlodzimierz Ogryczak,et al.  From stochastic dominance to mean-risk models: Semideviations as risk measures , 1999, Eur. J. Oper. Res..

[14]  R. Rockafellar,et al.  Deviation Measures in Risk Analysis and Optimization , 2002 .

[15]  Claude Lemaréchal,et al.  Bundle Methods in Stochastic Optimal Power Management: A Disaggregated Approach Using Preconditioners , 2001, Comput. Optim. Appl..

[16]  Alexander Shapiro,et al.  Optimization of Convex Risk Functions , 2006, Math. Oper. Res..

[17]  R. Tyrrell Rockafellar.  Conjugate Duality and Optimization , 1974 .

[18]  W. Römisch.  Stability of Stochastic Programming Problems , 2003 .

[19]  Rüdiger Schultz,et al.  Conditional Value-at-Risk in Stochastic Programs with Mixed-Integer Recourse , 2006, Math. Program..

[20]  K. Kiwiel,et al.  Power management in a hydro-thermal system under uncertainty by Lagrangian relaxation , 2002 .

[21]  Mark A. McComb.  Comparison Methods for Stochastic Models and Risks , 2003, Technometrics.

[22]  R. Rockafellar,et al.  Conditional Value-at-Risk for General Loss Distributions , 2001 .

[23]  Georg Ch. Pflug,et al.  A risk measure for income processes , 2004 .

[24]  Werner Römisch,et al.  Stochastic Lagrangian Relaxation Applied to Power Scheduling in a Hydro-Thermal System under Uncertainty , 2000, Ann. Oper. Res..

[25]  Wlodzimierz Ogryczak,et al.  Dual Stochastic Dominance and Related Mean-Risk Models , 2002, SIAM J. Optim..

[26]  Alexander Shapiro,et al.  On a Class of Minimax Stochastic Programs , 2004, SIAM J. Optim..

[27]  Jim Freeman.  Probability Metrics and the Stability of Stochastic Models , 1991 .

[28]  G. Pflug.  Some Remarks on the Value-at-Risk and the Conditional Value-at-Risk , 2000 .

[29]  Georg Ch. Pflug.  The Value of Perfect Information as a Risk Measure , 2004 .

[30]  Philippe Artzner,et al.  COHERENT MULTIPERIOD RISK MEASUREMENT , 2002 .

[31]  Alexander Schied,et al.  Convex measures of risk and trading constraints , 2002, Finance Stochastics.

[32]  Rüdiger Schultz,et al.  Risk Aversion via Excess Probabilities in Stochastic Programs with Mixed-Integer Recourse , 2003, SIAM J. Optim..

[33]  H. Föllmer,et al.  Stochastic Finance: An Introduction in Discrete Time , 2002 .

[34]  R. Wets,et al.  Stochastic programming , 1989 .

[35]  Takayuki Shiina,et al.  Stochastic Programming with Integer Variables , 2007, CSC.

[36]  Werner Römisch,et al.  Duality gaps in nonconvex stochastic optimization , 2004, Math. Program..

[37]  G. P. Szegö,et al.  Risk measures for the 21st century , 2004 .

[38]  Svetlozar T. Rachev,et al.  Quantitative Stability in Stochastic Programming: The Method of Probability Metrics , 2002, Math. Oper. Res..

[39]  Frank Riedel,et al.  Dynamic Coherent Risk Measures , 2003 .

[40]  F. Delbaen.  Coherent Risk Measures on General Probability Spaces , 2002 .