Time-consistent approximations of risk-averse multistage stochastic optimization problems

In this paper we study the concept of time consistency as it relates to multistage risk-averse stochastic optimization problems on finite scenario trees. We use dynamic time-consistent formulations to approximate problems having a single coherent risk measure applied to the aggregated costs over all time periods. The dual representation of coherent risk measures is used to create a time-consistent cutting plane algorithm. Additionally, we also develop methods for the construction of universal time-consistent upper bounds, when the objective function is the mean-semideviation measure of risk. Our numerical results indicate that the resulting dynamic formulations yield close approximations to the original problem.

[1]  Antonio Alonso Ayuso,et al.  Introduction to Stochastic Programming , 2009 .

[2]  M. Teboulle,et al.  AN OLD‐NEW CONCEPT OF CONVEX RISK MEASURES: THE OPTIMIZED CERTAINTY EQUIVALENT , 2007 .

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

[4]  Irina Penner.  Dynamic convex risk measures: time consistency, prudence, and sustainability , 2007 .

[5]  T. Ralphs,et al.  Decomposition Methods , 2010 .

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

[7]  Andrzej Ruszczynski,et al.  Two-stage portfolio optimization with higher-order conditional measures of risk , 2012, Ann. Oper. Res..

[8]  G. Pflug,et al.  Modeling, Measuring and Managing Risk , 2008 .

[9]  Vitor L. de Matos,et al.  Dynamic sampling algorithms for multi-stage stochastic programs with risk aversion , 2012, Eur. J. Oper. Res..

[10]  Andrzej Ruszczynski,et al.  Scenario decomposition of risk-averse multistage stochastic programming problems , 2012, Ann. Oper. Res..

[11]  B. Roorda,et al.  COHERENT ACCEPTABILITY MEASURES IN MULTIPERIOD MODELS , 2005 .

[12]  A. Ruszczynski,et al.  Nonlinear Optimization , 2006 .

[13]  Marco Frittelli,et al.  Dynamic convex risk measures , 2004 .

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

[15]  Alexandre Street,et al.  Time consistency and risk averse dynamic decision models: Definition, interpretation and practical consequences , 2014, Eur. J. Oper. Res..

[16]  Giacomo Scandolo,et al.  Conditional and dynamic convex risk measures , 2005, Finance Stochastics.

[17]  Martin L. Puterman,et al.  Markov Decision Processes: Discrete Stochastic Dynamic Programming , 1994 .

[18]  Patrick Cheridito,et al.  Time-Inconsistency of VaR and Time-Consistent Alternatives , 2007 .

[19]  Alexander Shapiro,et al.  Bounds for nested law invariant coherent risk measures , 2012, Oper. Res. Lett..

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

[21]  M. Frittelli,et al.  RISK MEASURES AND CAPITAL REQUIREMENTS FOR PROCESSES , 2006 .

[22]  Andrzej Ruszczynski,et al.  Risk-averse dynamic programming for Markov decision processes , 2010, Math. Program..

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

[24]  Melvyn Sim,et al.  Constructing Risk Measures from Uncertainty Sets , 2009, Oper. Res..

[25]  Gennady Samorodnitsky,et al.  Subadditivity Re–Examined: the Case for Value-at-Risk , 2005 .

[26]  Zdzisław Denkowski,et al.  Set-Valued Analysis , 2021 .

[27]  M. V. F. Pereira,et al.  Multi-stage stochastic optimization applied to energy planning , 1991, Math. Program..

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

[29]  Erlon Cristian Finardi,et al.  On Solving Multistage Stochastic Programs with Coherent Risk Measures , 2013, Oper. Res..

[30]  Werner Römisch,et al.  Sampling-Based Decomposition Methods for Multistage Stochastic Programs Based on Extended Polyhedral Risk Measures , 2012, SIAM J. Optim..

[31]  F. Delbaen.  Coherent risk measures , 2000 .

[32]  S. Weber,et al.  DISTRIBUTION‐INVARIANT RISK MEASURES, INFORMATION, AND DYNAMIC CONSISTENCY , 2006 .

[33]  Alexander Shapiro,et al.  On Kusuoka Representation of Law Invariant Risk Measures , 2013, Math. Oper. Res..

[34]  Richard W. Zurek,et al.  Interannual variability of planet-encircling dust storms on Mars , 1993 .

[35]  Alexander Shapiro,et al.  Conditional Risk Mappings , 2005, Math. Oper. Res..

[36]  Alexander Shapiro,et al.  On a time consistency concept in risk averse multistage stochastic programming , 2009, Oper. Res. Lett..

[37]  R. Rockafellar.  Conjugate Duality and Optimization , 1987 .

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

[39]  V. Kozmík,et al.  Risk-Averse Stochastic Dual Dynamic Programming , 2013 .

[40]  Alexander Shapiro,et al.  Analysis of stochastic dual dynamic programming method , 2011, Eur. J. Oper. Res..

[41]  M. Teboulle,et al.  Expected Utility, Penalty Functions, and Duality in Stochastic Nonlinear Programming , 1986 .

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

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

[44]  Gábor Rudolf,et al.  Optimization with Multivariate Conditional Value-at-Risk Constraints , 2013, Oper. Res..

[45]  Darinka Dentcheva,et al.  Kusuoka representation of higher order dual risk measures , 2010, Ann. Oper. Res..

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

[47]  Terry J. Lyons,et al.  Stochastic finance. an introduction in discrete time , 2004 .

[48]  John R. Birge,et al.  Introduction to Stochastic programming (2nd edition), Springer verlag, New York , 2011 .

[49]  R. Ibragimov.  Portfolio diversification and value at risk under thick-tailedness , 2004 .

[50]  Berend Roorda,et al.  Time Consistency Conditions for Acceptability Measures, with an Application to Tail Value at Risk , 2007 .

[51]  Marek Petrik,et al.  Tight Approximations of Dynamic Risk Measures , 2011, Math. Oper. Res..

[52]  Toshinao Yoshiba,et al.  On the Validity of Value-at-Risk: Comparative Analyses with Expected Shortfall , 2002 .

[53]  Kumpati S. Narendra,et al.  Identification and control of dynamical systems using neural networks , 1990, IEEE Trans. Neural Networks.

[54]  A. Stuart,et al.  Portfolio Selection: Efficient Diversification of Investments. , 1960 .

[55]  A. Stuart,et al.  Portfolio Selection: Efficient Diversification of Investments , 1959 .

[56]  M. Frittelli,et al.  Putting order in risk measures , 2002 .

[57]  Naomi Miller,et al.  Risk-Averse Two-Stage Stochastic Linear Programming: Modeling and Decomposition , 2011, Oper. Res..

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

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

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

[61]  Claudia Sagastizábal,et al.  Optimization of real asset portfolio using a coherent risk measure: application to oil and energy industries , 2011 .

[62]  Leslie Lamport,et al.  Latex : A Document Preparation System , 1985 .

[63]  Stanislav Uryasev,et al.  Conditional Value-at-Risk for General Loss Distributions , 2002 .

[64]  Alexander Shapiro,et al.  Lectures on Stochastic Programming - Modeling and Theory, Second Edition , 2014, MOS-SIAM Series on Optimization.

[65]  D. Tasche,et al.  Expected Shortfall: a natural coherent alternative to Value at Risk , 2001, cond-mat/0105191.

[66]  Peter Kall,et al.  Stochastic Linear Programming , 1975 .

[67]  Georg Ch. Pflug,et al.  Time-Consistent Decisions and Temporal Decomposition of Coherent Risk Functionals , 2016, Math. Oper. Res..

[68]  F. Delbaen,et al.  Dynamic Monetary Risk Measures for Bounded Discrete-Time Processes , 2004, math/0410453.