Advances in Risk-Averse Optimization

We discuss main ideas of formalizing risk-averse preferences and using them in opti- mization problems: utility theories, risk measures, and stochastic order constraints. We also present methods for solving risk-averse optimization problems. Multistage models are discussed as well.

[1]  A. Shiryaev,et al.  Limit Theorems for Stochastic Processes , 1987 .

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

[3]  E. Lehmann.  Ordered Families of Distributions , 1955 .

[4]  J. Neumann,et al.  Theory of games and economic behavior , 1945, 100 Years of Math Milestones.

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

[6]  Gábor Rudolf,et al.  Optimization Problems with Second Order Stochastic Dominance Constraints: Duality, Compact Formulations, and Cut Generation Methods , 2008, SIAM J. Optim..

[7]  D. Schmeidler.  Integral representation without additivity , 1986 .

[8]  Darinka Dentcheva,et al.  Inverse stochastic dominance constraints and rank dependent expected utility theory , 2006, Math. Program..

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

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

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

[12]  Alexander Shapiro,et al.  Lectures on Stochastic Programming: Modeling and Theory , 2009 .

[13]  Shabbir Ahmed,et al.  Convexity and decomposition of mean-risk stochastic programs , 2006, Math. Program..

[14]  P. Krokhmal.  Higher moment coherent risk measures , 2007 .

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

[16]  P. Fishburn.  The Foundations Of Expected Utility , 2010 .

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

[18]  András Prékopa.  Static Stochastic Programming Models , 1995 .

[19]  Alexander Shapiro,et al.  Uniqueness of Kusuoka Representations , 2013 .

[20]  Darinka Dentcheva,et al.  Optimization with Stochastic Dominance Constraints , 2003, SIAM J. Optim..

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

[22]  G. Debreu.  Mathematical Economics: Representation of a preference ordering by a numerical function , 1983 .

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

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

[25]  A. Skorokhod.  Limit Theorems for Stochastic Processes , 1956 .

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

[27]  A. Ruszczynski,et al.  Semi-infinite probabilistic optimization: first-order stochastic dominance constrain , 2004 .

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

[29]  J. Aubin.  Set-valued analysis , 1990 .

[30]  T. Rader,et al.  The Existence of a Utility Function to Represent Preferences , 1963 .

[31]  G. Debreu.  Mathematical Economics: Continuity properties of Paretian utility , 1964 .

[32]  Josef Hadar,et al.  Rules for Ordering Uncertain Prospects , 1969 .

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

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

[35]  R. Rockafellar,et al.  Optimization of conditional value-at risk , 2000 .

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

[37]  M. Rothschild,et al.  Increasing risk: I. A definition , 1970 .

[38]  Masaaki Kijima,et al.  Mean-risk analysis of risk aversion and wealth effects on optimal portfolios with multiple investment opportunities , 1993, Ann. Oper. Res..

[39]  Darinka Dentcheva,et al.  Optimality and duality theory for stochastic optimization problems with nonlinear dominance constraints , 2004, Math. Program..

[40]  Alexandra Künzi-Bay,et al.  Computational aspects of minimizing conditional value-at-risk , 2006, Comput. Manag. Sci..

[41]  Maria Gabriela Martinez,et al.  Two-stage stochastic optimization problems with stochastic ordering constraints on the recourse , 2012, Eur. J. Oper. Res..

[42]  Gautam Mitra,et al.  Processing second-order stochastic dominance models using cutting-plane representations , 2011, Math. Program..

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

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

[45]  James R. Luedtke.  New Formulations for Optimization under Stochastic Dominance Constraints , 2008, SIAM J. Optim..

[46]  H. Föllmer,et al.  Convex risk measures and the dynamics of their penalty functions , 2006 .

[47]  J. Quirk,et al.  Admissibility and Measurable Utility Functions , 1962 .

[48]  Maarten H. van der Vlerk,et al.  Integrated Chance Constraints: Reduced Forms and an Algorithm , 2006, Comput. Manag. Sci..

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

[50]  John M. Wilson,et al.  Introduction to Stochastic Programming , 1998, J. Oper. Res. Soc..

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

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

[53]  V. Strassen.  The Existence of Probability Measures with Given Marginals , 1965 .

[54]  A. Müller,et al.  Comparison Methods for Stochastic Models and Risks , 2002 .

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

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

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

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

[59]  J. Jaffray.  Existence of a Continuous Utility Function: An Elementary Proof , 1975 .

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

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

[62]  M. Yaari.  The Dual Theory of Choice under Risk , 1987 .

[63]  S. Eilenberg.  Ordered Topological Spaces , 1941 .

[64]  Stan Uryasev,et al.  Modeling and optimization of risk , 2011 .

[65]  Darinka Dentcheva,et al.  Common Mathematical Foundations of Expected Utility and Dual Utility Theories , 2012, SIAM J. Optim..

[66]  Dudley,et al.  Real Analysis and Probability: Integration , 2002 .

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

[68]  Nilay Noyan,et al.  Valid inequalities and restrictions for stochastic programming problems with first order stochastic dominance constraints , 2008, Math. Program..

[69]  R. Rockafellar,et al.  Generalized Deviations in Risk Analysis , 2004 .

[70]  G. Debreu.  ON THE CONTINUITY PROPERTIES OF PARETIAN UTILITY , 1963 .

[71]  Darinka Dentcheva,et al.  Inverse cutting plane methods for optimization problems with second-order stochastic dominance constraints , 2010 .

[72]  Gábor Rudolf,et al.  Relaxations of linear programming problems with first order stochastic dominance constraints , 2006, Oper. Res. Lett..

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

[74]  Willem K. Klein Haneveld.  Integrated Chance Constraints , 1986 .

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