Multivariate robust second-order stochastic dominance and resulting risk-averse optimization

ABSTRACT By utilizing a min-biaffine scalarization function, we define the multivariate robust second-order stochastic dominance relationship to flexibly compare two random vectors. We discuss the basic properties of the multivariate robust second-order stochastic dominance and relate it to the nonpositiveness of a functional which is continuous and subdifferentiable everywhere. We study a stochastic optimization problem with multivariate robust second-order stochastic dominance constraints and develop the necessary and sufficient conditions of optimality in the convex case. After specifying an ambiguity set based on moments information, we approximate the ambiguity set by a series of sets consisting of discrete distributions. Furthermore, we design a convex approximation to the proposed stochastic optimization problem with multivariate robust second-order stochastic dominance constraints and establish its qualitative stability under Kantorovich metric and pseudo metric, respectively. All these results lay a theoretical foundation for the modelling and solution of complex stochastic decision-making problems with multivariate robust second-order stochastic dominance constraints.

[1]  Herbert E. Scarf,et al.  A Min-Max Solution of an Inventory Problem , 1957 .

[2]  Haim Levy,et al.  Stochastic Dominance, Efficiency Criteria, and Efficient Portfolios: The Multi-Period Case , 1973 .

[3]  J. Diestel,et al.  On vector measures , 1974 .

[4]  R. Phelps Convex Functions, Monotone Operators and Differentiability , 1989 .

[5]  Stephen P. Boyd,et al.  The worst-case risk of a portfolio , 2000 .

[6]  J. Frédéric Bonnans,et al.  Perturbation Analysis of Optimization Problems , 2000, Springer Series in Operations Research.

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

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

[9]  Laurent El Ghaoui,et al.  Worst-Case Value-At-Risk and Robust Portfolio Optimization: A Conic Programming Approach , 2003, Oper. Res..

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

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

[12]  A. Ruszczynski,et al.  Portfolio optimization with stochastic dominance constraints , 2006 .

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

[14]  Garud Iyengar,et al.  Ambiguous chance constrained problems and robust optimization , 2006, Math. Program..

[15]  Stochastic Orders , 2008 .

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

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

[18]  Darinka Dentcheva,et al.  Optimization with multivariate stochastic dominance constraints , 2008, SIAM J. Optim..

[19]  Sanjay Mehrotra,et al.  A Cutting-Surface Method for Uncertain Linear Programs with Polyhedral Stochastic Dominance Constraints , 2009, SIAM J. Optim..

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

[21]  H. Levy Stochastic Dominance: Investment Decision Making under Uncertainty , 2010 .

[22]  Darinka Dentcheva,et al.  Robust stochastic dominance and its application to risk-averse optimization , 2010, Math. Program..

[23]  M. Denuit,et al.  Generalized Increasing Convex and Directionally Convex Orders , 2010, Journal of Applied Probability.

[24]  Yinyu Ye,et al.  Distributionally Robust Optimization Under Moment Uncertainty with Application to Data-Driven Problems , 2010, Oper. Res..

[25]  Rüdiger Schultz,et al.  Risk Management with Stochastic Dominance Models in Energy Systems with Dispersed Generation , 2011 .

[26]  Rüdiger Schultz,et al.  A note on second-order stochastic dominance constraints induced by mixed-integer linear recourse , 2011, Math. Program..

[27]  Li Chen,et al.  Tight Bounds for Some Risk Measures, with Applications to Robust Portfolio Selection , 2011, Oper. Res..

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

[29]  Jian Hu,et al.  Sample Average Approximation for Stochastic Dominance Constrained Programs , 2009 .

[30]  Daniel Kuhn,et al.  Distributionally robust joint chance constraints with second-order moment information , 2011, Mathematical Programming.

[31]  G. Burton Sobolev Spaces , 2013 .

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

[33]  Yong Wang,et al.  Exact Penalization, Level Function Method, and Modified Cutting-Plane Method for Stochastic Programs with Second Order Stochastic Dominance Constraints , 2013, SIAM J. Optim..

[34]  Dimitri Drapkin Models and algorithms for dominance-constrained stochastic programs with recourse , 2014 .

[35]  Abdel Lisser,et al.  Distributionally Robust Stochastic Knapsack Problem , 2014, SIAM J. Optim..

[36]  James R. Luedtke,et al.  Models and formulations for multivariate dominance-constrained stochastic programs , 2015 .

[37]  Thierry Post,et al.  A general test for SSD portfolio efficiency , 2015, OR Spectr..

[38]  Güzin Bayraksan,et al.  Data-Driven Stochastic Programming Using Phi-Divergences , 2015 .

[39]  Ruiwei Jiang,et al.  Data-driven chance constrained stochastic program , 2015, Mathematical Programming.

[40]  Nilay Noyan,et al.  Cut generation for optimization problems with multivariate risk constraints , 2015, Mathematical Programming.

[41]  A. Kleywegt,et al.  Distributionally Robust Stochastic Optimization with Wasserstein Distance , 2016, Math. Oper. Res..

[42]  J. Li Closed-Form Solutions for Worst-Case Law Invariant Risk Measures with Application to Robust Portfolio Optimization , 2016, 1609.04065.

[43]  Liwei Zhang,et al.  Probability approximation schemes for stochastic programs with distributionally robust second-order dominance constraints , 2017, Optim. Methods Softw..

[44]  Yongchao Liu,et al.  Distributionally robust optimization with matrix moment constraints: Lagrange duality and cutting plane methods , 2017, Mathematical Programming.

[45]  Gábor Rudolf,et al.  Optimization with Stochastic Preferences Based on a General Class of Scalarization Functions , 2018, Oper. Res..

[46]  Daniel Kuhn,et al.  Data-driven distributionally robust optimization using the Wasserstein metric: performance guarantees and tractable reformulations , 2015, Mathematical Programming.

[47]  Jonathan Yu-Meng Li,et al.  Closed-Form Solutions for Worst-Case Law Invariant Risk Measures with Application to Robust Portfolio Optimization , 2016, Oper. Res..

[48]  G. Hunanyan,et al.  Portfolio Selection , 2019, Finanzwirtschaft, Banken und Bankmanagement I Finance, Banks and Bank Management.

[49]  Huifu Xu,et al.  Quantitative stability analysis for minimax distributionally robust risk optimization , 2018, Mathematical Programming.