Risk-Averse Two-Stage Stochastic Program with Distributional Ambiguity

In this paper, we develop a risk-averse two-stage stochastic program (RTSP) that explicitly incorporates the distributional ambiguity covering both discrete and continuous distributions. We formulate RTSP from the perspective of distributional robustness by hedging against the worst-case distribution within an ambiguity set and considering the corresponding expected total cost. In particular, we derive an equivalent reformulation for RTSP that indicates that each worst-case expectation over an L1-norm-based ambiguity set reduces to a convex combination of a conditional value-at-risk and an essential supremum. Our reformulation explicitly shows how additional data can help reduce the conservatism of the problem from the traditional two-stage robust optimization to the traditional risk-neutral two-stage stochastic program (TSP).Accordingly, we develop solution algorithms for the reformulations of RTSP based on the sample average approximation method. We also extend the studies to ambiguity sets based on L∞-...

[1]  Guang Li,et al.  Two-stage network constrained robust unit commitment problem , 2014, Eur. J. Oper. Res..

[2]  Daniel Kuhn,et al.  Worst-Case Value at Risk of Nonlinear Portfolios , 2013, Manag. Sci..

[3]  Marc Goetschalckx,et al.  A stochastic programming approach for supply chain network design under uncertainty , 2004, Eur. J. Oper. Res..

[4]  Robert D. Doverspike,et al.  Network planning with random demand , 1994, Telecommun. Syst..

[5]  He Zhang,et al.  Models and algorithms for distributionally robust least squares problems , 2013, Mathematical Programming.

[6]  Lihua Yu,et al.  Asset price modeling: decision aids for scheduling and hedging (DASH) in deregulated electricity markets: a stochastic programming approach to power portfolio optimization , 2002, WSC '02.

[7]  Daniel Kuhn,et al.  Primal and dual linear decision rules in stochastic and robust optimization , 2011, Math. Program..

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

[9]  Alexander Shapiro,et al.  The Sample Average Approximation Method for Stochastic Discrete Optimization , 2002, SIAM J. Optim..

[10]  M. Sion.  On general minimax theorems , 1958 .

[11]  David P. Morton,et al.  A Sequential Sampling Procedure for Stochastic Programming , 2011, Oper. Res..

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

[13]  David P. Morton,et al.  Monte Carlo bounding techniques for determining solution quality in stochastic programs , 1999, Oper. Res. Lett..

[14]  John R. Birge,et al.  A stochastic model for the unit commitment problem , 1996 .

[15]  A. Ben-Tal,et al.  Adjustable robust solutions of uncertain linear programs , 2004, Math. Program..

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

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

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

[19]  Giuseppe Carlo Calafiore,et al.  Ambiguous Risk Measures and Optimal Robust Portfolios , 2007, SIAM J. Optim..

[20]  R. Wets,et al.  L-SHAPED LINEAR PROGRAMS WITH APPLICATIONS TO OPTIMAL CONTROL AND STOCHASTIC PROGRAMMING. , 1969 .

[21]  Alexander Shapiro,et al.  Sample Average Approximation Method for Chance Constrained Programming: Theory and Applications , 2009, J. Optimization Theory and Applications.

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

[23]  Dimitris Bertsimas,et al.  On the power and limitations of affine policies in two-stage adaptive optimization , 2012, Math. Program..

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

[25]  Anja De Waegenaere,et al.  Robust Solutions of Optimization Problems Affected by Uncertain Probabilities , 2011, Manag. Sci..

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

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

[28]  G. Pflug,et al.  Ambiguity in portfolio selection , 2007 .

[29]  A. Shapiro,et al.  The Sample Average Approximation Method for Stochastic Programs with Integer Recourse , 2002 .

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

[31]  Diego Klabjan,et al.  Robust Stochastic Lot-Sizing by Means of Histograms , 2013 .

[32]  Stergios B. Fotopoulos,et al.  All of Nonparametric Statistics , 2007, Technometrics.

[33]  R. Bass,et al.  Review: P. Billingsley, Convergence of probability measures , 1971 .

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

[35]  E. Beale.  ON MINIMIZING A CONVEX FUNCTION SUBJECT TO LINEAR INEQUALITIES , 1955 .

[36]  Güzin Bayraksan,et al.  Phi-Divergence Constrained Ambiguous Stochastic Programs for Data-Driven Optimization , 2016 .

[37]  James R. Luedtke,et al.  A Sample Approximation Approach for Optimization with Probabilistic Constraints , 2008, SIAM J. Optim..

[38]  Fanwen Meng,et al.  Two-stage stochastic linear programs with incomplete information on uncertainty , 2014, Eur. J. Oper. Res..

[39]  George B. Dantzig,et al.  Linear Programming Under Uncertainty , 2004, Manag. Sci..

[40]  Julia L. Higle,et al.  Stochastic Decomposition: An Algorithm for Two-Stage Linear Programs with Recourse , 1991, Math. Oper. Res..