Conic Programming Reformulations of Two-Stage Distributionally Robust Linear Programs over Wasserstein Balls

Adaptive robust optimization problems are usually solved approximately by restricting the adaptive decisions to simple parametric decision rules. However, the corresponding approximation error can be substantial. In this paper we show that two-stage robust and distributionally robust linear programs can often be reformulated exactly as conic programs that scale polynomially with the problem dimensions. Specifically, when the ambiguity set constitutes a 2-Wasserstein ball centered at a discrete distribution, then the distributionally robust linear program is equivalent to a copositive program (if the problem has complete recourse) or can be approximated arbitrarily closely by a sequence of copositive programs (if the problem has sufficiently expensive recourse). These results directly extend to the classical robust setting and motivate strong tractable approximations of two-stage problems based on semidefinite approximations of the copositive cone. We also demonstrate that the two-stage distributionally robust optimization problem is equivalent to a tractable linear program when the ambiguity set constitutes a 1-Wasserstein ball centered at a discrete distribution and there are no support constraints.

[1]  Trevor Hastie,et al.  The Elements of Statistical Learning , 2001 .

[2]  Daniel Kuhn,et al.  A scenario approach for estimating the suboptimality of linear decision rules in two-stage robust optimization , 2011, IEEE Conference on Decision and Control and European Control Conference.

[3]  Daniel Kuhn,et al.  Distributionally robust multi-item newsvendor problems with multimodal demand distributions , 2014, Mathematical Programming.

[4]  Jean-Philippe Vial,et al.  Robust Optimization , 2021, ICORES.

[5]  Samuel Burer,et al.  On the copositive representation of binary and continuous nonconvex quadratic programs , 2009, Math. Program..

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

[7]  Long Zhao,et al.  Solving two-stage robust optimization problems using a column-and-constraint generation method , 2013, Oper. Res. Lett..

[8]  Grani Adiwena Hanasusanto,et al.  Decision making under uncertainty: robust and data-driven approaches , 2015 .

[9]  A. Shapiro ON DUALITY THEORY OF CONIC LINEAR PROBLEMS , 2001 .

[10]  Melvyn Sim,et al.  Distributionally Robust Optimization and Its Tractable Approximations , 2010, Oper. Res..

[11]  Daureen Steinberg COMPUTATION OF MATRIX NORMS WITH APPLICATIONS TO ROBUST OPTIMIZATION , 2007 .

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

[13]  P. Lenk,et al.  Hierarchical Bayes Conjoint Analysis: Recovery of Partworth Heterogeneity from Reduced Experimental Designs , 1996 .

[14]  Etienne de Klerk,et al.  Approximation of the Stability Number of a Graph via Copositive Programming , 2002, SIAM J. Optim..

[15]  Xuan Vinh Doan,et al.  Models for Minimax Stochastic Linear Optimization Problems with Risk Aversion , 2010, Math. Oper. Res..

[16]  Chung-Piaw Teo,et al.  Mixed 0-1 Linear Programs Under Objective Uncertainty: A Completely Positive Representation , 2009, Oper. Res..

[17]  Chung-Piaw Teo,et al.  On reduced semidefinite programs for second order moment bounds with applications , 2017, Math. Program..

[18]  S. T. Buckland,et al.  An Introduction to the Bootstrap. , 1994 .

[19]  Florian Jarre,et al.  New results on the cp-rank and related properties of co(mpletely )positive matrices , 2015 .

[20]  Xiaobo Li,et al.  Distributionally Robust Mixed Integer Linear Programs: Persistency Models with Applications , 2013, Eur. J. Oper. Res..

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

[22]  C. Villani Optimal Transport: Old and New , 2008 .

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

[24]  Daniel Kuhn,et al.  K-adaptability in two-stage distributionally robust binary programming , 2015, Oper. Res. Lett..

[25]  Jonathan Baxter,et al.  A Model of Inductive Bias Learning , 2000, J. Artif. Intell. Res..

[26]  Ruiwei Jiang,et al.  Risk-Averse Two-Stage Stochastic Program with Distributional Ambiguity , 2018, Oper. Res..

[27]  Bastian Goldlücke,et al.  Variational Analysis , 2014, Computer Vision, A Reference Guide.

[28]  Samuel Burer,et al.  A copositive approach for two-stage adjustable robust optimization with uncertain right-hand sides , 2016, Comput. Optim. Appl..

[29]  Katta G. Murty,et al.  Some NP-complete problems in quadratic and nonlinear programming , 1987, Math. Program..

[30]  Panos M. Pardalos,et al.  Convex optimization theory , 2010, Optim. Methods Softw..

[31]  Alexander Shapiro,et al.  Minimax analysis of stochastic problems , 2002, Optim. Methods Softw..

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

[33]  A. Shapiro Monte Carlo Sampling Methods , 2003 .

[34]  Erick Delage,et al.  Linearized Robust Counterparts of Two-Stage Robust Optimization Problems with Applications in Operations Management , 2016, INFORMS J. Comput..

[35]  P. Parrilo Structured semidefinite programs and semialgebraic geometry methods in robustness and optimization , 2000 .

[36]  Jean B. Lasserre,et al.  Convexity in SemiAlgebraic Geometry and Polynomial Optimization , 2008, SIAM J. Optim..

[37]  S. Karlin,et al.  Studies in the Mathematical Theory of Inventory and Production, by K.J. Arrow, S. Karlin, H. Scarf with contributions by M.J. Beckmann, J. Gessford, R.F. Muth. Stanford, California, Stanford University Press, 1958, X p.340p., $ 8.75. , 1959, Bulletin de l'Institut de recherches économiques et sociales.

[38]  G. Pflug,et al.  Multistage Stochastic Optimization , 2014 .

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

[40]  I. Gilboa,et al.  Maxmin Expected Utility with Non-Unique Prior , 1989 .

[41]  Etienne de Klerk,et al.  Solving Standard Quadratic Optimization Problems via Linear, Semidefinite and Copositive Programming , 2002, J. Glob. Optim..

[42]  Yang Kang,et al.  Sample Out-of-Sample Inference Based on Wasserstein Distance , 2016, Oper. Res..

[43]  Elana Guslitser UNCERTAINTY- IMMUNIZED SOLUTIONS IN LINEAR PROGRAMMING , 2002 .

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

[45]  Daniel Kuhn,et al.  A comment on “computational complexity of stochastic programming problems” , 2016, Math. Program..

[46]  Daniel Kuhn,et al.  Distributionally Robust Convex Optimization , 2014, Oper. Res..

[47]  Daniel Kuhn,et al.  Distributionally Robust Logistic Regression , 2015, NIPS.

[48]  C. Jack,et al.  Alzheimer's Disease Neuroimaging Initiative , 2008 .

[49]  Massimiliano Pontil,et al.  Multi-task Learning , 2020, Transfer Learning.

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

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

[52]  Daniel Kuhn,et al.  Ambiguous Joint Chance Constraints Under Mean and Dispersion Information , 2017, Oper. Res..

[53]  R. Tibshirani Regression Shrinkage and Selection via the Lasso , 1996 .

[54]  Qingxia Kong,et al.  Scheduling Arrivals to a Stochastic Service Delivery System Using Copositive Cones , 2010, Oper. Res..

[55]  Daniel Kuhn,et al.  Generalized decision rule approximations for stochastic programming via liftings , 2014, Mathematical Programming.

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

[57]  C. Jack,et al.  Alzheimer's Disease Neuroimaging Initiative , 2008 .

[58]  JiangRuiwei,et al.  Risk-Averse Two-Stage Stochastic Program with Distributional Ambiguity , 2018 .

[59]  Michael D. Gordon,et al.  Regularized Least Absolute Deviations Regression and an Efficient Algorithm for Parameter Tuning , 2006, Sixth International Conference on Data Mining (ICDM'06).

[60]  David Wozabal,et al.  A framework for optimization under ambiguity , 2012, Ann. Oper. Res..

[61]  M. Kenward,et al.  An Introduction to the Bootstrap , 2007 .

[62]  Jason Weston,et al.  A unified architecture for natural language processing: deep neural networks with multitask learning , 2008, ICML '08.

[63]  J. Lofberg,et al.  YALMIP : a toolbox for modeling and optimization in MATLAB , 2004, 2004 IEEE International Conference on Robotics and Automation (IEEE Cat. No.04CH37508).

[64]  Tara L. Terry,et al.  Robust Linear Optimization With Recourse , 2009 .

[65]  P. H. Diananda On non-negative forms in real variables some or all of which are non-negative , 1962, Mathematical Proceedings of the Cambridge Philosophical Society.

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

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

[68]  Vishal Gupta,et al.  Robust sample average approximation , 2014, Math. Program..

[69]  Eric R. Ziegel,et al.  The Elements of Statistical Learning , 2003, Technometrics.

[70]  Yongpei Guan,et al.  Data-driven risk-averse stochastic optimization with Wasserstein metric , 2018, Oper. Res. Lett..