INFORMS doi 10.1287/xxxx.0000.0000 c○0000 INFORMS Price of Correlations in Stochastic Optimization

When decisions are made in the presence of high-dimensional stochastic data, handling joint distribution of correlated random variables can present a formidable task, both in terms of sampling and estimation as well as algorithmic complexity. A common heuristic is to estimate only marginal distributions and substitute joint distribution by independent (product) distribution. In this paper, we study possible loss incurred on ignoring correlations through a distributionally robust stochastic programming model, and we quantify that loss as price of correlations (POC). Using techniques of cost sharing from game theory, we identify a wide class of problems for which POC has a small upper bound. To our interest, this class will include many stochastic convex programs, uncapacitated facility location, Steiner tree, and submodular functions, suggesting that the intuitive approach of assuming independent distribution acceptably approximates the robust model for these stochastic optimization problems. Additionally, we demonstrate hardness of bounding POC via examples of subadditive and supermodular functions that have large POC. We find that our results are also useful for solving many deterministic optimization problems like welfare maximization, k-dimensional matching, and transportation problems, under certain conditions.

[1]  Piotr Berman,et al.  A d/2 Approximation for Maximum Weight Independent Set in d-Claw Free Graphs , 2000, Nord. J. Comput..

[2]  Peng Sun,et al.  A Robust Optimization Perspective on Stochastic Programming , 2007, Oper. Res..

[3]  L. Lovász,et al.  Geometric Algorithms and Combinatorial Optimization , 1981 .

[4]  Ioana Popescu,et al.  Robust Mean-Covariance Solutions for Stochastic Optimization , 2007, Oper. Res..

[5]  Chaitanya Swamy,et al.  Sampling-Based Approximation Algorithms for Multistage Stochastic Optimization , 2012, SIAM J. Comput..

[6]  Faruk Gul,et al.  WALRASIAN EQUILIBRIUM WITH GROSS SUBSTITUTES , 1999 .

[7]  Panos M. Pardalos,et al.  Encyclopedia of Optimization , 2006 .

[8]  Arkadi Nemirovski,et al.  Robust Convex Optimization , 1998, Math. Oper. Res..

[9]  Rolf H. Möhring,et al.  Approximation in stochastic scheduling: the power of LP-based priority policies , 1999, JACM.

[10]  R. Ravi,et al.  Boosted sampling: approximation algorithms for stochastic optimization , 2004, STOC '04.

[11]  Arkadi Nemirovski,et al.  Robust solutions of Linear Programming problems contaminated with uncertain data , 2000, Math. Program..

[12]  Jack Edmonds,et al.  Submodular Functions, Matroids, and Certain Polyhedra , 2001, Combinatorial Optimization.

[13]  Dimitris Bertsimas,et al.  Probabilistic Combinatorial Optimization: Moments, Semidefinite Programming, and Asymptotic Bounds , 2004, SIAM J. Optim..

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

[15]  Peter Brucker,et al.  A Monge Property for the D-dimensional Transportation Problem , 1995, Discret. Appl. Math..

[16]  Jan Vondrák,et al.  Maximizing a Submodular Set Function Subject to a Matroid Constraint (Extended Abstract) , 2007, IPCO.

[17]  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.

[18]  Arkadi Nemirovski,et al.  Robust optimization – methodology and applications , 2002, Math. Program..

[19]  S. Bikhchandani,et al.  Competitive Equilibrium in an Exchange Economy with Indivisibilities , 1997 .

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

[21]  V. Crawford,et al.  Job Matching, Coalition Formation, and Gross Substitutes , 1982 .

[22]  Uriel Feige,et al.  On maximizing welfare when utility functions are subadditive , 2006, STOC '06.

[23]  Jochen Könemann,et al.  Group-Strategyproof Mechanisms for Network Design Games via Primal-Dual Algorithms , 2008 .

[24]  Melvyn Sim,et al.  The Price of Robustness , 2004, Oper. Res..

[25]  Rainer E. Burkard,et al.  Perspectives of Monge Properties in Optimization , 1996, Discret. Appl. Math..

[26]  Michael Jünger,et al.  Combinatorial optimization - Eureka, you shrink! , 2003 .

[27]  Oded Schwartz,et al.  On the complexity of approximating k-set packing , 2006, computational complexity.

[28]  Stefano Leonardi,et al.  Cross-monotonic cost-sharing methods for connected facility location games , 2004, EC '04.

[29]  Ulrich Derigs.  On two methods for solving the bottleneck matching problem , 1980 .

[30]  Jan Vondrák,et al.  Optimal approximation for the submodular welfare problem in the value oracle model , 2008, STOC.

[31]  Mohammad Mahdian,et al.  Universal Facility Location , 2003, ESA.

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

[33]  Vahab S. Mirrokni,et al.  Tight information-theoretic lower bounds for welfare maximization in combinatorial auctions , 2008, EC '08.

[34]  W. Rogosinski.  Moments of non-negative mass , 1958, Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences.

[35]  H. Moulin.  Incremental cost sharing: Characterization by coalition strategy-proofness , 1999 .

[36]  Yuval Rabani,et al.  Allocating bandwidth for bursty connections , 1997, STOC '97.

[37]  Chaitanya Swamy,et al.  Sampling-based approximation algorithms for multi-stage stochastic optimization , 2005, 46th Annual IEEE Symposium on Foundations of Computer Science (FOCS'05).

[38]  Tim Roughgarden,et al.  New trade-offs in cost-sharing mechanisms , 2006, STOC '06.

[39]  Alexander Shapiro,et al.  Worst-case distribution analysis of stochastic programs , 2006, Math. Program..

[40]  J. Dupacová.  The minimax approach to stochastic programming and an illustrative application , 1987 .

[41]  Jochen Könemann,et al.  A group-strategyproof mechanism for Steiner forests , 2005, SODA '05.

[42]  Moses Charikar,et al.  Sampling Bounds for Stochastic Optimization , 2005, APPROX-RANDOM.

[43]  Jan Vondrák,et al.  Matroid matching: the power of local search , 2010, STOC '10.

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

[45]  Min-Chiang Wang,et al.  Expected Value of Distribution Information for the Newsvendor Problem , 2006, Oper. Res..

[46]  Alexei A. Gaivoronski,et al.  A numerical method for solving stochastic programming problems with moment constraints on a distribution function , 1991, Ann. Oper. Res..

[47]  Alexander Schrijver,et al.  On the Size of Systems of Sets Every t of Which Have an SDR, with an Application to the Worst-Case Ratio of Heuristics for Packing Problems , 1989, SIAM J. Discret. Math..

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

[49]  Jitka Dupacová,et al.  Stochastic Programming: Minimax Approach , 2009, Encyclopedia of Optimization.

[50]  A. C. Kimber.  A note on Poisson maxima , 1983 .

[51]  Ioana Popescu,et al.  Optimal Inequalities in Probability Theory: A Convex Optimization Approach , 2005, SIAM J. Optim..

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

[53]  Nicole Immorlica,et al.  Limitations of cross-monotonic cost sharing schemes , 2005, SODA '05.

[54]  Boris Lavrič.  Continuity of monotone functions , 1993 .

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

[56]  H. Moulin,et al.  Strategyproof sharing of submodular costs:budget balance versus efficiency , 2001 .

[57]  Amin Saberi,et al.  Correlation robust stochastic optimization , 2009, SODA '10.

[58]  H. Landau.  Moments in mathematics , 1987 .

[59]  B. R. Barmish,et al.  Distributionally Robust Monte Carlo Simulation: A Tutorial Survey , 2002 .

[60]  Tim Roughgarden,et al.  Algorithmic Game Theory , 2007 .

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

[62]  Y. Ermoliev,et al.  Stochastic Optimization Problems with Incomplete Information on Distribution Functions , 1985 .

[63]  W. K. Haneveld.  Robustness against dependence in PERT: An application of duality and distributions with known marginals , 1986 .