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Approximation algorithms for 2-stage stochastic optimization problems

Published: 01 March 2006 Publication History

Abstract

Uncertainty is a facet of many decision environments and might arise for various reasons, such as unpredictable information revealed in the future, or inherent fluctuations caused by noise. Stochastic optimization provides a means to handle uncertainty by modeling it by a probability distribution over possible realizations of the actual data, called scenarios. The field of stochastic optimization, or stochastic programming, has its roots in the work of Dantzig [4] and Beale [1] in the 1950s, and has since increasingly found application in a wide variety of areas, including transportation models, logistics, financial instruments, and network design. An important and widely-used model in stochastic programming is the 2-stage recourse model: first, given only distributional information about (some of) the data, one commits on initial (first-stage) actions. Then, once the actual data is realized according to the distribution, further recourse actions can be taken (in the second stage) to augment the earlier solution and satisfy the revealed requirements. The aim is to choose the initial actions so as to minimize the expected total cost incurred. The recourse actions typically entail making decisions in rapid reaction to the observed scenario, and are therefore more costly than decisions made ahead of time. Thus there is a trade-off between committing initially, having only imprecise information while incurring a lower cost, and deferring decisions to the second-stage, when we know the input precisely but the costs are higher. Many applications can be modeled this way, and much of the textbook of Birge and Louveaux [2] is devoted to models and algorithms for this class of problems. A commonly cited example involves a setting where a company has to decide where to set up facilities to serve client demands. Typically the demand pattern is not known precisely at the outset, but one might be able to obtain, through simulation models or surveys, statistical information about the demands. This motivates the following 2-step decision process: in the first-stage, given only distributional information about the demands (and deterministic data for the facility opening costs), one must decide which facilities to open initially; once the client demands are realized according to this distribution, we can extend the solution by opening more facilities, incurring a recourse cost, and we have to assign the realized demands to open facilities. This is the 2-stage stochastic uncapacitated facility location problem. The recourse costs are usually higher than the original ones (because opening a facility later would involve deploying resources with a small lead time); these costs could be different for the different facilities, and could even depend on the realized scenario.

References

[1]
E. M. L. Beale. On minimizing a convex function subject to linear inequalities. Journal of the Royal Statistical Society, Series B, 17:173--184; discussion 194--203, 1955.
[2]
J. R. Birge and F. V. Louveaux. Introduction to Stochastic Programming. Springer-Verlag, NY, 1997.
[3]
M. Charikar, C. Chekuri, and M. Pál. Sampling bounds for stochastic optimization. Proceedings, 9th RANDOM, pages 257--269, 2005.
[4]
G. B. Dantzig. Linear programming under uncertainty. Mgmt. Sc., 1:197--206, 1955.
[5]
B. Dean, M. Goemans, and J. Vondrak. Approximating the stochastic knapsack problem: the benefit of adaptivity. Proceedings, 45th Annual IEEE Symposium on Foundations of Computer Science, pages 208--217, 2004.
[6]
K. Dhamdhere, V. Goyal, R. Ravi, and M Singh. How to pay, come what may: approximation algorithms for demand-robust covering problems. Proceedings, 46th Annual IEEE Symposium on Foundations of Computer Science, pages 367--378, 2005.
[7]
K. Dhamdhere, R. Ravi, and M. Singh. On two-stage stochastic minimum spanning trees. Proceedings, 11th IPCO, pages 321--334, 2005.
[8]
S. Dye, L. Stougie, and A. Tomasgard. The stochastic single resource service-provision problem. Naval Research Logistics, 50(8):869--887, 2003. Also appeared as "The stochastic single node service provision problem", COSOR-Memorandum 99--13, Dept. of Mathematics and Computer Science, Eindhoven Technical University, Eindhoven, 1999.
[9]
M. Dyer, R. Kannan, and L. Stougie. A simple randomised algorithm for convex optimisation. SPOR-Report 2002-05, Dept. of Mathematics and Computer Science, Eindhoven Technical University, Eindhoven, 2002.
[10]
M. Dyer and L. Stougie. Computational complexity of stochastic programming problems. SPOR-Report 2005-11, Dept. of Mathematics and Computer Science, Eindhoven Technical University, Eindhoven, 2005.
[11]
M. Goemans and D. Williamson. A general approximation technique for constrained forest problems. SIAM Journal on Computing, 24:296--317, 1995.
[12]
A. Gupta and M. Pál. Stochastic Steiner trees without a root. Proceedings, 32nd ICALP, pages 1051--1063, 2005.
[13]
A. Gupta, M. Pál, R. Ravi, and A. Sinha. Boosted sampling: approximation algorithms for stochastic optimization. Proceedings, 36th Annual ACM Symposium on Theory of Computing, pages 417--426, 2004.
[14]
A. Gupta, M. Pál, R. Ravi, & A. Sinha. What about Wednesday? Approximation algorithms for multistage stochastic optimization. Proceedings, 8th APPROX, pages 86--98, 2005.
[15]
A. Gupta, R. Ravi, and A. Sinha. An edge in time saves nine: LP rounding approximation algorithms for stochastic network design. Proceedings, 45th Annual IEEE Symposium on Foundations of Computer Science, pages 218--227, 2004.
[16]
N. Immorlica, D. Karger, M. Minkoff, and V. Mirrokni. On the costs and benefits of procrastination: approximation algorithms for stochastic combinatorial optimization problems. Proceedings, 15th Annual ACM-SIAM Symposium on Discrete Algorithms, pages 684--693, 2004.
[17]
K. Jain and V. V. Vazirani. Approximation algorithms for metric facility location and k-median problems using the primal-dual schema and Lagrangian relaxation. JACM, 48(2):274--296, 2001.
[18]
J. Kleinberg, Y. Rabani, and É. Tardos. Allocating bandwidth for bursty connections. SIAM Journal on Computing, 30(1):191--217, 2000.
[19]
A. J. Kleywegt, A. Shapiro, and T. Homem-De-Mello. The sample average approximation method for stochastic discrete optimization. SIAM Journal of Optimization, 12:479--502, 2001.
[20]
R. Levi, M. Pál, R. Roundy, and D. B. Shmoys. Approximation algorithms, for stochastic inventory control models Proceedings, 11th IPCO, pages 306--320, 2005.
[21]
R. Levi, R. Roundy, and D. B. Shmoys. Provably near-optimal sampling-based policies for stochastic inventory control models. To appear in Proceedings, 38th Annual ACM Symposium on Theory of Computing, 2006.
[22]
J. Linderoth, A. Shapiro, and R. K. Wright. The empirical behavior of sampling methods for stochastic programming. Annals of Operations Research, to appear.
[23]
M. Mahdian. Facility Location and the Analysis of Algorithms through Factor-revealing Programs. Ph.D. thesis, MIT, Cambridge, MA, 2004.
[24]
M. Mahdian, Y. Ye, and J. Zhang. Improved approximation algorithms for metric facility location. Proceedings, 5th APPROX, pages 229--242, 2002.
[25]
R. Möhring, A. Schulz, and M. Uetz. Approximation in stochastic scheduling: the power of LP based priority policies. JACM, 46:924--942, 1999.
[26]
Y. Nesterov and J.-Ph. Vial. Confidence level solutions for stochastic programming. CORE Discussion Papers, 2000. http://www.core.ucl.ac.be/services/psfiles/ dp00/dp2000-13.pdf.
[27]
R. Ravi and A. Sinha. Hedging uncertainty: approximation algorithms for stochastic optimization problems. Proceedings, 10th IPCO, pages 101--115, 2004.
[28]
A. Ruszczynski and A. Shapiro. Editors, Stochastic Programming, volume 10 of Handbooks in Operations Research and Mgmt. Sc., North-Holland, Amsterdam, 2003.
[29]
A. Shapiro. Monte Carlo sampling methods. In A. Ruszczynski and A. Shapiro, editors, Stochastic Programming, volume 10 of Handbooks in Operations Research and Mgmt. Sc., North-Holland, Amsterdam, 2003.
[30]
A. Shapiro and A. Nemirovski. On complexity of stochastic programming problems. Published electronically in Optimization Online, 2004. http://www.optimization-online.org/DB_FILE/2004/10/978.pdf.
[31]
D. B. Shmoys and C. Swamy. An approximation scheme for stochastic linear programming and its application to stochastic integer programs. JACM, to appear. Preliminary version appeared as "Stochastic optimization is (almost) as easy as deterministic optimization" in Proceedings, 45th Annual IEEE FOCS, pages 228--237, 2004.
[32]
C. Swamy. Approximation Algorithms for Clustering Problems. Ph.D. thesis, Cornell University, Ithaca, NY, 2004.
[33]
C. Swamy and D. B. Shmoys. The sample average approximation method for 2-stage stochastic optimization. November 2004. http://ist.caltech.edu/~cswamy/papers/SAAproof.pdf.
[34]
C. Swamy and D. B. Shmoys. Sampling-based approximation algorithms for multi-stage stochastic optimization. Proceedings of the 46th Annual IEEE Symposium on Foundations of Computer Science, pages 357--366, 2005.
[35]
B. Verweij, S. Ahmed, A. J. Kleywegt, G. L. Nemhauser, and A. Shapiro. The sample average approximation method applied to stochastic routing problems: a computational study. Computational Optimization and Applications, 24:289--333, 2003.

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Published In

cover image ACM SIGACT News
ACM SIGACT News  Volume 37, Issue 1
March 2006
93 pages
ISSN:0163-5700
DOI:10.1145/1122480
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Association for Computing Machinery

New York, NY, United States

Publication History

Published: 01 March 2006
Published in SIGACT Volume 37, Issue 1

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