Abstract
Stochastic optimization is a leading approach to model optimization problems in which there is uncertainty in the input data, whether from measurement noise or an inability to know the future. In this survey, we outline some recent progress in the design of polynomial-time algorithms with performance guarantees on the quality of the solutions found for an important class of stochastic programming problems — 2-stage problems with recourse. In particular, we show that for a number of concrete problems, algorithmic approaches that have been applied for their deterministic analogues are also effective in this more challenging domain. More specifically, this work highlights the role of tools from linear programming, rounding techniques, primal-dual algorithms, and the role of randomization more generally.
Preview
Unable to display preview. Download preview PDF.
Similar content being viewed by others
References
Beale, E.M.L.: On minimizing a convex function subject to linear inequalities. J. Royal Stat. Soc., Series B. 17, 173–184; discussion 194–203 (1955)
Birge, J.R., Louveaux, F.V.: Introduction to Stochastic Programming. Springer, New York (1997)
Charikar, M., Chekuri, C., Pál, M.: Sampling bounds for stochastic optimization. In: Chekuri, C., Jansen, K., Rolim, J.D.P., Trevisan, L. (eds.) APPROX 2005 and RANDOM 2005. LNCS, vol. 3624, pp. 257–269. Springer, Heidelberg (2005)
Dantzig, G.B.: Linear programming under uncertainty. Management Sci. 1, 197–206 (1955)
Dean, B., Goemans, M.X., Vondrak, J.: Approximating the stochastic knapsack problem: the benefit of adaptivity. In: Proc. 45th Annual IEEE Symposium on Foundations of Computer Science, pp. 208–217 (2004)
Dye, S., Stougie, L., Tomasgard, A.: The stochastic single resource service-provision problem. Naval Res. Logistics 50, 869–887 (2003); Also appeared as, The stochastic single node service provision problem. COSOR-Memorandum 99-13, Dept. Math. & Comp. Sc., Eindhoven Tech. Univ., Eindhoven (1999)
Dyer, M., Kannan, R., Stougie, L.: A simple randomised algorithm for convex optimisation. SPOR-Report 2002-05, Dept. Math. & Comp. Sc., Eindhoven Tech. Univ., Eindhoven (2002)
Dyer, M., Stougie, L.: Computational complexity of stochastic programming problems. SPOR-Report 2005-11, Dept. Math. & Comp. Sc., Eindhoven Tech. Univ., Eindhoven (2005)
Goemans, M.X., Williamson, D.P.: A general approximation technique for constrained forest problems. SIAM J. Comput. 24, 296–317 (1995)
Gupta, A., Pál, M., Ravi, R., Sinha, A.: Boosted sampling: approximation algorithms for stochastic optimization. In: Proc. 36th ACM STOC, pp. 417–426 (2004)
Immorlica, N., Karger, D., Minkoff, M., Mirrokni, V.: On the costs and benefits of procrastination: approximation algorithms for stochastic combinatorial optimization problems. In: Proc. 15th SODA, pp. 684–693 (2004)
Jain, K., Vazirani, V.V.: Approximation algorithms for metric facility location and k-median problems using the primal-dual schema and Lagrangian relaxation. J. ACM 48, 274–296 (2001)
Kleinberg, J., Rabani, Y., Tardos, É.: Allocating bandwidth for bursty connections. SIAM J. Comput. 30, 191–217 (2000)
Kleywegt, A.J., Shapiro, A., Homem-De-Mello, T.: The sample average approximation method for stochastic discrete optimization. SIAM J. Optimization 12, 479–502 (2001)
Linderoth, J., Shapiro, A., Wright, S.: The empirical behavior of sampling methods for stochastic programming. Annals Oper. Res. (to appear)
Mahdian, M.: Facility Location and the Analysis of Algorithms through Factor-revealing Programs. Ph.D. thesis, MIT, Cambridge, MA (2004)
Mahdian, M., Ye, Y., Zhang, J.: Improved approximation algorithms for metric facility location problems. In: Jansen, K., Leonardi, S., Vazirani, V.V. (eds.) APPROX 2002. LNCS, vol. 2462, p. 229. Springer, Heidelberg (2002)
Möhring, R., Schulz, A., Uetz, M.: Approximation in stochastic scheduling: the power of LP based priority policies. JACM 46, 924–942 (1999)
Nemirovski, A., Shapiro, A.: On complexity of Shmoys–Swamy class of two-stage linear stochastic programming problems (2006) Optimization Online, http://www.optimization-online.org/DB_FILE/2006/07/1434.pdf
Nesterov, Y., Vial, J.-P.: Confidence level solutions for stochastic programming. CORE Discussion Papers (2000), http://www.core.ucl.ac.be/services/psfiles/dp00/dp2000-13.pdf
Ravi, R., Sinha, A.: Hedging uncertainty: Approximation algorithms for stochastic optimization problems. In: Bienstock, D., Nemhauser, G.L. (eds.) IPCO 2004. LNCS, vol. 3064, pp. 101–115. Springer, Heidelberg (2004)
Ruszczynski, A., Shapiro, A. (eds.): Stochastic Programming. Handbooks in Oper. Res. & Mgmt. Sc., vol. 10. North-Holland, Amsterdam (2003)
Shapiro, A.: Monte Carlo sampling methods. In: Ruszczynski, A., Shapiro, A. (eds.) Stochastic Programming. Handbooks in Oper. Res. & Mgmt. Sc., vol. 10, North-Holland, Amsterdam (2003)
Shapiro, A., Nemirovski, A.: On complexity of stochastic programming problems (2004), Optimization Online, http://www.optimization-online.org/DB_FILE/2004/10/978.pdf
Shmoys, D.B., Swamy, C.: An approximation scheme for stochastic linear programming and its application to stochastic integer programs. J. ACM (to appear); Preliminary version appeared as Stochastic optimization is (almost) as easy as deterministic optimization. In: Proc. 45th Annual IEEE FOCS, pp. 228–237 (2004)
Swamy, C.: Approximation Algorithms for Clustering Problems. Ph.D. thesis, Cornell Univ., Ithaca, NY (May 2004), http://www.math.uwaterloo.ca/cswamy/theses/master.pdf
Swamy, C., Shmoys, D.B.: The sample average approximation method for 2-stage stochastic optimization (November 2004), http://www.math.uwaterloo.ca/cswamy/papers/SAAproof.pdf
Verweij, B., Ahmed, S., Kleywegt, A.J., Nemhauser, G.L., Shapiro, A.: The sample average approximation method applied to stochastic routing problems: a computational study. Comp. Opt. Appl. 24, 289–333 (2003)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2006 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Swamy, C., Shmoys, D.B. (2006). Approximation Algorithms for 2-Stage Stochastic Optimization Problems. In: Arun-Kumar, S., Garg, N. (eds) FSTTCS 2006: Foundations of Software Technology and Theoretical Computer Science. FSTTCS 2006. Lecture Notes in Computer Science, vol 4337. Springer, Berlin, Heidelberg. https://doi.org/10.1007/11944836_3
Download citation
DOI: https://doi.org/10.1007/11944836_3
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-49994-7
Online ISBN: 978-3-540-49995-4
eBook Packages: Computer ScienceComputer Science (R0)