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An ADMM algorithm for two-stage stochastic programming problems

  • Sebastián Arpón
  • Tito Homem-de-MelloEmail author
  • Bernardo K. Pagnoncelli
S.I.: CLAIO 2016
  • 50 Downloads

Abstract

The alternate direction method of multipliers (ADMM) has received significant attention recently as a powerful algorithm to solve convex problems with a block structure. The vast majority of applications focus on deterministic problems. In this paper we show that ADMM can be applied to solve two-stage stochastic programming problems, and we propose an implementation in three blocks with or without proximal terms. We present numerical results for large scale instances, and extend our findings for risk averse formulations using utility functions.

Notes

Acknowledgements

The authors thank two anonymous referees for their constructive comments, which in particular allowed us to provide a generic formulation of the algorithm that includes the cases with and without proximal terms. This research was supported by FONDECYT Project No. 1171145.

References

  1. Birge, J. R., & Louveaux, F. (2011). Introduction to stochastic programming. Berlin: Springer.CrossRefGoogle Scholar
  2. Boyd, S., Parikh, N., Chu, E., Peleato, B., & Eckstein, J. (2011). Distributed optimization and statistical learning via the alternating direction method of multipliers. Foundations and Trends® in Machine Learning, 3(1), 1–122.CrossRefGoogle Scholar
  3. Chen, C., He, B., Ye, Y., & Yuan, X. (2014). The direct extension of ADMM for multi-block convex minimization problems is not necessarily convergent. Mathematical Programming, 155, 57–79.CrossRefGoogle Scholar
  4. Dantzig, G. B. (1955). Linear programming under uncertainty. Management Science, 1(3–4), 197–206.CrossRefGoogle Scholar
  5. Du, Y., Lin, X., & Ruszczyński, A. (2017). A selective linearization method for multiblock convex optimization. SIAM Journal on Optimization, 27(2), 1102–1117.CrossRefGoogle Scholar
  6. Eckstein, J. (2012). Augmented Lagrangian and alternating direction methods for convex optimization: A tutorial and some illustrative computational results (Technical report No. RRR 32-2012). RUTCOR, Rutgers University.Google Scholar
  7. Fazel, M., Pong, T. K., Sun, D., & Tseng, P. (2013). Hankel matrix rank minimization with applications to system identification and realization. SIAM Journal on Matrix Analysis and Applications, 34(3), 946–977.CrossRefGoogle Scholar
  8. Gabay, D., & Mercier, B. (1976). A dual algorithm for the solution of nonlinear variational problems via finite element approximation. Computers and Mathematics with Applications, 2(1), 17–40.CrossRefGoogle Scholar
  9. Homem-de-Mello, T., & Bayraksan, G. (2014). Monte Carlo sampling-based methods for stochastic optimization. Surveys in Operations Research and Management Science, 19, 56–85.CrossRefGoogle Scholar
  10. Kleywegt, A. J., Shapiro, A., & Homem-de-Mello, T. (2002). The sample average approximation method for stochastic discrete optimization. SIAM Journal on Optimization, 12(2), 479–502.CrossRefGoogle Scholar
  11. Kulkarni, A. A., & Shanbhag, U. V. (2012). Recourse-based stochastic nonlinear programming: Properties and Benders-SQP algorithms. Computational Optimization and Applications, 51(1), 77–123.CrossRefGoogle Scholar
  12. Lam, X. Y., Marron, J., Sun, D., & Toh, K. C. (2017). Fast algorithms for large scale generalized distance weighted discrimination fast algorithms for large scale generalized distance weighted discrimination. Journal of Computational and Graphical Statistics, 27, 368–379.CrossRefGoogle Scholar
  13. Lin, T., Ma, S., & Zhang, S. (2015). On the global linear convergence of the ADMM with multi-block variables. SIAM Journal on Optimization, 25(3), 1478–1497.CrossRefGoogle Scholar
  14. Linderoth, J., Shapiro, A., & Wright, S. (2006). The empirical behavior of sampling methods for stochastic programming. Annals of Operations Research, 142(1), 215–241.CrossRefGoogle Scholar
  15. Linderoth, J., & Wright, S. J. (2003). Implementing a decomposition algorithm for stochastic programming on a computational grid. Computational Optimization and Applications, 24, 207–250. (Special Issue on Stochastic Programming).CrossRefGoogle Scholar
  16. McKay, M. D., Beckman, R. J., & Conover, W. J. (1979). A comparison of three methods for selecting values of input variables in the analysis of output from a computer code. Technometrics, 21, 239–245.Google Scholar
  17. Mulvey, J. M., & Vladimirou, H. (1991). Applying the progressive hedging algorithm to stochastic generalized networks. Annals of Operations Research, 31(1), 399–424.CrossRefGoogle Scholar
  18. Parikh, N., & Boyd, S. (2013). Proximal algorithms. Foundations and Trends in Optimization, 1(3), 123–231.Google Scholar
  19. Phan, D., & Ghosh, S. (2014). Two-stage stochastic optimization for optimal power flow under renewable generation uncertainty. ACM Transactions on Modeling and Computer Simulation (TOMACS), 24(1), 2.CrossRefGoogle Scholar
  20. Rockafellar, R. T. (1970). Convex analysis convex analysis. Princeton: Princeton University Press.CrossRefGoogle Scholar
  21. Rockafellar, R. T. (1976). Monotone operators and the proximal point algorithm. SIAM Journal on Control and Optimization, 14(5), 877–898.CrossRefGoogle Scholar
  22. Rockafellar, R. T., & Wets, R. J. B. (1991). Scenarios and policy aggregation in optimization under uncertainty. Mathematics of Operations Research, 16(1), 119–147.CrossRefGoogle Scholar
  23. Ruszczyński, A. (1986). A regularized decomposition method for minimizing a sum of polyhedral functions. Mathematical Programming, 35, 309–333.CrossRefGoogle Scholar
  24. Ruszczyński, A. (2003). Decomposition methods. In A. Ruszczyński & A. Shapiro (Eds.), Handbook of stochastic optimization. Amsterdam: Elsevier.Google Scholar
  25. Ryan, S. M., Wets, R. J. B., Woodruff, D. L., Silva-Monroy, C., & Watson, J. P. (2013). Toward scalable, parallel progressive hedging for stochastic unit commitment. In 2013 IEEE Power and energy society general meeting (PES) (pp. 1–5).Google Scholar
  26. Schütz, P., Tomasgard, A., & Ahmed, S. (2009). Supply chain design under uncertainty using sample average approximation and dual decomposition. European Journal of Operational Research, 199(2), 409–419.CrossRefGoogle Scholar
  27. Shapiro, A. (2003). Monte Carlo sampling methods. In A. Ruszczynski & A. Shapiro (Eds.), Stochastic programming (Vol. 10). Amsterdam: Elsevier.CrossRefGoogle Scholar
  28. Shen, L., Pan, S. (2015). A corrected semi-proximal admm for multi-block convex optimization and its application to DNN-SDPS. arXiv preprint arXiv:1502.03194.
  29. Shenoy, S., Gorinevsky, D., Boyd, S. (2015). Non-parametric regression modeling for stochastic optimization of power grid load forecast. In American control conference (ACC) (pp. 1010–1015). IEEE.Google Scholar
  30. Sun, D., Toh, K. C., & Yang, L. (2015). A convergent 3-block semiproximal alternating direction method of multipliers for conic programming with 4-type constraints. SIAM Journal on Optimization, 25(2), 882–915.CrossRefGoogle Scholar
  31. Van Slyke, R., & Wets, R. J. B. (1969). L-shaped linear programs with application to optimal control and stochastic programming. SIAM Journal on Applied Mathematics, 17, 638–663.CrossRefGoogle Scholar
  32. Vayá, M. G., Andersson, G., & Boyd, S. (2014). Decentralized control of plug-in electric vehicles under driving uncertainty. In IEEE PES Innovative Smart Grid Technologies Conferece, Istanbul, Turkey.Google Scholar
  33. Verweij, B., Ahmed, S., Kleywegt, A. J., Nemhauser, G., & Shapiro, A. (2003). The sample average approximation method applied to stochastic routing problems: A computational study. Computational Optimization and Applications, 24(2–3), 289–333.CrossRefGoogle Scholar
  34. Wallace, S. W., & Ziemba, W. T. (2005). Applications of stochastic programming applications of stochastic programming (Vol. 5). Philadelphia: SIAM.CrossRefGoogle Scholar
  35. Xu, L., Yu, B., & Zhang, Y. (2017). An alternating direction and projection algorithm for structure-enforced matrix factorization. Computational Optimization and Applications, 68(2), 333–362.CrossRefGoogle Scholar

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© Springer Science+Business Media, LLC, part of Springer Nature 2019

Authors and Affiliations

  1. 1.School of BusinessUniversidad Adolfo IbáñezSantiagoChile

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