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An ADMM algorithm for two-stage stochastic programming problems

  • S.I.: CLAIO 2016
  • Published:
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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.

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Notes

  1. Note that in this case it would be more appropriate to call it a “disutility function” in line with the literature, but we keep our terminology for simplicity.

References

  • Birge, J. R., & Louveaux, F. (2011). Introduction to stochastic programming. Berlin: Springer.

    Book  Google Scholar 

  • 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.

    Article  Google Scholar 

  • 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.

    Article  Google Scholar 

  • Dantzig, G. B. (1955). Linear programming under uncertainty. Management Science, 1(3–4), 197–206.

    Article  Google Scholar 

  • Du, Y., Lin, X., & Ruszczyński, A. (2017). A selective linearization method for multiblock convex optimization. SIAM Journal on Optimization, 27(2), 1102–1117.

    Article  Google Scholar 

  • 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.

  • 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.

    Article  Google Scholar 

  • 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.

    Article  Google Scholar 

  • 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.

    Article  Google Scholar 

  • 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.

    Article  Google Scholar 

  • 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.

    Article  Google Scholar 

  • 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.

    Article  Google Scholar 

  • 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.

    Article  Google Scholar 

  • Linderoth, J., Shapiro, A., & Wright, S. (2006). The empirical behavior of sampling methods for stochastic programming. Annals of Operations Research, 142(1), 215–241.

    Article  Google Scholar 

  • 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).

    Article  Google Scholar 

  • 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 

  • Mulvey, J. M., & Vladimirou, H. (1991). Applying the progressive hedging algorithm to stochastic generalized networks. Annals of Operations Research, 31(1), 399–424.

    Article  Google Scholar 

  • Parikh, N., & Boyd, S. (2013). Proximal algorithms. Foundations and Trends in Optimization, 1(3), 123–231.

    Google Scholar 

  • 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.

    Article  Google Scholar 

  • Rockafellar, R. T. (1970). Convex analysis convex analysis. Princeton: Princeton University Press.

    Book  Google Scholar 

  • Rockafellar, R. T. (1976). Monotone operators and the proximal point algorithm. SIAM Journal on Control and Optimization, 14(5), 877–898.

    Article  Google Scholar 

  • Rockafellar, R. T., & Wets, R. J. B. (1991). Scenarios and policy aggregation in optimization under uncertainty. Mathematics of Operations Research, 16(1), 119–147.

    Article  Google Scholar 

  • Ruszczyński, A. (1986). A regularized decomposition method for minimizing a sum of polyhedral functions. Mathematical Programming, 35, 309–333.

    Article  Google Scholar 

  • Ruszczyński, A. (2003). Decomposition methods. In A. Ruszczyński & A. Shapiro (Eds.), Handbook of stochastic optimization. Amsterdam: Elsevier.

    Google Scholar 

  • 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).

  • 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.

    Article  Google Scholar 

  • Shapiro, A. (2003). Monte Carlo sampling methods. In A. Ruszczynski & A. Shapiro (Eds.), Stochastic programming (Vol. 10). Amsterdam: Elsevier.

    Chapter  Google Scholar 

  • 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.

  • 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.

  • 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.

    Article  Google Scholar 

  • 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.

    Article  Google Scholar 

  • 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.

  • 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.

    Article  Google Scholar 

  • Wallace, S. W., & Ziemba, W. T. (2005). Applications of stochastic programming applications of stochastic programming (Vol. 5). Philadelphia: SIAM.

    Book  Google Scholar 

  • 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.

    Article  Google Scholar 

Download references

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.

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Authors and Affiliations

  1. School of Business, Universidad Adolfo Ibáñez, Santiago, Chile

    Sebastián Arpón, Tito Homem-de-Mello & Bernardo K. Pagnoncelli

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  1. Sebastián Arpón
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  2. Tito Homem-de-Mello
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  3. Bernardo K. Pagnoncelli
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Correspondence to Tito Homem-de-Mello.

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Arpón, S., Homem-de-Mello, T. & Pagnoncelli, B.K. An ADMM algorithm for two-stage stochastic programming problems. Ann Oper Res 286, 559–582 (2020). https://doi.org/10.1007/s10479-019-03471-0

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