Mathematical Programming

, Volume 130, Issue 1, pp 177–209 | Cite as

Primal and dual linear decision rules in stochastic and robust optimization

  • Daniel Kuhn
  • Wolfram Wiesemann
  • Angelos Georghiou
Full Length Paper Series A


Linear stochastic programming provides a flexible toolbox for analyzing real-life decision situations, but it can become computationally cumbersome when recourse decisions are involved. The latter are usually modeled as decision rules, i.e., functions of the uncertain problem data. It has recently been argued that stochastic programs can quite generally be made tractable by restricting the space of decision rules to those that exhibit a linear data dependence. In this paper, we propose an efficient method to estimate the approximation error introduced by this rather drastic means of complexity reduction: we apply the linear decision rule restriction not only to the primal but also to a dual version of the stochastic program. By employing techniques that are commonly used in modern robust optimization, we show that both arising approximate problems are equivalent to tractable linear or semidefinite programs of moderate sizes. The gap between their optimal values estimates the loss of optimality incurred by the linear decision rule approximation. Our method remains applicable if the stochastic program has random recourse and multiple decision stages. It also extends to cases involving ambiguous probability distributions.


Linear decision rules Stochastic optimization Robust optimization Error bounds Semidefinite programming 

Mathematics Subject Classification (2000)

90C15 (Stochastic Programming) 


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  1. 1.
    Ang, M., Chou, M., Sim, M., So, K.: A robust optimization framework for analyzing distribution systems with transshipments. Working paper. National University of Singapore, Singapore (2008)Google Scholar
  2. 2.
    Ben-Tal A., Boyd S., Nemirovski A.: Extending scope of robust optimization: comprehensive robust counterparts of uncertain problems. Math. Program. 107(1–2 Ser. B), 63–89 (2006)MathSciNetMATHCrossRefGoogle Scholar
  3. 3.
    Ben-Tal A., Golany B., Nemirovski A., Vial J.: Supplier-retailer flexible commitments contracts: a robust optimization approach. Manuf. Serv. Oper. Manag. 73, 248–273 (2005)CrossRefGoogle Scholar
  4. 4.
    Ben-Tal A., Goryashko A., Guslitzer E., Nemirovski A.: Adjustable robust solutions of uncertain linear programs. Math. Program. 99(2, Ser. A), 351–376 (2004)MathSciNetMATHCrossRefGoogle Scholar
  5. 5.
    Ben-Tal A., Nemirovski A.: Robust convex optimization. Math. Oper. Res. 23(4), 769–805 (1998)MathSciNetMATHCrossRefGoogle Scholar
  6. 6.
    Ben-Tal A., Nemirovski A.: Robust solutions of uncertain linear programs. Oper. Res. Lett. 25(1), 1–13 (1999)MathSciNetMATHCrossRefGoogle Scholar
  7. 7.
    Ben-Tal A., Nemirovski A., Roos C.: Robust solutions of uncertain quadratic and conic quadratic problems. SIAM J. Optim. 13(2), 535–560 (2002)MathSciNetMATHCrossRefGoogle Scholar
  8. 8.
    Bertsekas D.: Dynamic Programming and Optimal Control, Volumes I and II. Athena Scientific, Belmont, MA (2001)Google Scholar
  9. 9.
    Bertsimas D., Sim M.: The price of robustness. Oper. Res. 52(1), 35–53 (2004)MathSciNetMATHCrossRefGoogle Scholar
  10. 10.
    Billingsley P.: Convergence of Probability Measures. Wiley, New York (1968)MATHGoogle Scholar
  11. 11.
    Boyd S., El Ghaoui L., Feron E., Balakrishnan V.: Linear Matrix Inequalities in System and Control Theory, vol. 15 of SIAM Studies in Applied Mathematics. Society for Industrial and Applied Mathematics (SIAM), Philadelphia (1994)Google Scholar
  12. 12.
    Calafiore G.: Multi-period portfolio optimization with linear control policies. Automatica 44(10), 2463–2473 (2008)MathSciNetMATHCrossRefGoogle Scholar
  13. 13.
    Chen X., Sim M., Sun P.: A robust optimization perspective on stochastic programming. Oper. Res. 55(6), 1058–1071 (2007)MathSciNetMATHCrossRefGoogle Scholar
  14. 14.
    Chen X., Sim M., Sun P., Zhang J.: A linear decision-based approximation approach to stochastic programming. Oper. Res. 56(2), 344–357 (2008)MathSciNetMATHCrossRefGoogle Scholar
  15. 15.
    Chen, X., Zhang, Y.: Uncertain linear programs: extended affinely adjustable robust counterparts. Oper. Res. (2009), opre.1080.0605Google Scholar
  16. 16.
    Dyer M., Stougie L.: Computational complexity of stochastic programming problems. Math. Program. A 106(3), 423–432 (2006)MathSciNetMATHCrossRefGoogle Scholar
  17. 17.
    El Ghaoui L., Oustry F., Lebret H.: Robust solutions to uncertain semidefinite programs. SIAM J. Optim. 9(1), 33–52 (1998)MathSciNetMATHCrossRefGoogle Scholar
  18. 18.
    Garstka S.J., Wets R.J.-B.: On decision rules in stochastic programming. Math. Program. 7, 117–143 (1974)MathSciNetMATHCrossRefGoogle Scholar
  19. 19.
    Klein Haneveld W.: Duality in Stochastic linear and dynamic programming, vol. 274 of Lecture Notes in Economics and Mathematical Systems. Springer, Berlin (1985)Google Scholar
  20. 20.
    Kuhn D.: An information-based approximation scheme for stochastic optimization problems in continuous time. Math. Oper. Res. 34(2), 428–444 (2009)MathSciNetMATHCrossRefGoogle Scholar
  21. 21.
    Ordóñez F., Zhao J.: Robust capacity expansion of network flows. Networks 50(2), 136–145 (2007)MathSciNetMATHCrossRefGoogle Scholar
  22. 22.
    Rockafellar R., Uryasev S.: Optimization of conditional value-at-risk. J. Risk 2(3), 21–41 (2000)Google Scholar
  23. 23.
    Rockafellar R., Wets R.-B.: Variational Analysis, vol. 317 of A Series of Comprehensive Studies in Mathematics. Springer-Verlag, New York (1998)Google Scholar
  24. 24.
    Shapiro, A.: On duality theory of conic linear problems. In Semi-Infinite Programming. Kluwer Academic Publishers, pp. 135–165 (2001)Google Scholar
  25. 25.
    Shapiro A.: Inference of statistical bounds for multistage stochastic programming problems. Math. Methods Oper. Res. 58(1), 57–68 (2003)MathSciNetMATHCrossRefGoogle Scholar
  26. 26.
    Shapiro A., Nemirovski A.: On complexity of stochastic programming problems. In: Jeyakumar, V., Rubinov, A. (eds) Continuous Optimization: Current Trends and Applications, pp. 111–144. Springer, Berlin (2005)Google Scholar
  27. 27.
    Vandenberghe L., Boyd S.: Semidefinite programming. SIAM Rev. 38(1), 49–95 (1996)MathSciNetMATHCrossRefGoogle Scholar
  28. 28.
    Wright S.: Primal-dual aggregation and disaggregation for stochastic linear programs. Math. Oper. Res. 19(4), 893–908 (1994)MathSciNetMATHCrossRefGoogle Scholar
  29. 29.
    Ye Y.: Interior Point Algorithms: Theory and Analysis. Wiley, New York (1997)MATHGoogle Scholar

Copyright information

© Springer and Mathematical Programming Society 2009

Authors and Affiliations

  • Daniel Kuhn
    • 1
  • Wolfram Wiesemann
    • 1
  • Angelos Georghiou
    • 1
  1. 1.Department of ComputingImperial College of Science, Technology, and MedicineLondonUK

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