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Primal and dual linear decision rules in stochastic and robust optimization

Abstract

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.

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References

  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)

  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)

  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)

  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)

  5. 5

    Ben-Tal A., Nemirovski A.: Robust convex optimization. Math. Oper. Res. 23(4), 769–805 (1998)

  6. 6

    Ben-Tal A., Nemirovski A.: Robust solutions of uncertain linear programs. Oper. Res. Lett. 25(1), 1–13 (1999)

  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)

  8. 8

    Bertsekas D.: Dynamic Programming and Optimal Control, Volumes I and II. Athena Scientific, Belmont, MA (2001)

  9. 9

    Bertsimas D., Sim M.: The price of robustness. Oper. Res. 52(1), 35–53 (2004)

  10. 10

    Billingsley P.: Convergence of Probability Measures. Wiley, New York (1968)

  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)

  12. 12

    Calafiore G.: Multi-period portfolio optimization with linear control policies. Automatica 44(10), 2463–2473 (2008)

  13. 13

    Chen X., Sim M., Sun P.: A robust optimization perspective on stochastic programming. Oper. Res. 55(6), 1058–1071 (2007)

  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)

  15. 15

    Chen, X., Zhang, Y.: Uncertain linear programs: extended affinely adjustable robust counterparts. Oper. Res. (2009), opre.1080.0605

  16. 16

    Dyer M., Stougie L.: Computational complexity of stochastic programming problems. Math. Program. A 106(3), 423–432 (2006)

  17. 17

    El Ghaoui L., Oustry F., Lebret H.: Robust solutions to uncertain semidefinite programs. SIAM J. Optim. 9(1), 33–52 (1998)

  18. 18

    Garstka S.J., Wets R.J.-B.: On decision rules in stochastic programming. Math. Program. 7, 117–143 (1974)

  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)

  20. 20

    Kuhn D.: An information-based approximation scheme for stochastic optimization problems in continuous time. Math. Oper. Res. 34(2), 428–444 (2009)

  21. 21

    Ordóñez F., Zhao J.: Robust capacity expansion of network flows. Networks 50(2), 136–145 (2007)

  22. 22

    Rockafellar R., Uryasev S.: Optimization of conditional value-at-risk. J. Risk 2(3), 21–41 (2000)

  23. 23

    Rockafellar R., Wets R.-B.: Variational Analysis, vol. 317 of A Series of Comprehensive Studies in Mathematics. Springer-Verlag, New York (1998)

  24. 24

    Shapiro, A.: On duality theory of conic linear problems. In Semi-Infinite Programming. Kluwer Academic Publishers, pp. 135–165 (2001)

  25. 25

    Shapiro A.: Inference of statistical bounds for multistage stochastic programming problems. Math. Methods Oper. Res. 58(1), 57–68 (2003)

  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)

  27. 27

    Vandenberghe L., Boyd S.: Semidefinite programming. SIAM Rev. 38(1), 49–95 (1996)

  28. 28

    Wright S.: Primal-dual aggregation and disaggregation for stochastic linear programs. Math. Oper. Res. 19(4), 893–908 (1994)

  29. 29

    Ye Y.: Interior Point Algorithms: Theory and Analysis. Wiley, New York (1997)

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Correspondence to Daniel Kuhn.

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Kuhn, D., Wiesemann, W. & Georghiou, A. Primal and dual linear decision rules in stochastic and robust optimization. Math. Program. 130, 177–209 (2011). https://doi.org/10.1007/s10107-009-0331-4

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Keywords

  • Linear decision rules
  • Stochastic optimization
  • Robust optimization
  • Error bounds
  • Semidefinite programming

Mathematics Subject Classification (2000)

  • 90C15 (Stochastic Programming)