Journal of Optimization Theory and Applications

, Volume 158, Issue 2, pp 576–589 | Cite as

A Polynomial-Time Solution Scheme for Quadratic Stochastic Programs

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Abstract

We consider quadratic stochastic programs with random recourse—a class of problems which is perceived to be computationally demanding. Instead of using mainstream scenario tree-based techniques, we reduce computational complexity by restricting the space of recourse decisions to those linear and quadratic in the observations, thereby obtaining an upper bound on the original problem. To estimate the loss of accuracy of this approach, we further derive a lower bound by dualizing the original problem and solving it in linear and quadratic recourse decisions. By employing robust optimization techniques, we show that both bounding problems may be approximated by tractable conic programs.

Keywords

Decision rule approximation Robust optimization Quadratic stochastic programming Conic programming 

References

  1. 1.
    Van Slyke, R., Wets, R.: L-Shaped linear programs with applications to optimal control and stochastic programming. SIAM J. Appl. Math. 17(4), 638–663 (1969) MathSciNetMATHCrossRefGoogle Scholar
  2. 2.
    Römisch, W., Wets, R.: Stability of ε-approximate solutions to convex stochastic programs. SIAM J. Optim. 18(3), 961–979 (2007) MathSciNetMATHCrossRefGoogle Scholar
  3. 3.
    Shapiro, A., Dentcheva, D., Ruszczyński, A.: Lectures on Stochastic Programming: Modeling and Theory. SIAM, Philadelphia (2009) CrossRefGoogle Scholar
  4. 4.
    Ben-Tal, A., Goryashko, A., Guslitzer, E., Nemirovski, A.: Adjustable robust solutions of uncertain linear programs. Math. Program. 99(2), 351–376 (2004) MathSciNetMATHCrossRefGoogle Scholar
  5. 5.
    Goh, J., Sim, M.: Distributionally robust optimization and its tractable approximations. Oper. Res. 58(4), 902–917 (2010) MathSciNetMATHCrossRefGoogle Scholar
  6. 6.
    Shapiro, A., Nemirovski, A.: On complexity of stochastic programming problems. Appl. Optim. 99, 111–146 (2005) MathSciNetCrossRefGoogle Scholar
  7. 7.
    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
  8. 8.
    Georghiou, A., Wiesemann, W., Kuhn, D.: Generalized decision rule approximations for stochastic programming via liftings (2010). Available on Optimization Online, submitted for publication Google Scholar
  9. 9.
    Bampou, D., Kuhn, D.: Polynomial approximations for continuous linear programs. SIAM J. Optim. 22(2), 628–648 (2012) MathSciNetMATHCrossRefGoogle Scholar
  10. 10.
    Bertsimas, D., Iancu, D., Parrilo, P.: A hierarchy of near-optimal policies for multistage adaptive optimization. IEEE Trans. Autom. Control 56(12), 2809–2824 (2011) MathSciNetCrossRefGoogle Scholar
  11. 11.
    Kuhn, D., Wiesemann, W., Georghiou, A.: Primal and dual linear decision rules in stochastic and robust optimization. Math. Program. 130(1), 177–209 (2011) MathSciNetMATHCrossRefGoogle Scholar
  12. 12.
    Ben-Tal, A., El Ghaoui, L., Nemirovski, A.: Robust Optimization. Princeton University Press, Princeton (2009) MATHGoogle Scholar
  13. 13.
    Ye, Y.: Interior Point Algorithms: Theory and Analysis. Wiley, New York (1997) MATHCrossRefGoogle Scholar
  14. 14.
    Wiesemann, W., Kuhn, D., Rustem, B.: Robust Markov decision processes. Math. Oper. Res. (Forthcoming) Google Scholar
  15. 15.
    Alizadeh, F., Goldfarb, D.: Second-order cone programming. Math. Program. 95(1), 3–51 (2003) MathSciNetMATHCrossRefGoogle Scholar
  16. 16.
    Lobo, M., Vandenberghe, L., Boyd, S., Lebret, H.: Applications of second-order cone programming. Linear Algebra Appl. 284(1–3), 193–228 (1998) MathSciNetMATHCrossRefGoogle Scholar
  17. 17.
    Wright, S.: Primal-dual aggregation and disaggregation for stochastic linear programs. Math. Oper. Res. 19(4), 893–908 (1994) MathSciNetMATHCrossRefGoogle Scholar
  18. 18.
    Luenberger, D.: Optimization by Vector Space Methods. Wiley, New York (1969) MATHGoogle Scholar
  19. 19.
    Rocha, P.: Medium-term planning in deregulated energy markets with decision rules. Ph.D. thesis, Imperial College, London (2012) Google Scholar

Copyright information

© Springer Science+Business Media New York 2013

Authors and Affiliations

  1. 1.Department of ComputingImperial College LondonLondonUK

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