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Generalized Decision Rule Approximations for Stochastic Programming via Liftings

Author

Listed:
  • Angelos Georghiou
  • Wolfram Wiesemann
  • Daniel Kuhn
Abstract
Stochastic programming provides a versatile framework for decision-making under uncertainty, but the resulting optimization problems can be computationally demanding. It has recently been shown that, primal and dual linear decision rule approximations can yield tractable upper and lower bounds on the optimal value of a stochastic program. Unfortunately, linear decision rules often provide crude approximations that result in loose bounds. To address this problem, we propose a lifting technique that maps a given stochastic program to an equivalent problem on a higherdimensional probability space. We prove that solving the lifted problem in primal and dual linear decision rules provides tighter bounds than those obtained from applying linear decision rules to the original problem. We also show that there is a one-to-one correspondence between linear decision rules in the lifted problem and families of non-linear decision rules in the original problem. Finally, we identify structured liftings that give rise to highly flexible piecewise linear decision rules and assess their performance in the context of a stylized investment planning problem.

Suggested Citation

  • Angelos Georghiou & Wolfram Wiesemann & Daniel Kuhn, 2010. "Generalized Decision Rule Approximations for Stochastic Programming via Liftings," Working Papers 043, COMISEF.
    • Handle: RePEc:com:wpaper:043
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    File URL: http://comisef.eu/files/wps043.pdf
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    References listed on IDEAS

    as
    1. Xin Chen & Melvyn Sim & Peng Sun & Jiawei Zhang, 2008. "A Linear Decision-Based Approximation Approach to Stochastic Programming," Operations Research, INFORMS, vol. 56(2), pages 344-357, April.
    2. Xin Chen & Yuhan Zhang, 2009. "Uncertain Linear Programs: Extended Affinely Adjustable Robust Counterparts," Operations Research, INFORMS, vol. 57(6), pages 1469-1482, December.
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    Cited by:

    1. Robin Vujanic & Paul Goulart & Manfred Morari, 2016. "Robust Optimization of Schedules Affected by Uncertain Events," Journal of Optimization Theory and Applications, Springer, vol. 171(3), pages 1033-1054, December.
    2. Paula Rocha & Daniel Kuhn, 2013. "A Polynomial-Time Solution Scheme for Quadratic Stochastic Programs," Journal of Optimization Theory and Applications, Springer, vol. 158(2), pages 576-589, August.
    3. Shao-Wei Lam & Tsan Sheng Ng & Melvyn Sim & Jin-Hwa Song, 2013. "Multiple Objectives Satisficing Under Uncertainty," Operations Research, INFORMS, vol. 61(1), pages 214-227, February.
    4. Rocha, Paula & Kuhn, Daniel, 2012. "Multistage stochastic portfolio optimisation in deregulated electricity markets using linear decision rules," European Journal of Operational Research, Elsevier, vol. 216(2), pages 397-408.

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