Mathematical Programming

, Volume 152, Issue 1–2, pp 301–338

Generalized decision rule approximations for stochastic programming via liftings

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

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 higher-dimensional 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 nonlinear decision rules in the original problem. Finally, we identify structured liftings that give rise to highly flexible piecewise linear and nonlinear decision rules, and we assess their performance in the context of a dynamic production planning problem.

Mathematics Subject Classification

90C15 (Stochastic Programming) 

Supplementary material

10107_2014_789_MOESM1_ESM.pdf (97 kb)
Supplementary material 1 (pdf 97 KB)

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Copyright information

© Springer-Verlag Berlin Heidelberg and Mathematical Optimization Society 2014

Authors and Affiliations

  • Angelos Georghiou
    • 1
  • Wolfram Wiesemann
    • 2
  • Daniel Kuhn
    • 3
  1. 1.Process Systems Engineering LaboratoryMassachusetts Institute of TechnologyCambridgeUSA
  2. 2.Imperial College Business SchoolImperial College LondonLondonUK
  3. 3.Risk Analytics and Optimization ChairÉcole Polytechnique Fédérale de LausanneLausanneSwitzerland

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