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Duality gaps in nonconvex stochastic optimization

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Abstract.

We consider multistage stochastic optimization models containing nonconvex constraints, e.g., due to logical or integrality requirements. We study three variants of Lagrangian relaxations and of the corresponding decomposition schemes, namely, scenario, nodal and geographical decomposition. Based on convex equivalents for the Lagrangian duals, we compare the duality gaps for these decomposition schemes. The first main result states that scenario decomposition provides a smaller or equal duality gap than nodal decomposition. The second group of results concerns large stochastic optimization models with loosely coupled components. The results provide conditions implying relations between the duality gaps of geographical decomposition and the duality gaps for scenario and nodal decomposition, respectively.

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Authors and Affiliations

  1. Department of Mathematical Sciences, Stevens Institute of Technology, Hoboken, NJ, 07030, U.S.A

    Darinka Dentcheva

  2. Humboldt University Berlin, Institute of Mathematics, D-10099, Berlin, Germany

    Werner Römisch

Authors
  1. Darinka Dentcheva
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  2. Werner Römisch
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Correspondence to Darinka Dentcheva.

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Mathematics Subject Classification (1991): 90C15

Acknowledgments. This work was supported by the Priority Programme Online Optimization of Large Scale Systems of the Deutsche Forschungsgemeinschaft. The authors wish to thank Andrzej Ruszczyński (Rutgers University) for helpful discussions.

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Dentcheva, D., Römisch, W. Duality gaps in nonconvex stochastic optimization. Math. Program., Ser. A 101, 515–535 (2004). https://doi.org/10.1007/s10107-003-0496-1

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