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A branch-and-cut decomposition algorithm for solving chance-constrained mathematical programs with finite support

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Abstract

We present a new approach for exactly solving chance-constrained mathematical programs having discrete distributions with finite support and random polyhedral constraints. Such problems have been notoriously difficult to solve due to nonconvexity of the feasible region, and most available methods are only able to find provably good solutions in certain very special cases. Our approach uses both decomposition, to enable processing subproblems corresponding to one possible outcome at a time, and integer programming techniques, to combine the results of these subproblems to yield strong valid inequalities. Computational results on a chance-constrained formulation of a resource planning problem inspired by a call center staffing application indicate the approach works significantly better than both an existing mixed-integer programming formulation and a simple decomposition approach that does not use strong valid inequalities. We also demonstrate how the approach can be used to efficiently solve for a sequence of risk levels, as would be done when solving for the efficient frontier of risk and cost.

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Notes

  1. The extension to more general discrete distributions of the form \(\mathbb P \{\omega = \omega ^k \} = p_k\), where \(p_k \ge 0\) and \(\sum _k p_k = 1\), is straightforward and is omitted to simplify exposition.

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Acknowledgments

The author thanks Shabbir Ahmed for the suggestion to compare the presented approach with a basic decomposition algorithm. The author also thanks the anonymous referees for helpful comments.

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Authors and Affiliations

  1. Department of Industrial and Systems Engineering, University of Wisconsin-Madison, Madison, WI, 53706, USA

    James Luedtke

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  1. James Luedtke
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Correspondence to James Luedtke.

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This research has been supported by NSF under grant CMMI-0952907.

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Luedtke, J. A branch-and-cut decomposition algorithm for solving chance-constrained mathematical programs with finite support. Math. Program. 146, 219–244 (2014). https://doi.org/10.1007/s10107-013-0684-6

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