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A solution algorithm for chance-constrained problems with integer second-stage recourse decisions

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Abstract

We study a class of chance-constrained two-stage stochastic optimization problems where the second-stage recourse decisions belong to mixed-integer convex sets. Due to the nonconvexity of the second-stage feasible sets, standard decomposition approaches cannot be applied. We develop a provably convergent branch-and-cut scheme that iteratively generates valid inequalities for the convex hull of the second-stage feasible sets, resorting to spatial branching when cutting no longer suffices. We show that this algorithm attains an approximate notion of convergence, whereby the feasible sets are relaxed by some positive tolerance \(\epsilon \). Computational results on chance-constrained resource planning problems indicate that our implementation of the proposed algorithm is highly effective in solving this class of problems, compared to a state-of-the-art MIP solver and to a naive decomposition scheme.

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Notes

  1. For clarity of presentation, this instruction is not explicitly given in Algorithm 2.0.1 but it is straightforward to include it.

  2. Instances are publicly available at https://github.com/paoloparonuzzi/CCP-INT-problems-instances.

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Acknowledgements

The authors are grateful to two anonymous reviewers for their constructive comments and remarks. Thanks are also due to Kibaek Kim for stimulating discussion on decomposition approaches for stochastic programming. Most of this research was performed when Andrea Lodi was Canada Excellence Research Chair at Polytechnique Montréal and the CERC support is strongly acknowledged. Enrico Malaguti, Michele Monaci, and Paolo Paronuzzi were funded by the Air Force Office of Scientific Research under awards number FA8655-20-1-7012 and FA8655-20-1-7019.

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Authors and Affiliations

  1. Jacobs Technion-Cornell Institute, Cornell Tech and Technion - IIT, New York, USA

    Andrea Lodi

  2. DEI “Guglielmo Marconi”, Università di Bologna, Bologna, Italy

    Enrico Malaguti, Michele Monaci & Paolo Paronuzzi

  3. Department of Industrial and Systems Engineering, University of Southern California, Los Angeles, USA

    Giacomo Nannicini

Authors
  1. Andrea Lodi
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  2. Enrico Malaguti
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  3. Michele Monaci
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  4. Giacomo Nannicini
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  5. Paolo Paronuzzi
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Correspondence to Andrea Lodi.

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Lodi, A., Malaguti, E., Monaci, M. et al. A solution algorithm for chance-constrained problems with integer second-stage recourse decisions. Math. Program. 205, 269–301 (2024). https://doi.org/10.1007/s10107-023-01984-y

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  • DOI: https://doi.org/10.1007/s10107-023-01984-y

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