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A linear programming approach for linear programs with probabilistic constraints

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  • Reich, Daniel

Abstract

We study a class of mixed-integer programs for solving linear programs with joint probabilistic constraints from random right-hand side vectors with finite distributions. We present greedy and dual heuristic algorithms that construct and solve a sequence of linear programs. We provide optimality gaps for our heuristic solutions via the linear programming relaxation of the extended mixed-integer formulation of Luedtke et al. (2010) [13] as well as via lower bounds produced by their cutting plane method. While we demonstrate through an extensive computational study the effectiveness and scalability of our heuristics, we also prove that the theoretical worst-case solution quality for these algorithms is arbitrarily far from optimal. Our computational study compares our heuristics against both the extended mixed-integer programming formulation and the cutting plane method of Luedtke et al. (2010) [13]. Our heuristics efficiently and consistently produce solutions with small optimality gaps, while for larger instances the extended formulation becomes intractable and the optimality gaps from the cutting plane method increase to over 5%.

Suggested Citation

  • Reich, Daniel, 2013. "A linear programming approach for linear programs with probabilistic constraints," European Journal of Operational Research, Elsevier, vol. 230(3), pages 487-494.
    • Handle: RePEc:eee:ejores:v:230:y:2013:i:3:p:487-494
    DOI: 10.1016/j.ejor.2013.04.049
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    References listed on IDEAS

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    1. Tanner, Matthew W. & Ntaimo, Lewis, 2010. "IIS branch-and-cut for joint chance-constrained stochastic programs and application to optimal vaccine allocation," European Journal of Operational Research, Elsevier, vol. 207(1), pages 290-296, November.
    2. B. K. Pagnoncelli & D. Reich & M. C. Campi, 2012. "Risk-Return Trade-off with the Scenario Approach in Practice: A Case Study in Portfolio Selection," Journal of Optimization Theory and Applications, Springer, vol. 155(2), pages 707-722, November.
    3. GÜNLÜK, Oktay & POCHET, Yves, 2001. "Mixing mixed-integer inequalities," LIDAM Reprints CORE 1504, Université catholique de Louvain, Center for Operations Research and Econometrics (CORE).
    4. MILLER, Andrew J. & WOLSEY, Laurence A., 2003. "Tight formulations for some simple mixed integer programs and convex objective integer programs," LIDAM Reprints CORE 1653, Université catholique de Louvain, Center for Operations Research and Econometrics (CORE).
    5. B. K. Pagnoncelli & S. Ahmed & A. Shapiro, 2009. "Sample Average Approximation Method for Chance Constrained Programming: Theory and Applications," Journal of Optimization Theory and Applications, Springer, vol. 142(2), pages 399-416, August.
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    2. Bentaha, Mohand Lounes & Battaïa, Olga & Dolgui, Alexandre & Hu, S. Jack, 2015. "Second order conic approximation for disassembly line design with joint probabilistic constraints," European Journal of Operational Research, Elsevier, vol. 247(3), pages 957-967.
    3. Xiaodi Bai & Jie Sun & Xiaojin Zheng, 2021. "An Augmented Lagrangian Decomposition Method for Chance-Constrained Optimization Problems," INFORMS Journal on Computing, INFORMS, vol. 33(3), pages 1056-1069, July.

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