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Fenchel decomposition for stochastic mixed-integer programming

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  • Lewis Ntaimo

Abstract

This paper introduces a new cutting plane method for two-stage stochastic mixed-integer programming (SMIP) called Fenchel decomposition (FD). FD uses a class of valid inequalities termed, FD cuts, which are derived based on Fenchel cutting planes from integer programming. First, we derive FD cuts based on both the first and second-stage variables, and devise an FD algorithm for SMIP and establish finite convergence for binary first-stage. Second, we derive FD cuts based on the second-stage variables only and use an idea from disjunctive programming to lift the cuts to the higher dimension space including the first-stage variables. We then devise an alternative algorithm (FD-L algorithm) based on the lifted FD cuts. Finally, we report on computational results based on several test instances from the literature involving the special structure of knapsack problems with nonnegative left-hand side coefficients. The results are promising and show that both algorithms can outperform a standard direct solver and a disjunctive decomposition algorithm on large-scale instances. Furthermore, the FD-L algorithm provides better performance than the FD algorithm in general. Since Fenchel cuts can be computationally expensive in general and are best suited for problems with special structure, both algorithms exploit the special structure of the test instances by reducing the size of the cut generation problems based on the number of nonzero components in the non-integer solution that needs to be cut off. Copyright Springer Science+Business Media, LLC. 2013

Suggested Citation

  • Lewis Ntaimo, 2013. "Fenchel decomposition for stochastic mixed-integer programming," Journal of Global Optimization, Springer, vol. 55(1), pages 141-163, January.
    • Handle: RePEc:spr:jglopt:v:55:y:2013:i:1:p:141-163
    DOI: 10.1007/s10898-011-9817-8
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    References listed on IDEAS

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    1. Caroe, Claus C. & Tind, Jorgen, 1997. "A cutting-plane approach to mixed 0-1 stochastic integer programs," European Journal of Operational Research, Elsevier, vol. 101(2), pages 306-316, September.
    2. Rüdiger Schultz, 1993. "Continuity Properties of Expectation Functions in Stochastic Integer Programming," Mathematics of Operations Research, INFORMS, vol. 18(3), pages 578-589, August.
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    Cited by:

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    2. Pavlo Glushko & Csaba I. Fábián & Achim Koberstein, 2022. "An L-shaped method with strengthened lift-and-project cuts," Computational Management Science, Springer, vol. 19(4), pages 539-565, October.
    3. Onur Tavaslıoğlu & Oleg A. Prokopyev & Andrew J. Schaefer, 2019. "Solving Stochastic and Bilevel Mixed-Integer Programs via a Generalized Value Function," Operations Research, INFORMS, vol. 67(6), pages 1659-1677, November.
    4. Valicka, Christopher G. & Garcia, Deanna & Staid, Andrea & Watson, Jean-Paul & Hackebeil, Gabriel & Rathinam, Sivakumar & Ntaimo, Lewis, 2019. "Mixed-integer programming models for optimal constellation scheduling given cloud cover uncertainty," European Journal of Operational Research, Elsevier, vol. 275(2), pages 431-445.
    5. Cheng Guo & Merve Bodur & Dionne M. Aleman & David R. Urbach, 2021. "Logic-Based Benders Decomposition and Binary Decision Diagram Based Approaches for Stochastic Distributed Operating Room Scheduling," INFORMS Journal on Computing, INFORMS, vol. 33(4), pages 1551-1569, October.
    6. Kathryn M. Schumacher & Amy E. M. Cohn & Richard Li-Yang Chen, 2017. "Algorithm for the N -2 Security-Constrained Unit Commitment Problem with Transmission Switching," INFORMS Journal on Computing, INFORMS, vol. 29(4), pages 645-659, November.
    7. Rui Chen & James Luedtke, 2022. "On Generating Lagrangian Cuts for Two-Stage Stochastic Integer Programs," INFORMS Journal on Computing, INFORMS, vol. 34(4), pages 2332-2349, July.
    8. Niels Laan & Ward Romeijnders & Maarten H. Vlerk, 2018. "Higher-order total variation bounds for expectations of periodic functions and simple integer recourse approximations," Computational Management Science, Springer, vol. 15(3), pages 325-349, October.
    9. van der Laan, Niels & Romeijnders, Ward & van der Vlerk, M.H., 2017. "Higher-order total variation bounds for expectations of periodic functions and simple integer recourse approximations," Research Report 2017-014-EEF, University of Groningen, Research Institute SOM (Systems, Organisations and Management).
    10. Merve Bodur & Sanjeeb Dash & Oktay Günlük & James Luedtke, 2017. "Strengthened Benders Cuts for Stochastic Integer Programs with Continuous Recourse," INFORMS Journal on Computing, INFORMS, vol. 29(1), pages 77-91, February.
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