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Analysis of MILP Techniques for the Pooling Problem

Author

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  • Santanu S. Dey

    (School of Industrial and Systems Engineering, Georgia Institute of Technology, Atlanta, Georgia, 30332)

  • Akshay Gupte

    (Department of Mathematical Sciences, Clemson University, Clemson, South Carolina, 29634)

Abstract

The pq -relaxation for the pooling problem can be constructed by applying McCormick envelopes for each of the bilinear terms appearing in the so-called pq -formulation of the pooling problem. This relaxation can be strengthened by using piecewise-linear functions that over- and under-estimate each bilinear term. Although there is a significant amount of empirical evidence to show that such piecewise-linear relaxations, which can be written as mixed-integer linear programs (MILPs), yield good bounds for the pooling problem, to the best of our knowledge, no formal result regarding the quality of these relaxations is known. In this paper, we prove that the ratio of the upper bound obtained by solving piecewise-linear relaxations (objective function is maximization) to the optimal objective function value of the pooling problem is at most n , where n is the number of output nodes. Furthermore for any ϵ > 0 and for any piecewise-linear relaxation, there exists an instance where the ratio of the relaxation value to the optimal value is at least n − ϵ . This analysis naturally yields a polynomial-time n -approximation algorithm for the pooling problem. We also show that if there exists a polynomial-time approximation algorithm for the pooling problem with guarantee better than n 1− ϵ for any ϵ > 0, then NP-complete problems have randomized polynomial-time algorithms. Finally, motivated by the approximation algorithm, we design a heuristic that involves solving an MILP-based restriction of the pooling problem. This heuristic is guaranteed to provide solutions within a factor of n . On large-scale test instances and in significantly lesser time, this heuristic provides solutions that are often orders of magnitude better than those given by commercial local and global optimization solvers.

Suggested Citation

  • Santanu S. Dey & Akshay Gupte, 2015. "Analysis of MILP Techniques for the Pooling Problem," Operations Research, INFORMS, vol. 63(2), pages 412-427, April.
  • Handle: RePEc:inm:oropre:v:63:y:2015:i:2:p:412-427
    DOI: 10.1287/opre.2015.1357
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    References listed on IDEAS

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    1. Charles Audet & Jack Brimberg & Pierre Hansen & Sébastien Le Digabel & Nenad Mladenovi'{c}, 2004. "Pooling Problem: Alternate Formulations and Solution Methods," Management Science, INFORMS, vol. 50(6), pages 761-776, June.
    2. Mohammed Alfaki & Dag Haugland, 2013. "Strong formulations for the pooling problem," Journal of Global Optimization, Springer, vol. 56(3), pages 897-916, July.
    3. Thomas E. Baker & Leon S. Lasdon, 1985. "Successive Linear Programming at Exxon," Management Science, INFORMS, vol. 31(3), pages 264-274, March.
    4. Mohammed Alfaki & Dag Haugland, 2014. "A cost minimization heuristic for the pooling problem," Annals of Operations Research, Springer, vol. 222(1), pages 73-87, November.
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    Citations

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    Cited by:

    1. Benjamin Beach & Robert Hildebrand & Joey Huchette, 2022. "Compact mixed-integer programming formulations in quadratic optimization," Journal of Global Optimization, Springer, vol. 84(4), pages 869-912, December.
    2. Natashia Boland & Thomas Kalinowski & Fabian Rigterink, 2016. "New multi-commodity flow formulations for the pooling problem," Journal of Global Optimization, Springer, vol. 66(4), pages 669-710, December.
    3. Akshay Gupte & Shabbir Ahmed & Santanu S. Dey & Myun Seok Cheon, 2017. "Relaxations and discretizations for the pooling problem," Journal of Global Optimization, Springer, vol. 67(3), pages 631-669, March.
    4. Natashia Boland & Thomas Kalinowski & Fabian Rigterink, 2017. "A polynomially solvable case of the pooling problem," Journal of Global Optimization, Springer, vol. 67(3), pages 621-630, March.
    5. Marandi, Ahmadreza & Dahl, Joachim & de Klerk, Etienne, 2018. "A numerical evaluation of the bounded degree sum-of-squares hierarchy of Lasserre, Toh, and Yang on the pooling problem," Other publications TiSEM 981f1428-4d42-4d3f-9a7a-7, Tilburg University, School of Economics and Management.
    6. Radu Baltean-Lugojan & Ruth Misener, 2018. "Piecewise parametric structure in the pooling problem: from sparse strongly-polynomial solutions to NP-hardness," Journal of Global Optimization, Springer, vol. 71(4), pages 655-690, August.
    7. Ahmadreza Marandi & Joachim Dahl & Etienne Klerk, 2018. "A numerical evaluation of the bounded degree sum-of-squares hierarchy of Lasserre, Toh, and Yang on the pooling problem," Annals of Operations Research, Springer, vol. 265(1), pages 67-92, June.
    8. Fischetti, Matteo & Monaci, Michele, 2020. "A branch-and-cut algorithm for Mixed-Integer Bilinear Programming," European Journal of Operational Research, Elsevier, vol. 282(2), pages 506-514.
    9. Dag Haugland & Eligius M. T. Hendrix, 2016. "Pooling Problems with Polynomial-Time Algorithms," Journal of Optimization Theory and Applications, Springer, vol. 170(2), pages 591-615, August.
    10. Santanu S. Dey & Burak Kocuk & Asteroide Santana, 2020. "Convexifications of rank-one-based substructures in QCQPs and applications to the pooling problem," Journal of Global Optimization, Springer, vol. 77(2), pages 227-272, June.

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