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Convexifications of rank-one-based substructures in QCQPs and applications to the pooling problem

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  • Santanu S. Dey

    (Georgia Institute of Technology)

  • Burak Kocuk

    (Sabancı University)

  • Asteroide Santana

    (Georgia Institute of Technology)

Abstract

We study sets defined as the intersection of a rank-one constraint with different choices of linear side constraints. We identify different conditions on the linear side constraints, under which the convex hull of the rank-one set is polyhedral or second-order cone representable. In all these cases, we also show that a linear objective can be optimized in polynomial time over these sets. Towards the application side, we show how these sets relate to commonly occurring substructures of a general quadratically constrained quadratic program. To further illustrate the benefit of studying quadratically constrained quadratic programs from a rank-one perspective, we propose new rank-one formulations for the generalized pooling problem and use our convexification results to obtain several new convex relaxations for the pooling problem. Finally, we run a comprehensive set of computational experiments and show that our convexification results together with discretization significantly help in improving dual bounds for the generalized pooling problem.

Suggested Citation

  • 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.
    • Handle: RePEc:spr:jglopt:v:77:y:2020:i:2:d:10.1007_s10898-019-00844-4
    DOI: 10.1007/s10898-019-00844-4
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    References listed on IDEAS

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    1. 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.
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    5. 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.
    6. 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.
    7. 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.
    8. 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.
    9. 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.
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