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An Interior Point–Inspired Algorithm for Linear Programs Arising in Discrete Optimal Transport

Author

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  • Filippo Zanetti

    (School of Mathematics and Maxwell Institute for Mathematical Sciences, The University of Edinburgh, Edinburgh EH9 3JZ, United Kingdom)

  • Jacek Gondzio

    (School of Mathematics and Maxwell Institute for Mathematical Sciences, The University of Edinburgh, Edinburgh EH9 3JZ, United Kingdom)

Abstract

Discrete optimal transport problems give rise to very large linear programs (LPs) with a particular structure of the constraint matrix. In this paper, we present a hybrid algorithm that mixes an interior point method (IPM) and column generation, specialized for the LP originating from the Kantorovich optimal transport problem. Knowing that optimal solutions of such problems display a high degree of sparsity, we propose a column generation–like technique to force all intermediate iterates to be as sparse as possible. The algorithm is implemented nearly matrix-free. Indeed, most of the computations avoid forming the huge matrices involved and solve the Newton system using only a much smaller Schur complement of the normal equations. We prove theoretical results about the sparsity pattern of the optimal solution, exploiting the graph structure of the underlying problem. We use these results to mix iterative and direct linear solvers efficiently in a way that avoids producing preconditioners or factorizations with excessive fill-in and at the same time guaranteeing a low number of conjugate gradient iterations. We compare the proposed method with two state-of-the-art solvers and show that it can compete with the best network optimization tools in terms of computational time and memory use. We perform experiments with problems reaching more than four billion variables and demonstrate the robustness of the proposed method.

Suggested Citation

  • Filippo Zanetti & Jacek Gondzio, 2023. "An Interior Point–Inspired Algorithm for Linear Programs Arising in Discrete Optimal Transport," INFORMS Journal on Computing, INFORMS, vol. 35(5), pages 1061-1078, September.
  • Handle: RePEc:inm:orijoc:v:35:y:2023:i:5:p:1061-1078
    DOI: 10.1287/ijoc.2022.0184
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    References listed on IDEAS

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    1. Marco E. Lübbecke & Jacques Desrosiers, 2005. "Selected Topics in Column Generation," Operations Research, INFORMS, vol. 53(6), pages 1007-1023, December.
    2. François Vanderbeck, 2005. "Implementing Mixed Integer Column Generation," Springer Books, in: Guy Desaulniers & Jacques Desrosiers & Marius M. Solomon (ed.), Column Generation, chapter 0, pages 331-358, Springer.
    3. Luca Bergamaschi & Jacek Gondzio & Manolo Venturin & Giovanni Zilli, 2007. "Inexact constraint preconditioners for linear systems arising in interior point methods," Computational Optimization and Applications, Springer, vol. 36(2), pages 137-147, April.
    4. Jacek Gondzio, 2012. "Matrix-free interior point method," Computational Optimization and Applications, Springer, vol. 51(2), pages 457-480, March.
    5. Castro, Jordi & Nasini, Stefano, 2021. "A specialized interior-point algorithm for huge minimum convex cost flows in bipartite networks," European Journal of Operational Research, Elsevier, vol. 290(3), pages 857-869.
    6. Carsten Gottschlich & Dominic Schuhmacher, 2014. "The Shortlist Method for Fast Computation of the Earth Mover's Distance and Finding Optimal Solutions to Transportation Problems," PLOS ONE, Public Library of Science, vol. 9(10), pages 1-10, October.
    7. Gondzio, Jacek & González-Brevis, Pablo & Munari, Pedro, 2013. "New developments in the primal–dual column generation technique," European Journal of Operational Research, Elsevier, vol. 224(1), pages 41-51.
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