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The blockwise coordinate descent method for integer programs

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Abstract

Blockwise coordinate descent methods have a long tradition in continuous optimization and are also frequently used in discrete optimization under various names. New interest in blockwise coordinate descent methods arises for improving sequential solutions for problems which consist of several planning stages. In this paper we systematically formulate and analyze the blockwise coordinate descent method for integer programming problems. We discuss convergence of the method and properties of the resulting solutions. We extend the notion of Pareto optimality for blockwise coordinate descent to the case that the blocks do not form a partition and compare Pareto optimal solutions to blockwise optimal and to global optimal solutions. Among others we derive a condition which ensures that the solution obtained by blockwise coordinate descent is weakly Pareto optimal and we confirm convergence of the blockwise coordinate descent to a global optimum in matroid polytopes. The results are interpreted in the context of multi-stage linear integer programming problems and illustrated for integrated planning in public transportation.

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Authors and Affiliations

  1. Institut für Mathematik, Technische Universität Berlin, Berlin, Germany

    Sven Jäger

  2. Fachbereich Mathematik, TU Kaiserslautern, Kaiserslautern, Germany

    Anita Schöbel

  3. Fraunhofer Institut für Techno- und Wirtschaftsmathematik ITWM, Kaiserslautern, Germany

    Anita Schöbel

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  1. Sven Jäger
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  2. Anita Schöbel
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Correspondence to Anita Schöbel.

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Jäger, S., Schöbel, A. The blockwise coordinate descent method for integer programs. Math Meth Oper Res 91, 357–381 (2020). https://doi.org/10.1007/s00186-019-00673-x

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