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Graver basis and proximity techniques for block-structured separable convex integer minimization problems

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Abstract

We consider \(N\)-fold \(4\)-block decomposable integer programs, which simultaneously generalize \(N\)-fold integer programs and two-stage stochastic integer programs with \(N\) scenarios. In previous work (Hemmecke et al. in Integer programming and combinatorial optimization. Springer, Berlin, 2010), it was proved that for fixed blocks but variable \(N\), these integer programs are polynomial-time solvable for any linear objective. We extend this result to the minimization of separable convex objective functions. Our algorithm combines Graver basis techniques with a proximity result (Hochbaum and Shanthikumar in J. ACM 37:843–862,1990), which allows us to use convex continuous optimization as a subroutine.

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Acknowledgments

We wish to thank Rüdiger Schultz for valuable comments and for pointing us to [4]. We also would like to thank Shmuel Onn for pointing us toward the paper by Hochbaum and Shanthikumar. The second author was supported by grant DMS-0914873 of the National Science Foundation. The third author was partially supported by the German Science Foundation, grant SFB/Transregio 63 InPROMPT. A part of this work was completed during a stay of the three authors at BIRS.

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Authors and Affiliations

  1. Zentrum Mathematik, M9, Technische Universität München, Boltzmannstr. 3, 85747 , Garching, Germany

    Raymond Hemmecke

  2. Department of Mathematics, University of California, Davis, One Shields Avenue, Davis, CA, 95616, USA

    Matthias Köppe

  3. Institute for Operations Research, ETH Zürich, Rämistrasse 101, 8092 , Zurich, Switzerland

    Robert Weismantel

Authors
  1. Raymond Hemmecke
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  2. Matthias Köppe
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  3. Robert Weismantel
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Correspondence to Matthias Köppe.

Additional information

We dedicate this paper to the memory of Uri Rothblum. His paper [19] has been an inspiration for our work on nonlinear discrete optimization. As a coauthor of R.H. and R.W. in [1], Uri contributed to the application of Graver basis techniques for block-structured problems. We believe that the present paper continues the theme of research at the interface of algebra, geometry, combinatorics, and optimization that Uri appreciated.

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Hemmecke, R., Köppe, M. & Weismantel, R. Graver basis and proximity techniques for block-structured separable convex integer minimization problems. Math. Program. 145, 1–18 (2014). https://doi.org/10.1007/s10107-013-0638-z

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  • DOI: https://doi.org/10.1007/s10107-013-0638-z

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