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Characterization and Approximation of Strong General Dual Feasible Functions

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Book cover Combinatorial Optimization (ISCO 2018)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 10856))

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Abstract

Dual feasible functions (DFFs) have been used to provide bounds for standard packing problems and valid inequalities for integer optimization problems. In this paper, the connection between general DFFs and a particular family of cut-generating functions is explored. We find the characterization of (restricted/strongly) maximal general DFFs and prove a 2-slope theorem for extreme general DFFs. We show that any restricted maximal general DFF can be well approximated by an extreme general DFF.

The authors gratefully acknowledge partial support from the National Science Foundation through grant DMS-1320051 (M. Köppe). A part of this work was done while the first author (M. Köppe) was visiting the Simons Institute for the Theory of Computing. It was partially supported by the DIMACS/Simons Collaboration on Bridging Continuous and Discrete Optimization through NSF grant CCF-1740425.

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Notes

  1. 1.

    We will use the term “piecewise linear” throughout the paper without explanation. We refer readers to [13] for precise definitions of “piecewise linear” functions in both continuous and discontinuous cases.

  2. 2.

    In this paper, a function name shown in typewriter font is the name of the function in our SageMath program [12]. At the time of writing, the function is available on the feature branch Later it will be merged into the branch.

  3. 3.

    See the constructor .

  4. 4.

    Interested readers are referred to the function in order to check the claimed properties of \(\phi _{s,\delta }\).

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Authors and Affiliations

  1. Department of Mathematics, University of California, Davis, Davis, USA

    Matthias Köppe & Jiawei Wang

Authors
  1. Matthias Köppe
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  2. Jiawei Wang
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Correspondence to Matthias Köppe .

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Editors and Affiliations

  1. Department of Industrial and Operations Engineering, University of Michigan, Ann Arbor, MI, USA

    Jon Lee

  2. Istituto di Analisi dei Sistemi ed Informatica “A. Ruberti” (IASI), CNR, Rome, Italy

    Giovanni Rinaldi

  3. LAMSADE, Université Paris-Dauphine, Paris, France

    A. Ridha Mahjoub

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Köppe, M., Wang, J. (2018). Characterization and Approximation of Strong General Dual Feasible Functions. In: Lee, J., Rinaldi, G., Mahjoub, A. (eds) Combinatorial Optimization. ISCO 2018. Lecture Notes in Computer Science(), vol 10856. Springer, Cham. https://doi.org/10.1007/978-3-319-96151-4_23

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  • DOI: https://doi.org/10.1007/978-3-319-96151-4_23

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  • Online ISBN: 978-3-319-96151-4

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