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Discrete Midpoint Convexity

Author

Listed:
  • Satoko Moriguchi

    (Department of Economics and Business Administration, Tokyo Metropolitan University, Tokyo 192-0397, Japan;)

  • Kazuo Murota

    (Department of Economics and Business Administration, Tokyo Metropolitan University, Tokyo 192-0397, Japan;)

  • Akihisa Tamura

    (Department of Mathematics, Keio University, Yokohama 223-8522, Japan;)

  • Fabio Tardella

    (Department of Methods and Models for Economics, Territory and Finance, Sapienza University of Rome, Rome 00161, Italy)

Abstract

For a function defined on the integer lattice, we consider discrete versions of midpoint convexity, which offer a unifying framework for discrete convexity of functions, including integral convexity, L ♮ -convexity, and submodularity. By considering discrete midpoint convexity for all pairs at ℓ ∞ -distance equal to 2 or not smaller than 2, we identify new classes of discrete convex functions, called locally and globally discrete midpoint convex functions . These functions enjoy nice structural properties. They are stable under scaling and addition and satisfy a family of inequalities named parallelogram inequalities . Furthermore, they admit a proximity theorem with the same small proximity bound as that for L ♮ -convex functions. These structural properties allow us to develop an algorithm for the minimization of locally and globally discrete midpoint convex functions based on the proximity-scaling approach and on a novel 2-neighborhood steepest descent algorithm.

Suggested Citation

  • Satoko Moriguchi & Kazuo Murota & Akihisa Tamura & Fabio Tardella, 2020. "Discrete Midpoint Convexity," Mathematics of Operations Research, INFORMS, vol. 45(1), pages 99-128, February.
    • Handle: RePEc:inm:ormoor:v:45:y:2020:i:1:p:99-128
    DOI: 10.1287/moor.2018.0984
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    References listed on IDEAS

    as
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