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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
    1. Mehmet A. Begen & Maurice Queyranne, 2011. "Appointment Scheduling with Discrete Random Durations," Mathematics of Operations Research, INFORMS, vol. 36(2), pages 240-257, May.
    2. Kazuo Murota, 2016. "Discrete convex analysis: A tool for economics and game theory," The Journal of Mechanism and Institution Design, Society for the Promotion of Mechanism and Institution Design, University of York, vol. 1(1), pages 151-273, December.
    3. Dorit Hochbaum, 2007. "Complexity and algorithms for nonlinear optimization problems," Annals of Operations Research, Springer, vol. 153(1), pages 257-296, September.
    4. Iimura, Takuya & Murota, Kazuo & Tamura, Akihisa, 2005. "Discrete fixed point theorem reconsidered," Journal of Mathematical Economics, Elsevier, vol. 41(8), pages 1030-1036, December.
    5. Lehmann, Benny & Lehmann, Daniel & Nisan, Noam, 2006. "Combinatorial auctions with decreasing marginal utilities," Games and Economic Behavior, Elsevier, vol. 55(2), pages 270-296, May.
    6. Paul Zipkin, 2008. "On the Structure of Lost-Sales Inventory Models," Operations Research, INFORMS, vol. 56(4), pages 937-944, August.
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    Cited by:

    1. Shaomin Li & Haoyu Wei & Xiaoyu Lei, 2022. "Heterogeneous Overdispersed Count Data Regressions via Double-Penalized Estimations," Mathematics, MDPI, vol. 10(10), pages 1-25, May.

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