Discrete midpoint convexity

Satoko Moriguchi, Kazuo Murota, Akihisa Tamura, Fabio Tardella

Research output: Contribution to journalArticle

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.

Original languageEnglish
Pages (from-to)99-128
Number of pages30
JournalMathematics of Operations Research
Volume45
Issue number1
DOIs
Publication statusPublished - 2020 Feb

Keywords

  • Discrete convex function
  • Integral convexity
  • L-convexity
  • Midpoint convexity
  • Proximity theorem
  • Scaling algorithm

ASJC Scopus subject areas

  • Mathematics(all)
  • Computer Science Applications
  • Management Science and Operations Research

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