Directed discrete midpoint convexity

Akihisa Tamura, Kazuya Tsurumi

Research output: Contribution to journalArticle


For continuous functions, midpoint convexity characterizes convex functions. By considering discrete versions of midpoint convexity, several types of discrete convexities of functions, including integral convexity, L-convexity and global/local discrete midpoint convexity, have been studied. We propose a new type of discrete midpoint convexity that lies between L-convexity and integral convexity and is independent of global/local discrete midpoint convexity. The new convexity, named DDM-convexity, has nice properties satisfied by L-convexity and global/local discrete midpoint convexity. DDM-convex functions are stable under scaling, satisfy the so-called parallelogram inequality and a proximity theorem with the same small proximity bound as that for L-convex functions. Several characterizations of DDM-convexity are given and algorithms for DDM-convex function minimization are developed. We also propose DDM-convexity in continuous variables and give proximity theorems on these functions.

Original languageEnglish
JournalJapan Journal of Industrial and Applied Mathematics
Publication statusAccepted/In press - 2020 Jan 1


  • Discrete midpoint convexity
  • Integral convexity
  • L-convexity
  • Midpoint convexity
  • Proximity theorem
  • Scaling algorithm

ASJC Scopus subject areas

  • Engineering(all)
  • Applied Mathematics

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