Integrality of subgradients and biconjugates of integrally convex functions

  • Kazuo Murota
  • Akihisa TamuraEmail author
Original Paper


Integrally convex functions constitute a fundamental function class in discrete convex analysis. This paper shows that an integer-valued integrally convex function admits an integral subgradient and that the integral biconjugate of an integer-valued integrally convex function coincides with itself. The proof is based on the Fourier–Motzkin elimination. The latter result provides a unified proof of integral biconjugacy for various classes of integer-valued discrete convex functions, including L-convex, M-convex, L\(_{2}\)-convex, M\(_{2}\)-convex, BS-convex, and UJ-convex functions as well as multimodular functions. Our results of integral subdifferentiability and integral biconjugacy make it possible to extend the theory of discrete DC (difference of convex) functions developed for L- and M-convex functions to that for integrally convex functions, including an analogue of the Toland–Singer duality for integrally convex functions.


Discrete convex analysis Integrally convex function Subgradient Biconjugate function Integrality Fourier–Motzkin elimination 



The authors thank Satoru Fujishige and Hiroshi Hirai for helpful comments. This work was supported by CREST, JST, Grant Number JPMJCR14D2, Japan; JSPS KAKENHI Grant Numbers 26280004, 16K00023; and JSPS Core-to-core program “Foundation of a Global Research Cooperative Center in Mathematics Focused on Number Theory and Geometry.”


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Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Department of Economics and Business AdministrationTokyo Metropolitan UniversityTokyoJapan
  2. 2.Department of MathematicsKeio UniversityYokohamaJapan

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