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Integrality of subgradients and biconjugates of integrally convex functions

  • Kazuo Murota
  • Akihisa TamuraEmail author
Original Paper

Abstract

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.

Keywords

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

Notes

Acknowledgements

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.”

References

  1. 1.
    Favati, P., Tardella, F.: Convexity in nonlinear integer programming. Ric. Oper. 53, 3–44 (1990)Google Scholar
  2. 2.
    Fujishige, S.: Submodular Functions and Optimization. Annals of Discrete Mathematics, vol. 58, 2nd edn. Elsevier, Amsterdam (2005)Google Scholar
  3. 3.
    Fujishige, S.: Bisubmodular polyhedra, simplicial divisions, and discrete convexity. Discrete Optim. 12, 115–120 (2014)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Hajek, B.: Extremal splittings of point processes. Math. Oper. Res. 10, 543–556 (1985)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Hiriart-Urruty, J.-B., Lemaréchal, C.: Fundamentals of Convex Analysis. Springer, Berlin (2001)CrossRefGoogle Scholar
  6. 6.
    Iimura, T., Murota, K., Tamura, A.: Discrete fixed point theorem reconsidered. J. Math. Econ. 41, 1030–1036 (2005)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Maehara, T., Murota, K.: A framework of discrete DC programming by discrete convex analysis. Math. Program. Ser. A 152, 435–466 (2015)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Moriguchi, S., Murota, K.: Projection and convolution operations for integrally convex functions. Discrete Appl. Math. 255, 283–298 (2019)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Moriguchi, S., Murota, K., Tamura, A., Tardella, F.: Scaling, proximity, and optimization of integrally convex functions. Math. Program. 175, 119–154 (2019)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Murota, K.: Discrete convex analysis. Math. Program. 83, 313–371 (1998)MathSciNetzbMATHGoogle Scholar
  11. 11.
    Murota, K.: Discrete Convex Analysis. Society for Industrial and Applied Mathematics, Philadelphia (2003)CrossRefGoogle Scholar
  12. 12.
    Murota, K.: Note on multimodularity and L-convexity. Math. Oper. Res. 30, 658–661 (2005)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Murota, K.: Primer of Discrete Convex Analysis—Discrete Versus Continuous Optimization. Kyoritsu Publishing Co., Tokyo (2007). (in Japanese)Google Scholar
  14. 14.
    Murota, K.: Discrete convex analysis: a tool for economics and game theory. J. Mech. Inst. Des. 1, 151–273 (2016)Google Scholar
  15. 15.
    Murota, K., Shioura, A.: Relationship of M-/L-convex functions with discrete convex functions by Miller and by Favati-Tardella. Discrete Appl. Math. 115, 151–176 (2001)MathSciNetCrossRefGoogle Scholar
  16. 16.
    Schrijver, A.: Theory of Linear and Integer Programming. Wiley, New York (1986)zbMATHGoogle Scholar
  17. 17.
    Singer, I.: A Fenchel-Rockafellar type duality theorem for maximization. Bull. Aust. Math. Soc. 20, 193–198 (1979)MathSciNetCrossRefGoogle Scholar
  18. 18.
    Toland, J.F.: A duality principle for non-convex optimisation and the calculus of variations. Arch. Ration. Mech. Anal. 71, 41–61 (1979)CrossRefGoogle Scholar
  19. 19.
    Yang, Z.: Discrete fixed point analysis and its applications. J. Fixed Point Theory Appl. 6, 351–371 (2009)MathSciNetCrossRefGoogle Scholar

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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