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Conjugacy relationship between M-convex and L-convex functions in continuous variables

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Abstract.

By extracting combinatorial structures in well-solved nonlinear combinatorial optimization problems, Murota (1996,1998) introduced the concepts of M-convexity and L-convexity to functions defined over the integer lattice. Recently, Murota–Shioura (2000, 2001) extended these concepts to polyhedral convex functions and quadratic functions in continuous variables. In this paper, we consider a further extension to more general convex functions defined over the real space, and provide a proof for the conjugacy relationship between general M-convex and L-convex functions.

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References

  1. Aczel, J.: Lectures on Functional Equations and Their Applications. Academic Press, New York, 1966

  2. Camerini, P.M., Conforti, M., Naddef, D.: Some easily solvable nonlinear integer programs. Ricerca Operativa 50, 11–25 (1989)

    Google Scholar 

  3. Danilov, V., Koshevoy, G., Murota, K.: Discrete convexity and equilibria in economies with indivisible goods and money. Math. Social Sci. 41, 251–273 (2001)

    Article  MATH  Google Scholar 

  4. Favati, P., Tardella, F.: Convexity in nonlinear integer programming. Ricerca Operativa 53, 3–44 (1990)

    Google Scholar 

  5. Fujishige, S., Murota, K.: Notes on L-/M-convex functions and the separation theorems. Math. Program. 88, 129–146 (2000)

    Article  Google Scholar 

  6. Iri, M.: Network Flow, Transportation and Scheduling – Theory and Algorithms. Academic Press, New York, 1969

  7. Lovász, L.: Submodular functions and convexity. In: Bachem, A., Grötschel, M., Korte, B., (eds.), Mathematical Programming – The State of the Art. Springer, Berlin, 1983, pp. 235–257

  8. Miller, B.L.: On minimizing nonseparable functions defined on the integers with an inventory application. SIAM J. Appl. Math. 21, 166–185 (1971)

    MATH  Google Scholar 

  9. Murota, K.: Convexity and Steinitz’s exchange property. Adv. Math. 124, 272–311 (1996)

    Article  MATH  Google Scholar 

  10. Murota, K.: Discrete convex analysis. Math. Program. 83, 313–371 (1998)

    Article  Google Scholar 

  11. Murota, K.: Matrices and Matroids for Systems Analysis. Springer, Berlin, 2000

  12. Murota, K.: Discrete Convex Analysis – An Introduction. Kyoritsu Publishing Co., Tokyo [In Japanese], 2001

  13. Murota, K.: Discrete Convex Analysis. Society for Industrial and Applied Mathematics, Philadelphia, 2003

  14. Murota, K., Shioura, A.: M-convex function on generalized polymatroid. Math. Oper. Res. 24, 95–105 (1999)

    MATH  Google Scholar 

  15. Murota, K., Shioura, A.: Extension of M-convexity and L-convexity to polyhedral convex functions. Adv. Appl. Math. 25, 352–427 (2000)

    Article  MATH  Google Scholar 

  16. Murota, K., Shioura, A.: Quadratic M-convex and L-convex functions. RIMS preprint, No. 1326, Kyoto University. Adv. Appl. Math., to appear (2004)

  17. Murota, K., Shioura, A.: Fundamental properties of M-convex and L-convex functions in continuous variables. IEICE Trans. Fundamentals, to appear

  18. Murota, K., Tamura, A.: New characterizations of M-convex functions and their applications to economic equilibrium models with indivisibilities. Discrete Appl. Math. 131, 495–512 (2003)

    Article  MATH  Google Scholar 

  19. Murota, K., Tamura, A.: Application of M-convex submodular flow problem to mathematical economics. Japan J. Indust. Appl. Math. 20, 257–277 (2003)

    MATH  Google Scholar 

  20. Rockafellar, R.T.: Convex Analysis. Princeton Univ. Press, Princeton, 1970

  21. Rockafellar, R.T.: Network Flows and Monotropic Optimization. Wiley, New York, 1984

  22. Roberts, A.W., Varberg, D.E.: Convex Functions. Academic Press, New York, 1973

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Authors and Affiliations

  1. Graduate School of Information Science and Technology, University of Tokyo, Tokyo, 113-8656, Japan

    Kazuo Murota

  2. Graduate School of Information Sciences, Tohoku University, Sendai, 980-8579, Japan

    Akiyoshi Shioura

  3. PRESTO, JST, Tokyo, Japan

    Kazuo Murota

Authors
  1. Kazuo Murota
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  2. Akiyoshi Shioura
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Corresponding author

Correspondence to Akiyoshi Shioura.

Additional information

Mathematics Subject Classification (1991): 90C10, 90C25, 90C27, 90C35

This work is supported by Grant-in-Aid of the Ministry of Education, Culture, Sports, Science and Technology of Japan

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Murota, K., Shioura, A. Conjugacy relationship between M-convex and L-convex functions in continuous variables. Math. Program., Ser. A 101, 415–433 (2004). https://doi.org/10.1007/s10107-003-0478-3

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  • DOI: https://doi.org/10.1007/s10107-003-0478-3

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