Advertisement

Convexity and Optimization of Condense Discrete Functions

  • Emre Tokgöz
  • Sara Nourazari
  • Hillel Kumin
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 6630)

Abstract

A function with one integer variable is defined to be integer convex by Fox \(\left[ 3\right] \) and Denardo \(\left[ 1\right] \) if its second forward differences are positive. In this paper, condense discrete convexity of nonlinear discrete multivariable functions with their corresponding Hessian matrices is introduced which is a generalization of the integer convexity definition of Fox \(\left[ 3\right] \) and Denardo \(\left[ 1\right] \) to higher dimensional space ℤn. In addition, optimization results are proven for C 1 condense discrete convex functions assuming that the given condense discrete convex function is C 1. Yüceer \(\left[ 17\right] \) proves convexity results for a certain class of discrete convex functions and shows that the restriction of the adaptation of Rosenbrook’s function from real variables to discrete variables does not yield a discretely convex function. Here it is shown that the adaptation of Rosenbrook’s function considered in \(\left[ 17\right] \) is a condense discrete convex function where the set of local minimums is also the the set of global minimums.

Keywords

Integer programming mathematical programming discrete convex function real convex function 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Denardo, E.V.: Dynamic Programming. Prentice-Hall, Englewood Cliffs (1982)zbMATHGoogle Scholar
  2. 2.
    Favati, P., Tardella, F.: Convexity in Nonlinear Integer Programming. Ricerca Operativa 53, 3–44 (1990)Google Scholar
  3. 3.
    Fox, B.: Discrete optimization via marginal analysis. Management Sci. 13, 210–216 (1966)CrossRefzbMATHGoogle Scholar
  4. 4.
    Fujishige, S., Murota, K.: Notes on L-/M-convex functions and the separation theorems. Math. Prog. 88, 129–146 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Hirai, H., Murota, K.: M-convex functions and tree metrics. Japan J. Industrial Applied Math. 21, 391–403 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Kiselman, C.O., Christer, O.: Local minima, marginal functions, and separating hyperplanes in discrete optimization. In: Bhatia, R. (ed.) Abstracts: Short Communications; Posters. International Congress of Mathematicians, Hyderabad, August 19-27, pp. 572–573 (2010)Google Scholar
  7. 7.
    Kiselman, C.O., Acad, C. R.: Local minima, marginal functions, and separating hyperplanes in discrete optimization. Sci. Paris, Ser. I, (or Three problems in digital convexity: local minima, marginal functions, and separating hyperplanes - The case of two variables, by C.O. Kiselman, Manuscript) (2008)Google Scholar
  8. 8.
    Kiselman, C. O., Samieinia S.: Convexity of marginal functions in the discrete case. manuscript (2010), http://www2.math.uu.se/~kiselman/papersix.pdf
  9. 9.
    Miller, B.L.: On minimizing nonseparable functions defined on the integers with an inventory application. SIAM J. Appl. Math. 21, 166–185 (1971)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Moriguchi, S., Murota, K.: Discrete Hessian matrix for L-convex functions. IECE Trans. Fundamentals, E88-A (2005)Google Scholar
  11. 11.
    Murota, K.: Convexity and Steinitz’s exchange property. Adv. Math., 272–311 (1996)Google Scholar
  12. 12.
    Murota, K.: Discrete convex analysis. Math. Prog. 83, 313–371 (1998)MathSciNetzbMATHGoogle Scholar
  13. 13.
    Murota, K.: Discrete convex analysis. Society for Industrial and Applied Mathematics, Philadelphia (2003)CrossRefzbMATHGoogle Scholar
  14. 14.
    Murota, K., Shioura, A.: M-convex function on generalized polymatroid. Math. Oper. Res. 24, 95–105 (1999)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Rockafellar, R.T.: Convex Analysis. Princten University Press, Princeton (1970)CrossRefzbMATHGoogle Scholar
  16. 16.
    Ui, T.: A note on discrete convexity and local optimality. Japan J. Indust. Appl. Math. 23, 21–29 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Yüceer, U.: Discrete convexity: convexity for functions defined on discrete spaces. Disc. Appl. Math. 119, 297–304 (2002)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2011

Authors and Affiliations

  • Emre Tokgöz
    • 1
    • 2
  • Sara Nourazari
    • 3
  • Hillel Kumin
    • 1
  1. 1.School of Industrial EngineeringUniversity of OklahomaNormanU.S.A.
  2. 2.Department of MathematicsUniversity of OklahomaNormanU.S.A.
  3. 3.Department of Systems Engineering and Operations ResearchGeorge Mason UniversityFairfaxU.S.A.

Personalised recommendations