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)


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.


Integer programming mathematical programming discrete convex function real convex function 


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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.

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