Convexity and optimization of condense discrete functions

  • Authors:

  • Emre Tokgöz;Sara Nourazari;Hillel Kumin

  • Affiliations:

  • School of Industrial Engineering, University of Oklahoma, Department of Mathematics, Norman, OK;George Mason University, Department of Systems Engineering and Operations Research, Fairfax, Virginia;School of Industrial Engineering, University of Oklahoma, Norman, OK

  • Venue:
  • SEA'11 Proceedings of the 10th international conference on Experimental algorithms
  • Year:
  • 2011

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Abstract

A function with one integer variable is defined to be integer convex by Fox [3] and Denardo [1] 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 [3] and Denardo [1] to higher dimensional space Zn. In addition, optimization results are proven for C1 condense discrete convex functions assuming that the given condense discrete convex function is C1. Yüceer [17] 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 [17] is a condense discrete convex function where the set of local minimums is also the the set of global minimums.