Convexity and Optimization of Condense Discrete Functions
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.
KeywordsInteger programming mathematical programming discrete convex function real convex function
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