Discrete convexity: convexity for functions defined on discrete spaces

Authors:
Ümit Yüceer
Affiliations:
Department of Industrial Engineering, Eastern Mediterranean University, Gazimagusa, T.R.N.C., via Mersin 10 Turkey
Venue:
Discrete Applied Mathematics
Year:
2002

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Abstract

The concept of discrete convexity for a real-valued function defined on a discrete space is an extension of the convexity definition of continuous functions. The equivalence of discrete convexity to the conventional definition of increasing (non-decreasing) first forward differences of functions of single variables is established. A further extension of the discrete convexity with submodularity yields the concept of strong discrete convexity. A function with the property of strong discrete convexity has a positive semi-definite matrix of second forward differences.