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
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
Denardo, E.V.: Dynamic Programming. Prentice-Hall, Englewood Cliffs (1982)
Favati, P., Tardella, F.: Convexity in Nonlinear Integer Programming. Ricerca Operativa 53, 3–44 (1990)
Fox, B.: Discrete optimization via marginal analysis. Management Sci. 13, 210–216 (1966)
Fujishige, S., Murota, K.: Notes on L-/M-convex functions and the separation theorems. Math. Prog. 88, 129–146 (2000)
Hirai, H., Murota, K.: M-convex functions and tree metrics. Japan J. Industrial Applied Math. 21, 391–403 (2004)
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)
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)
Kiselman, C. O., Samieinia S.: Convexity of marginal functions in the discrete case. manuscript (2010), http://www2.math.uu.se/~kiselman/papersix.pdf
Miller, B.L.: On minimizing nonseparable functions defined on the integers with an inventory application. SIAM J. Appl. Math. 21, 166–185 (1971)
Moriguchi, S., Murota, K.: Discrete Hessian matrix for L-convex functions. IECE Trans. Fundamentals, E88-A (2005)
Murota, K.: Convexity and Steinitz’s exchange property. Adv. Math., 272–311 (1996)
Murota, K.: Discrete convex analysis. Math. Prog. 83, 313–371 (1998)
Murota, K.: Discrete convex analysis. Society for Industrial and Applied Mathematics, Philadelphia (2003)
Murota, K., Shioura, A.: M-convex function on generalized polymatroid. Math. Oper. Res. 24, 95–105 (1999)
Rockafellar, R.T.: Convex Analysis. Princten University Press, Princeton (1970)
Ui, T.: A note on discrete convexity and local optimality. Japan J. Indust. Appl. Math. 23, 21–29 (2006)
Yüceer, U.: Discrete convexity: convexity for functions defined on discrete spaces. Disc. Appl. Math. 119, 297–304 (2002)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2011 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Tokgöz, E., Nourazari, S., Kumin, H. (2011). Convexity and Optimization of Condense Discrete Functions. In: Pardalos, P.M., Rebennack, S. (eds) Experimental Algorithms. SEA 2011. Lecture Notes in Computer Science, vol 6630. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-20662-7_3
Download citation
DOI: https://doi.org/10.1007/978-3-642-20662-7_3
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-20661-0
Online ISBN: 978-3-642-20662-7
eBook Packages: Computer ScienceComputer Science (R0)