Abstract
LetA 1,⋯,A n be distinctk-dimensional vectors. We consider the problem of partitioning these vectors intom sets so as to maximize an objective which is a quasi-convex function of the sum of vectors in each set. We show that there exists an optimal partition whose sets have (pairwise) disjoint conic hulls. We also show that if the number of vectors in each of the sets is constrained, then a weaker conclusion holds, namely, there exists an optimal partition whose sets have (pairwise) disjoint convex hulls. The results rely on deriving necessary and sufficient conditions for a point to be an extreme point of a corresponding polytope.
Similar content being viewed by others
References
E.R. Barnes, “An algorithm for partitioning the nodes of a graph,”SIAM Journal of Algebraic and Discrete Mathematics 3 (1982) 541–550.
E.R. Barnes, “Partitioning the nodes of a graph,” manuscript (1985).
E.R. Barnes and A.J. Hoffman, “Partitioning, spectra and linear programming,” in: W.E. Pulleyblank, ed.,Progress in Combinatorial Optimization (Academic Press, New York, 1984).
E.R. Barnes, A. Vannelli and J.O. Walker, “A new procedure for partitioning the nodes of a graph,” IBM Research Report RC 10561 (June 1984).
A.K. Chakravarty, J.B. Orlin and U.G. Rothblum, “A partitioning problem with additive objective with an application to optimal inventory groupings for joint replenishment,”Operations Research 30 (1982) 1018–1022.
A.K. Chakravarty, J.B. Orlin and U.G. Rothblum, “Consecutive optimizors for a partitioning problem with applications to optimal inventory groupings for joint replenishment,”Operations Research 33 (1985) 820–834.
E.V. Denardo,Dynamic Programming: Models and Applications (Prentice Hall, Englewood Cliffs, NJ 1982).
W.E. Donath and A.J. Hoffman, “Lower bounds for partitioning of graphs,”IBM Journal of Research and Development 17 (1973) 420–425.
F.K. Hwang, “Optimal partitions,”Journal on Optimization Theory and Applications 34 (1981) 1–10.
F.K. Hwang, J. Sun and E.Y. Yao, “Optimal set partitioning,”SIAM Journal on Algebraic and Discrete Mathematics 6 (1985) 163–170.
G. Ramfos, “Generating hyperplane partitions ofn points ink dimensions,” Bachelor's Thesis, National Technical University of Athens (Athens, 1983). [In Greek.]
A. Schrijver,Theory of Linear and Integer Programming (Wiley, New York, 1986).
Author information
Authors and Affiliations
Additional information
Research of this author was partially supported by NSF Grant ECS-83-10213 and by a Grant for the Promotion of Research at the Technion.
Rights and permissions
About this article
Cite this article
Barnes, E.R., Hoffman, A.J. & Rothblum, U.G. Optimal partitions having disjoint convex and conic hulls. Mathematical Programming 54, 69–86 (1992). https://doi.org/10.1007/BF01586042
Received:
Revised:
Issue Date:
DOI: https://doi.org/10.1007/BF01586042