Multimodularity, Convexity and Optimization Properties

Eitan Altman 1 Bruno Gaujal 2 Arie Hordijk 3
2 MESCAL - Middleware efficiently scalable
ID-IMAG - Informatique et Distribution, Inria Grenoble - Rhône-Alpes
Abstract : We investigate in this paper the properties of multimodular functions. In doing so we give elementary proofs for properties already established by Hajek, and we generalize some of his results. In particular, we extend the relation bewteen convexity and multimodularity to some convex subsets of the integers numbers. We also obtain general optimizatnio results for average costs related to a sequence of multimodular functions rather than to a single functions. Under this general context, we show that the expected average cost problem is optimized by ising balanced sequences. We finally illustrate the usefulness of this theory in admission control into a D/D/1 queue with fixed bathch arrivals, with no state information. We show that the balanced policy minimizes the average queue length for the case of an intinite queeu, but not for the case of a finite queue. When further adding a constraint on the losses, it si show that a balanced policy is also optimal for the finite queue case.
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https://hal.inria.fr/inria-00113337
Contributor : Bruno Gaujal <>
Submitted on : Monday, November 13, 2006 - 10:49:34 AM
Last modification on : Wednesday, April 11, 2018 - 1:52:52 AM
Long-term archiving on : Tuesday, April 6, 2010 - 10:20:14 PM

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Eitan Altman, Bruno Gaujal, Arie Hordijk. Multimodularity, Convexity and Optimization Properties. Mathematics of Operations Research, INFORMS, 2000, 25 (2), pp.324-347. ⟨10.1287/moor.25.2.324.12230⟩. ⟨inria-00113337⟩

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