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Enhancements on the Hyperplanes Arrangements in Mixed-Integer Programming Techniques

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Abstract

This paper is concerned with improvements in constraints handling for mixed-integer optimization problems. The novel element is the reduction of the number of binary variables used for expressing the complement of a convex (polytopic) region. As a generalization, the problem of representing the complement of a possibly not connected union of such convex sets is detailed. In order to illustrate the benefits of the proposed improvements, a typical control application, the control of multiagent systems using receding horizon optimization techniques, is considered.

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Notes

  1. The relative interiors of these regions do not intersect, but their closures have as a common boundary the affine subspace \(\mathcal{H}_{i}\).

  2. There exists a finite M sufficiently large if and only if the polyhedra of type (1) are bounded. In the remaining portion of the paper, all the polyhedra of type (1) are assumed to be bounded for this reason.

  3. d denotes the degree of the facet, ranging from 0 for extreme points to N 0−1 for faces of the hypercube.

  4. The “+” superscript was chosen for the homogeneity of notation, equivalently one could have chosen any combination of signs in the half-space representation (4) in order to describe the polyhedral regions P l .

  5. The relative interior of any two cells is disjoint, but their boundaries may have one of the hyperplanes \(\mathcal{H}_{i}\) as a common element.

  6. We call a hyperplane arrangement to be “in general position” whenever any small change in the position of the composing hyperplanes does not change the number of cells.

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Acknowledgements

The research of Ionela Prodan is financially supported by the EADS Corporate Foundation (091-AO09-1006). Florin Stoican’s work was carried out in Supelec, during the tenure of a CARNOT C3S fellowship. The authors would like to thank Professor Panos M. Pardalos, as well as the anonymous reviewers for their useful comments and remarks that helped in improving the overall presentation of this paper.

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Correspondence to Ionela Prodan.

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Communicated by Panos M. Pardalos.

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Prodan, I., Stoican, F., Olaru, S. et al. Enhancements on the Hyperplanes Arrangements in Mixed-Integer Programming Techniques. J Optim Theory Appl 154, 549–572 (2012). https://doi.org/10.1007/s10957-012-0022-9

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