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Integer programming as projection

Williams, H. Paul and Hooker, J. N. (2014) Integer programming as projection. Working Paper, LSEOR 13.143. Management Science Group, Department of Management, The London School of Economics and Political Science, London, UK.

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Abstract

We generalise polyhedral projection (Fourier-Motzkin elimination) to integer programming (IP) and derive from this an alternative perspective on IP that parallels the classical theory. We first observe that projection of an IP yields an IP augmented with linear congruence relations and finite-domain variables, which we term a generalised IP. The projection algorithm can be converted to a branch-and-bound algorithm for generalised IP in which the search tree has bounded depth (as opposed to conventional branching, in which there is no bound). It also leads to valid inequalities that are analogous to Chv´atal-Gomory cuts but are derived from congruences rather than rounding, and whose rank is bounded by the number of variables. Finally, projection provides an alternative approach to IP duality. It yields a value function that consists of nested roundings as in the classical case, but in which ordinary rounding is replaced by rounding to the nearest multiple of an appropriate modulus, and the depth of nesting is again bounded by the number of variables.

Item Type: Monograph (Working Paper)
Official URL: http://www.lse.ac.uk/management/research/academic-...
Additional Information: © 2014 The London School of Economics and Political Science
Library of Congress subject classification: H Social Sciences > HD Industries. Land use. Labor
Q Science > QA Mathematics
Sets: Departments > Management
Research centres and groups > Management Science Group
Rights: http://www.lse.ac.uk/library/usingTheLibrary/academicSupport/OA/depositYourResearch.aspx
Identification Number: LSEOR 13.143
Date Deposited: 24 Jan 2014 10:46
URL: http://eprints.lse.ac.uk/55426/

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