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Integer programming as projection

Williams, H. Paul and Hooker, J. N. (2016) Integer programming as projection. Discrete Optimization, 22 (B). pp. 291-311. ISSN 1572-5286

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Identification Number: 10.1016/j.disopt.2016.08.004


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 Chvátal–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. For large perturbations of the right-hand sides, the value function is shift periodic and can be interpreted economically as yielding “average” shadow prices.

Item Type: Article
Official URL:
Additional Information: © 2016 Elsevier
Divisions: Management
Subjects: H Social Sciences > HD Industries. Land use. Labor
Q Science > QA Mathematics
Date Deposited: 06 Oct 2016 09:51
Last Modified: 20 Oct 2021 00:32

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