Abstract
Lifting is a procedure for deriving valid inequalities for mixed-integer sets from valid inequalities for suitable restrictions of those sets. Lifting has been shown to be very effective in developing strong valid inequalities for linear integer programming and it has been successfully used to solve such problems with branch-and-cut algorithms. Here we generalize the theory of lifting to conic integer programming, i.e., integer programs with conic constraints. We show how to derive conic valid inequalities for a conic integer program from conic inequalities valid for its lower-dimensional restrictions. In order to simplify the computations, we also discuss sequence-independent lifting for conic integer programs. When the cones are restricted to nonnegative orthants, conic lifting reduces to the lifting for linear integer programming as one may expect.
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We are thankful to an anonymous referee for several valuable comments, especially the ones that led to Propositions 6 and 7.
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This research is supported, in part, by the National Science Foundation Grant 0700203: Conic Integer Programming. A. Atamtürk is grateful to the hospitality of the Georgia Institute of Technology, where part of this research was conducted.
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Open Access This is an open access article distributed under the terms of the Creative Commons Attribution Noncommercial License (https://creativecommons.org/licenses/by-nc/2.0), which permits any noncommercial use, distribution, and reproduction in any medium, provided the original author(s) and source are credited.
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Atamtürk, A., Narayanan, V. Lifting for conic mixed-integer programming. Math. Program. 126, 351–363 (2011). https://doi.org/10.1007/s10107-009-0282-9
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DOI: https://doi.org/10.1007/s10107-009-0282-9