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RLT insights into lift-and-project closures

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Abstract

This paper examines relationships between the two methods for constructing polyhedral outer-approximations of mixed 0-1 linear programs known in the literature as the level-1 Reformulation–Linearization-Technique (RLT) and the elementary closure of lift-and-project (L&P) cuts based on the level-1 L&P. The latter can be obtained, via a transformation of variables, as a relaxed version of the former through the removal of constraints. Whereas both approaches derive formulations in lifted spaces by scaling problem constraints with suitable multipliers, a key difference is that the multipliers used by the RLT are defined in terms of the binary variables of the given problem. As a consequence, these multipliers, in contrast to those used in the L&P, portend important advantages, including the ability to handle nonlinear expressions, to recognize redundant inequalities, and to exploit special structures. This paper compares the two methods, and uses RLT constructs to provide insights into the L&P approach. Some particular insights are illustrated on the quadratic assignment and MAX-2SAT problems, with level-2 RLT and level-2 L&P comparisons also given for the latter problem.

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Acknowledgments

The authors gratefully acknowledge support from the National Science Foundation under grant numbers CMMI-0968909 and CMMI-0969169. We also thank two anonymous referees for providing valuable suggestions that improved the presentation of the paper.

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Authors and Affiliations

  1. Department of Mathematical Sciences, O-327 Martin Hall, Clemson University, Clemson, SC, 29634-0975, USA

    Warren P. Adams

  2. Grado Department of Industrial and Systems Engineering (0118), Virginia Tech, 250 Durham Hall, Blacksburg, VA, 24061, USA

    Hanif D. Sherali

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  1. Warren P. Adams
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  2. Hanif D. Sherali
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Correspondence to Warren P. Adams.

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Adams, W.P., Sherali, H.D. RLT insights into lift-and-project closures. Optim Lett 9, 19–39 (2015). https://doi.org/10.1007/s11590-014-0763-5

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