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A Hierarchy of Relaxations Leading to the Convex Hull Representation for General Discrete Optimization Problems

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Abstract

We consider linear mixed-integer programs where a subset of the variables are restricted to take on a finite number of general discrete values. For this class of problems, we develop a reformulation-linearization technique (RLT) to generate a hierarchy of linear programming relaxations that spans the spectrum from the continuous relaxation to the convex hull representation. This process involves a reformulation phase in which suitable products using a defined set of Lagrange interpolating polynomials (LIPs) are constructed, accompanied by the application of an identity that generalizes x(1−x) for the special case of a binary variable x. This is followed by a linearization phase that is based on variable substitutions. The constructs and arguments are distinct from those for the mixed 0-1 RLT, yet they encompass these earlier results. We illustrate the approach through some examples, emphasizing the polyhedral structure afforded by the linearized LIPs. We also consider polynomial mixed-integer programs, exploitation of structure, and conditional-logic enhancements, and provide insight into relationships with a special-structure RLT implementation.

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Authors and Affiliations

  1. Department of Mathematical Sciences, Clemson University, Clemson, South Carolina, 29634

    Warren P. Adams & 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. A Hierarchy of Relaxations Leading to the Convex Hull Representation for General Discrete Optimization Problems. Ann Oper Res 140, 21–47 (2005). https://doi.org/10.1007/s10479-005-3966-4

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