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The mixing-MIR set with divisible capacities

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Abstract

We study the set \(S = \{(x, y) \in \Re_{+} \times Z^{n} : x + B_{j} y_{j} \geq b_{j}, j = 1, \ldots, n\}\) , where \(B_{j}, b_{j} \in \Re_{+} - \{0\}\) , j =  1, ..., n, and B 1 | ... | B n . The set S generalizes the mixed-integer rounding (MIR) set of Nemhauser and Wolsey and the mixing-MIR set of Günlük and Pochet. In addition, it arises as a substructure in general mixed-integer programming (MIP), such as in lot-sizing. Despite its importance, a number of basic questions about S remain unanswered, including the tractability of optimization over S and how to efficiently find a most violated cutting plane valid for P =  conv (S). We address these questions by analyzing the extreme points and extreme rays of P. We give all extreme points and extreme rays of P. In the worst case, the number of extreme points grows exponentially with n. However, we show that, in some interesting cases, it is bounded by a polynomial of n. In such cases, it is possible to derive strong cutting planes for P efficiently. Finally, we use our results on the extreme points of P to give a polynomial-time algorithm for solving optimization over S.

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Authors and Affiliations

  1. State University of New York, New York, USA

    M. Zhao

  2. Department of Industrial and Systems Engineering, University at Buffalo, 438 Bell Hall, Buffalo, NY, 14260-2050, USA

    I. R. de Farias Jr

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  1. M. Zhao
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  2. I. R. de Farias Jr
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Correspondence to I. R. de Farias Jr.

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Zhao, M., de Farias, I.R. The mixing-MIR set with divisible capacities. Math. Program. 115, 73–103 (2008). https://doi.org/10.1007/s10107-007-0140-6

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  • DOI: https://doi.org/10.1007/s10107-007-0140-6

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