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Article Dans Une Revue Discrete Applied Mathematics Année : 2010

DRL*: A Hierarchy of Strong Block-Decomposable Linear Relaxations for 0-1 MIPs

Michel Minoux
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Hacène Ouzia
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Résumé

In this paper we introduce DRL*, a new hierarchy of linear relaxations for 0–1 mixed integer linear programs (MIPs), based on the idea of Reformulation–Linearization, and explore its links with the Lift-and-Project (L&P) hierarchy and the Sherali–Adams (RLT) hierarchy. The relaxations of the new hierarchy are shown to be intermediate in strength between L&P and RLT relaxations, and examples are shown for which it leads to significantly stronger bounds than those obtained from Lift-and-Project relaxations. On the other hand, as opposed to the RLT relaxations, a key advantage of the DRL* relaxations is that they feature a decomposable structure when formulated in extended space, therefore lending themselves to more efficient solution algorithms by properly exploiting decomposition. Links between DRL* and both the L&P and RLT hierarchies are further explored, and those constraints which should be added to the rank d L&P relaxation (resp to the rank d RLT relaxation) to make it coincide with the rank d DRL* relaxation (resp: to the rank d RLT relaxation) are identified. Furthermore, a full characterization of those 0–1 MIPs for which the DRL* and RLT relaxations coincide is obtained. As an application, we show that both the RLT and DRL* relaxations are the same up to rank d for the problem of optimizing a pseudoboolean function of degree d over a polyhedron. We report computational results comparing the strengths of the rank 2 L&P, DRL* and RLT relaxations. Impact on possible improved efficiency in computing some bounds for the quadratic assignment problem and other directions for future research are suggested in the conclusions.

Dates et versions

hal-01170343 , version 1 (01-07-2015)

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Citer

Michel Minoux, Hacène Ouzia. DRL*: A Hierarchy of Strong Block-Decomposable Linear Relaxations for 0-1 MIPs. Discrete Applied Mathematics, 2010, 158 (18), pp.2031-2048. ⟨10.1016/j.dam.2010.08.020⟩. ⟨hal-01170343⟩
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