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Advances in Operations Research
Volume 2015 (2015), Article ID 784817, 13 pages
http://dx.doi.org/10.1155/2015/784817
Research Article

Two New Reformulation Convexification Based Hierarchies for 0-1 MIPs

Sorbonne Universités, UPMC Univ Paris 06, LIP6 UMR 7606, 4 Place Jussieu, 75005 Paris, France

Received 28 July 2015; Accepted 5 October 2015

Academic Editor: Ching-Jong Liao

Copyright © 2015 Hacene Ouzia. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

First, we introduce two new reformulation convexification based hierarchies called RTC and RSC for which the rank continuous relaxations are denoted by and , respectively. These two hierarchies are obtained using two different convexification schemes: term convexification in the case of the RTC hierarchy and standard convexification in the case of the RSC hierarchy. Secondly, we compare the strength of these two hierarchies. We will prove that (i) the hierarchy RTC is equivalent to the RLT hierarchy of Sherali-Adams, (ii) the hierarchy RTC dominates the hierarchy RSC, and (iii) the hierarchy RSC is dominated by the Lift-and-Project hierarchy. Thirdly, for every rank , we will prove that and where the sets and are convex, while and are two nonconvex sets with empty interior (all these sets depend on the convexification step). The first inclusions allow, in some cases, an explicit characterization (in the space of the original variables) of the RLT relaxations. Finally, we will discuss weak version of both RTC and RSC hierarchies and we will emphasize some connections between them.