Two New Reformulation Convexification Based Hierarchies for 0-1 MIPs

Abstract

First, we introduce two new reformulation convexification based hierarchies called RTC and RSC for which the rank d continuous relaxations are denoted by P^RTCd and P^RSCd, 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 d, we will prove that convTd∩Etd⊆P^RTCd⊆Td and convSd∩Esd⊆P^RSCd⊆Sd where the sets Td and Sd are convex, while Etd and Esd 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.