Two New Reformulation Convexification Based Hierarchies for 0-1 MIPs
Résumé
Let n and m two nonnegative integers. Let X ⊆ R n+m be the set of feasible solutions of a mixed integer linear problem (n binary variables and m continuous variables). Without lost of generality, we will assume that X is a subset of the hypercube [0, 1] n+m. By hierarchy of relaxations (hierarchy for short) of the set X we mean a finite family of continuous relaxations indexed by an integer, called rank, such that : (i) the relaxation of rank 0 coincides with the continuous relaxation of X; (ii) for every integer d, the relaxation of rank d is always included in the relaxation of rank d − 1 and (iii) the relaxation of rank n (the number of binary variables) coincides with the convex hull of the set X. We will say that an hierarchy A dominates another hierarchy B if, for avery integer d, the rank d relaxation of the hierarchy A is included in the rank d relaxation of the hierarchy B. An hierarchy A is said to be equivalent to an hierarchy B if A and B dominate each other. The hierarchies we will address in this work are all defined using four steps : refor-mulation, convexification, linearization and projection. The reformulation we will consider was introduced by Sherali and Adams (see [2]). The linearization step consists in replacing, using new variables, the nonlinear terms appearing in the description of the set obtained after the reformulation (or convexification) step. Different linearizations are possible and, as proved in [1], this gives rise to different hierarchies. The linear description obtained after the linearization step is called extended linear description. The projection step consists in projecting back the extended linear description onto the space of the original variables.
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