DRL*: A hierarchy of strong block-decomposable linear relaxations for 0–1 MIPs

Two New Reformulation Convexification Based Hierarchies for 0-1 MIPs

“…The third step consists in projecting back the resulting polyhedron onto the original -space. As observed in [8] the linearization step can be performed in various ways, leading to various hierarchies of relaxations.…”

“…Starting from this semialgebraic reformulation, various linear relaxations can be constructed depending on the type of linearization considered (for more details see [8,13]).…”

“…This completes the proof. Now, we will compare the strength of the two hierarchies RSC and L&P. As shown in [8], the L&P hierarchy can be obtained from the semialgebraic set (6) using a suitable linearization (for more details, see [8]). Precisely, any rank-L&P relaxation is a rank-reformulation-linearization relaxation where the linearization is performed using the following substitutions: for every subset from , let , = ( , \ ) ∀ ∈ , ⊆ , , 0 = ( , \ ) ∀ ⊆ .…”

“…Optimizing even a linear function over the mixed integer set is an NP-hard problem in general (see [1][2][3][4]). A way of building strengthened linear relaxations is to use an approach combining reformulation, linearization, and projection such as those proposed in [5][6][7][8][9][10]. In such approach, one first reformulates the constraints defining the set of feasible solutions by introducing nonlinearities.…”

“…Many hierarchies were introduced. To mention a few, L&P hierarchy (Balas et al, see [5]), LS hierarchy (Lovász and Schrijver, see [7]), RLT hierarchy (Sherali and Admas, see [9,10]), LAS hierarchy (Lasserre, see [6]), BZ hierarchy (Bienstock and Zuckerberg, see [12]), and DRL * hierarchy (Minoux and Ouzia, see [8,13]). For more details, a set-theoretical interpretation of the reformulationlinearization approaches has been proposed in [14] and a theoretical comparative study between the RLT, LS, and LAS relaxations can be found in [15].…”