Comparisons and enhancement strategies for linearizing mixed 0-1 quadratic programs

Abstract

We present a linearization strategy for mixed 0-1 quadratic programs that produces small formulations with tight relaxations. It combines constructs from a classical method of Glover and a more recent reformulation-linearization technique (RLT). By using binary identities to rewrite the objective, a variant of the first method results in a concise formulation with the level-1 RLT strength. This variant is achieved as a modified surrogate dual of a Lagrangian subproblem to the RLT. Special structures can be exploited to obtain reductions in problem size, without forfeiting strength. Preliminary computational experience demonstrates the potential of the new representations.

Reformulations in Mathematical Programming: Definitions and Systematics

“…Liftings, for example (which consist in adding variables to the problem formulation), usually yield reformulations where an optimum in the original problem is mapped to a set of optima in the reformulated problem. Furthermore, it is sometimes noted how a reformulation in this sense is overkill because the reformulation only needs to hold at global optimality [1]. Furthermore, reformulations sometimes really refer to a change of variables, as is the case in [65].…”

“…It is important because it effectively reduces the number of quadratic terms in the problem (only those corresponding to the set N are added). This reformulation can be extended to work with sparse sets E [56], namely sets E whose cardinality is small with respect to 1 2 n(n + 1). Essentially, the constraints w ij = x i x j for (i, j) ∈ B are replaced by the RRCS ∀i ≤ n (Aw i = x i ), where w i = (w i1 , .…”

“…Binary Quadratic Programs (BQP) have attracted a considerable amount of attentions, mostly because of the fact that there is an easy exact linearization [22] that presents some practical drawbacks. Extensive work was carried out by research teams led by A. Billionnet [13], F. Glover [1], P. Hammer [35], P. Hansen [36] (see Sect. 4.1), Ph.…”

Non-convex mixed-integer nonlinear programming: A survey

“…See, e.g., [95,96] for related formulations. Chaovalitwongse et al [97] and Sherali and Smith [98] provide recent, conceptually different O(n) linearization approaches.…”

Decomposition algorithms for data placement problem based on Lagrangian relaxation and randomized rounding

“…This allows to obtain an equivalent binary linear program at the price of including a set of new constraints. One particularly useful linearization technique, specifically designed for (mixed) binary quadratic problems, was given and analyzed in Adams et al (2004 …”