Concise RLT forms of binary programs: A computational study of the quadratic knapsack problem

Abstract

Abstract:The reformulation-linearization technique (RLT) is a methodology for constructing tight linear programming relaxations of mixed discrete problems. A key construct is the multiplication of "product factors" of the discrete variables with problem constraints to form polynomial restrictions, which are subsequently linearized. For special problem forms, the structure of these linearized constraints tends to suggest that certain classes may be more beneficial than others. We examine the usefulness of subsets of constraints for a family of 0-1 quadratic multidimensional knapsack programs and perform extensive computational tests on a classical special case known as the 0-1 quadratic knapsack problem. We consider RLT forms both with and without these inequalities, and their comparisons with linearizations derived from published methods. Interestingly, the computational results depend in part upon the commercial software used.

Computational Comparison of Exact Solution Methods for 0-1 Quadratic Programs: Recommendations for Practitioners

“…As noted earlier, the two most common representations of the quadratic objective coefficients C ij in the literature are to either assume C ij = C ji for all i < j (so that the quadratic coefficient matrix C is symmetric) or assume C ij = 0 for all i ≥ j (so that the quadratic coefficient matrix C is upper triangular); both of which can be assumed without loss of generality. While the continuous relaxation strength of STD and STD ′ are not affected by the choice of objective function representation, the relaxation value of G1 is dependent on the manner in which the objective function of a BQP is expressed as shown in [9,10]. This follows from the fact that the quadratic objective coefficients C ij do not appear in the auxiliary constraints of STD or STD ′ , whereas they do appear in those of G1.…”

“…Quadratic Multidimensional Knapsack Problem. The next class of problems that we consider is the quadratic knapsack problem with multiple constraints, also known as the quadratic multidimensional knapsack problem (QMKP) (see [10,12]). These problems have the following form.…”

“…For our tests, we considered two values of n (4 and 5) and three values of m (10,12,15). For each of the six ðm, nÞ pairs, we generated 10 instances where the coefficients e ik and C ikjl are integers from a uniform distribution over the interval [−50,50] as in [1].…”

Optimal Decentralization of Early Infant Diagnosis of HIV in Resource-Limited Settings

“…Specifically, we use a linearization reformulation technique (Forrester et al 2010) instead of meta-heuristics (e.g., Tabu Search) and complex specialized algorithms (e.g., Lagrangean relaxation) used in the earlier papers.…”