Exact Solution Methods for the k-Item Quadratic Knapsack Problem

Abstract

The purpose of this paper is to solve the 0-1 k-item quadratic knapsack problem (kQKP ), a problem of maximizing a quadratic function subject to two linear constraints. We propose an exact method based on semidefinite optimization. The semidefinite relaxation used in our approach includes simple rank one constraints, which can be handled efficiently by interior point methods. Furthermore, we strengthen the relaxation by polyhedral constraints and obtain approximate solutions to this semidefinite problem by applying a bundle method. We review other exact solution methods and compare all these approaches by experimenting with instances of various sizes and densities.Keywords: quadratic programming, 0-1 knapsack, k-cluster, semidefinite programming either the value of the problem is equal to max i=1,...,n c ii (for k = 1), or the domain of (kQKP ) is empty (for k > k max ).(kQKP ) is an NP-hard problem as it includes two classical NP-hard subproblems, the k-cluster problem [6] by dropping constraint (1), and the quadratic knapsack problem [20] by dropping constraint (2). Even more, the work of Bhaskara et. al [4] indicates that approximating k-cluster within a polynomial factor might be a harder problem than Unique Games. Rader and Woeginger [16] state negative results concerning the approximability of QKP if negative cost coefficients are present.Applications of (kQKP ) cover those found in previous references for k-cluster or classical quadratic knapsack problems (e.g., task assignment problems in a client-server architecture with limited memory), but also multivariate linear regression and portfolio selection. Specific heuristic and exact methods including branch-and-bound and branch-and-cut with surrogate relaxations have been designed for these applications (see, e.g., [3,5,9,19,23]).The purpose of this paper is twofold.

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

“…Question 3: With regards to the overall computational effort to formulate and solve problem G1 to optimality using an MILP solver, should we compute the bounds U 1 j , U 0 j , L 1 j , and L 0 j using ( 15), (16), or (17)? As noted in [11], we can reduce the size of G1 by removing the constraints (11) and ( 13) that bound z j from below because of the maximization objective and the fact that the z j terms do not appear elsewhere in the problem.…”

“…For each i, U i and L i are upper and lower bounds, respectively, on ∑ n j=1,j≠i C ij x j , and can be computed in a similar fashion as in (15), (16), or (17). However, based upon preliminary computational results, we will construct SS using the bounds computed as in (16).…”

“…For each i, U i and L i are upper and lower bounds, respectively, on ∑ n j=1,j≠i C ij x j , and can be computed in a similar fashion as in (15), (16), or (17). However, based upon preliminary computational results, we will construct SS using the bounds computed as in (16). Note that the authors of [5] actually introduced three formulations, called BP, BP, and BP-strong, which all consider instances of BQP that include quadratic constraints.…”

“…Note that when Journal of Applied Mathematics formulating G1 and G2, we computed the bounds within the formulations as in ( 16) since preliminary computational experience showed this method provides the best performance. However, we will formally compare the bounds of (15), (16), and (17) in another set of tests. We also note that when computing solution times, we included the time needed to find the bounds within G1 and G2.…”

“…Figure 5 shows the performance profiles of G2, G2a, and G3b where we aggregate across all objective coefficient densities, sizes, and MILP solvers. As with our previous test cases, we computed the bounds within the formulations as in (16).…”