Lattice preconditioning for the real relaxation branch-and-bound approach for integer least squares problems

Abstract

The integer least squares problem is an important problem that arises in numerous applications. We propose a real relaxation-based branch-and-bound (RRBB) method for this problem. First, we define a quantity called the distance to integrality, propose it as a measure of the number of nodes in the RRBB enumeration tree, and provide computational evidence that the size of the RRBB tree is proportional to this distance. Since we cannot know the distance to integrality a priori, we prove that the norm of the Moore-Penrose generalized inverse of the matrix of coefficients is a key factor for bounding this distance, and then we propose a preconditioning method to reduce this norm using lattice reduction techniques. We also propose a set of valid box constraints that help accelerate the RRBB method. Our computational results show that the proposed preconditioning significantly reduces the size of the RRBB enumeration tree, that the preconditioning combined with the proposed set of box constraints can significantly reduce the computational time of RRBB, and that the resulting RRBB method can outperform the Schnorr and Eucher method, a widely used method for solving integer least squares problems, on some types of problem data.

On the KZ Reduction

“…This indicates that the upper bound in (18) is much sharper than that in (23). By (21), for i ≥ 9, we have…”

“…By the definition of α n in (13) and its upper bound (15) given in Theorem 2, we can see that (18) holds. Since A is KZ reduced, so are R i:j,i:j ∈ R (j−i+1)×(j−i+1) for 1 ≤ i < j ≤ n. Then according to (18), (19) holds.…”

“…In the following we compare our bounds (18) and (20) in Theorem 3 with (23) and (24), respectively. By (16), for i ≥ 9, we have…”

“…Then, in the second step, a search algorithm, typically the Schnorr-Euchner search strategy [16], which is an improvement of the Fincke-Pohst search strategy [17], is used to enumerate the integer vectors within a hyperellipsoid sphere (or equivalently, the lattice points within a hypersphere). The first step, which is also called as a preprocessing step, is carried out to make the second step faster (see, e.g., [15] [18]).…”

Convex reformulation for binary quadratic programming problems via average objective value maximization

“…The test problems are randomly generated with a method similar to the one proposed in [1]: for a given dimension n, we first generate an n×n matrix H , each element of H is a randomly generated Gaussian number with mean 0, and variance 1. Then we generate s ∈ {0, 1} n , with probability Prob(s i = 0) = Prob(s i = 1) = 0.5 for each element.…”