On the Structure of Reduced Kernel Lattice Bases

Aardal, Karen; Heymann, Frederik von

doi:10.1007/978-3-642-36694-9_1

Cited by 2 publications

(1 citation statement)

References 16 publications

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“…Since then a number of lattice-based approaches have been proposed for integer linear programming problems, see e.g. the surveys [3,1] and the recent papers [26,4,34,31,2] among many others. Recent research has also considered convex quadratic objectives (as is the case for the ILS problem), see e.g.…”

Section: Introductionmentioning

confidence: 99%

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

Anjos

Chang

2014

J Glob Optim

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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.

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