Hyperspheric Integer Programming

This paper develops an efficient method for finding the optimal solution to linear mathematical programs on 0–1 variables. It is shown that the lattice (0–1) points satisfying some linear constraint of dimension n can equally be represented by those lying in a hypersphere of the same dimension. The lattice points satisfying two linear constraints can be represented by a hypersphere which contains the intersection of the hyperspheres of the two constraints. The method for finding the optimal solution consists of enumerating lattice points which are close to the center of the hypersphere corresponding to the constraints. As soon as a better value of the objective function has been found, than some lower bound, we find a new hypersphere which contains the lattice points of the constraints at which the objective function remains higher than the best known value. We continue in this manner until we have at some stage enumerated all lattice points within a given hypersphere and found none which give a better value.