A polynomial-time recursive algorithm for some unconstrained quadratic optimization problems

Determining the cells of an arrangement of hyperplanes is a classical problem in combinatorial geometry. In this paper we present an efficient recursive procedure to solve it. In fact, by considering a connection between the latter problem and optimality conditions of unconstrained quadratic optimization problems in the following form: (QP) min^xtQx|x∈^-1,1n, with Q∈Rn×ⁿ, we can show the proposed algorithm solves problems of the form (QP) in polynomial time when the rank of the matrix Q is fixed and the number of its positive diagonal entries is O(log(n)).