Constructing copositive matrices from interior matrices

Abstract

Let A be an n by n symmetric matrix with real entries. Using the l₁-norm for vectors and letting S₁⁺= {x ∈ ℝⁿ|||x||1 = 1, x ≥ 0}, the matrix A is said to be interior if the quadratic form x^TAx achieves its minimum on S₁⁺ in the interior. Necessary and sufficient conditions are provided for a matrix to be interior. A copositive matrix is referred to as being exceptional if it is not the sum of a positive semidefinite matrix and a nonnegative matrix. A method is provided for constructing exceptional copositive matrices by completing a partial copositive matrix that has certain specified overlapping copositive interior principal submatrices.