The copositive completion problem

Abstract

An n × n real symmetric matrix A is called (strictly) copositive if x T Ax ⩾ 0 (>0) whenever x ∈ R n satisfies x ⩾ 0 ( x ⩾ 0 and x ≠ 0). The (strictly) copositive matrix completion problem asks which partial (strictly) copositive matrices have a completion to a (strictly) copositive matrix. We prove that every partial (strictly) copositive matrix has a (strictly) copositive matrix completion and give a lower bound on the values used in the completion. We answer affirmatively an open question whether an n × n copositive matrix A = ( a ij ) with all diagonal entries a ii = 1 stays copositive if each off-diagonal entry of A is replaced by min{ a ij , 1}.