On standard quadratic programs with exact and inexact doubly nonnegative relaxations

Abstract The problem of minimizing a (nonconvex) quadratic form over the unit simplex, referred to as standard program, admits an exact convex conic formulation computationally intractable cone completely positive matrices. Replacing in this by larger but tractable doubly nonnegative matrices, i.e., semidefinite and componentwise one obtains so-called relaxation, whose optimal value yields lower bound on that original problem. We present full algebraic characterization set instances programs admit relaxation. This algorithmic recipe for constructing such instance. In addition, we explicitly identify three families which relaxation is exact. establish several relations between convexity graph instance tightness also provide has gap show how construct using characterization.