  In this study, we examine the various extensions of the doubly nonnegative
(DNN) cone, frequently used in completely positive programming (CPP) to achieve
a tighter relaxation than the positive semidefinite cone. To provide tighter
relaxation for generalized CPP (GCPP) than the positive semidefinite cone,
inner-approximation hierarchies of the generalized copositive cone are
exploited to obtain two generalized DNN (GDNN) cones from the DNN cone. This
study conducts theoretical and numerical comparisons to assess the relaxation
strengths of the two GDNN cones over the direct products of a nonnegative
orthant and second-order or positive semidefinite cones. These comparisons also
include an analysis of the existing GDNN cone proposed by Burer and Dong. The
findings from solving several GDNN programming relaxation problems for a GCPP
problem demonstrate that the three GDNN cones provide significantly tighter
bounds for GCPP than the positive semidefinite cone.
