Generalizations of doubly nonnegative cones and their comparison

In this study, we theoretically and numerically compare several generalizations of the doubly nonnegative (DNN) cone, which is frequently used to provide a relaxation that is tighter than that of the positive semidefinite cone for completely positive programming (CPP). To provide tighter relaxation for generalized CPP (GCPP) than the positive semidefinite cone, we generalize the DNN cone and obtain Zuluaga--Vera--Pe\~{n}a (ZVP)- and Nishijima--Nakata (NN)-type generalized DNN (GDNN) cones by exploiting inner-approximation hierarchies for the generalized copositive cone. Our investigation primarily focuses on the inclusion relationship between the two GDNN cones over a direct product of a nonnegative orthant and second-order cones as well as the existing (BD-type) cone proposed by Burer and Dong. We found that the NN-type GDNN cone is theoretically included in the ZVP-type cone, and that no inclusion relationship obtains between the ZVP- and BD-type GDNN cones. The results of solving several GDNN programming relaxation problems for a GCPP problem are provided to demonstrate that the three GDNN cones, especially the ZVP- and NN-types, yielded much tighter bounds for GCPP than the positive semidefinite cone.



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