Generalizations of Doubly Nonnegative Cones and Their Comparison

Our objective was to generalize the well-known doubly nonnegative (DNN) cone, and compare it with a generalized cone proposed by Burer and Dong in 2012.
Several inner-approximation hierarchies for the generalized copositive cone over a closed cone have been proposed as generalizations of that proposed by Parrilo in 2000.
By exploiting these hierarchies, we generalized the DNN cone and obtained Zuluaga-Vera-Pe\~{n}a (ZVP)- and Nishijima-Nakata (NN)-type generalized DNN (GDNN) cones.
The proposed GDNN cones have a semidefinite representation in contrast with the existing Burer-Dong (BD)-type GDNN cone.
We focused our investigation on the inclusion relationship between the three GDNN cones over the direct product of a nonnegative orthant and second-order or semidefinite cones.
We found that the NN-type GDNN cone is included in the ZVP-type cone theoretically and the BD-type cone numerically.
Although there is no inclusion relationship between the ZVP- and BD-type GDNN cones theoretically, the result of solving GDNN programming relaxation problems of mixed 0--1 second-order cone programming shows that the ZVP-type GDNN cone yields a tighter bound than the BD-type cone in most cases.
Ultimately, the proposed GDNN cones exhibit theoretical and numerical superiority over the existing cone.



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