Generalization of Doubly Nonnegative Cone: Focusing on Inner-Approximation for Generalized Copositive Cone

We aim to provide better relaxation for generalized completely positive (copositive) programming. We first develop an inner-approximation hierarchy for the generalized copositive cone over a symmetric cone. Exploiting this hierarchy as well as the existing hierarchy proposed by Zuluaga et al. (SIAM J Optim 16(4):1076--1091, 2006), we then propose two (NN and ZVP) generalized doubly nonnegative (GDNN) cones. They are (if defined) always tractable, in contrast to the existing (BD) GDNN cone proposed by Burer and Dong (Oper Res Lett 40(3):203--206, 2012). We focus our investigation on the inclusion relationship between the three GDNN cones over a direct product of a nonnegative orthant and second-order cones or semidefinite cones. We find that the NN GDNN cone is included in the ZVP one theoretically and in the BD one numerically. Although there is no inclusion relationship between the ZVP and BD GDNN cones theoretically, the result of solving GDNN programming relaxation problems of mixed 0--1 second-order cone programming shows that the proposed GDNN cones provide a tighter bound than the existing one in most cases. To sum up, the proposed GDNN cones have theoretical and numerical superiority over the existing one.

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