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In this paper, we study subspace properties of the quadratically constrained quadratic program (QCQP). We prove that, if an appropriate subspace is chosen to satisfy subspace properties, then the solution of the QCQP lies in that subspace. So, we can solve the QCQP in that subspace rather than solve it in the original space. The computational cost could be reduced significantly if the dimension of the subspace is much smaller. We also show how to construct such subspaces and investigate their dimensions.

A symmetric matrix $A$ is completely positive (CP) if there exists an entrywise nonnegative matrix $V$ such that $A = VV ^T$. A real symmetric matrix is called completely positive separable (CPS) if it can be written as a sum of rank-1 Kronecker products of completely positive matrices. This paper studies the CPS problem. A criterion is given to determine whether a given matrix is CPS, and a specific CPS decomposition is constructed if the matrix is CPS.

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