Strengthened existence and uniqueness conditions for search directions in semidefinite programming

In this paper, we revisit and strengthen both of the above mentioned sufficient conditions. We include characterizations for existence and uniqueness in the matrix equations arising from the linearization of the optimality conditions. As well, we present new results on the relationship between the Kronecker product and the symmetric Kronecker product that arise from these matrix equations. We conclude with a proof that the existence and uniqueness of the AHO direction is a generic property for every SDP problem and extend the results to the general Monteiro-Zhang family of search directions.