A General Matrix Variable Optimization Framework for MIMO Assisted Wireless Communications

Complex matrix derivatives play an important role in matrix optimization, since they form a theoretical basis for the Karush-Kuhn-Tucker (KKT) conditions associated with matrix variables. We commence with a comprehensive discussion of complex matrix derivatives. First, some fundamental conclusions are presented for deriving the optimal structures of matrix variables from complex matrix derivatives. Then, some restrictions are imposed on complex matrix derivatives for ensuring that the resultant first order equations in the KKT conditions exploit symmetric properties. Accordingly, a specific family of symmetric matrix equations is proposed and their properties are unveiled. Using these symmetric matrix equations, the optimal structures of matrix variables are directly available, and thereby the original optimization problems can be significantly simplified. In addition, we take into account the positive semidefinite constraints imposed on matrix variables. In order to accommodate the positive semidefinitness of matrix variables, we introduce a matrix transformation technique by leveraging the symmetric matrix equations, which can dramatically simplify the KKT conditions based analysis albeit at the expense of destroying convexity. Moreover, this matrix transformation technique is valuable in practice, since it offers a more efficient means of computing the optimal solution based on the optimal structures derived directly from the KKT conditions.