Generalized Quadratic Matrix Programming: A Unified Framework for Linear Precoding With Arbitrary Input Distributions

This paper investigates a new class of nonconvex optimization, which provides a unified framework for linear precoding in single/multiuser multiple-input multiple-output channels with arbitrary input distributions. The new optimization is called generalized quadratic matrix programming (GQMP). Due to the nondeterministic polynomial time hardness of GQMP problems, instead of seeking globally optimal solutions, we propose an efficient algorithm that is guaranteed to converge to a Karush-Kuhn-Tucker point. The idea behind this algorithm is to construct explicit concave lower bounds for nonconvex objective and constraint functions, and then solve a sequence of concave maximization problems until convergence. In terms of application, we consider a downlink underlay secure cognitive radio network, where each node has multiple antennas. We design linear precoders to maximize the average secrecy (sum) rate with finite-alphabet inputs and statistical channel state information at the transmitter. The precoding problems under secure multicast/broadcast scenarios are GQMP problems, and thus, they can be solved efficiently by our proposed algorithm. Several numerical examples are provided to show the efficacy of our algorithm.