Nonsmooth Optimization for Efficient Beamforming in Cognitive Radio Multicast Transmission

It is known that the design of optimal transmit beamforming vectors for cognitive radio multicast transmission can be formulated as indefinite quadratic optimization programs. Given the challenges of such nonconvex problems, the conventional approach in literature is to recast them as convex semidefinite programs (SDPs) together with rank-one constraints. Then, these nonconvex and discontinuous constraints are dropped allowing for the realization of a pool of relaxed candidate solutions, from which various randomization techniques are utilized with the hope to recover the optimal solutions. However, it has been shown that such approach fails to deliver satisfactory outcomes in many practical settings, wherein the determined solutions are found to be unacceptably far from the actual optimality. On the contrary, we in this contribution tackle the aforementioned optimal beamforming problems differently by representing them as SDPs with additional reverse convex (but continuous) constraints. Non-smooth optimization algorithms are then proposed to locate the optimal solutions of such design problems in an efficient manner. Our thorough numerical examples verify that the proposed algorithms offer almost global optimality whilst requiring relatively low computational load.