Optimization for beamforming problems in wireless networks

Beamforming in wireless communications have gathered great interests in recent years due to its ability to enhanced the performance of networks significantly by exploiting intensively the spatial diversity. In this work, the objectives of beamforming design consist of several optimization targets ranging from minimizing the beamforming power subject to quality-of-service (QoS) constraints to maximizing the minimum QoS regarding fixed budgets of transmitting power. The design problems of beamforming are intrinsically challenging because their natural formulations are nonconvex optimization problems. Moreover several problems are proved to be non-deterministic polynomial-time hard (NP-hard) such as beamforming in multicast transmission. These problems are very difficult to solve at optimality in practical sense. Therefore, there is a strong motivation to convert the original design problems into a series of convex problems with desirable computational complexity by applying efficient optimization techniques. This dissertation contributes mainly in exploiting the convex optimization algorithms to solve nonconvex beamforming problems in several network settings. First, the transmit beamforming for downlink communication of multicast transmission with spectrum sharing is investigated. Secondly, the beamforming design is applied on amplify-forward (AF) wireless relaying systems using single-antenna relays. The key contribution is to derive the beamforming schemes applied on transmit antennas so that the beamforming power is minimized while all users' signal-to-interference-and-noise ratios (SINRs) are guaranteed. The formulation results in nonconvex optimization problems due to SINR constraints hence require to be converted into semidefinite programming (SDP) forms. The SDP problems are again nonconvex regarding the rank-one constraints on semidefinite variables. Conventionally, the rank-one constraints are relaxed hence the problems cannot be solved thoroughly. In this work, nonsmooth optimization techniques are employed to tackle with the nonconvex rank-one constraints and are successfully to deliver efficient solutions that can outperform the conventional methods. Finally the precoding design problems in mutiple-input multiple-output (MIMO) relaying scenarios are considered. The difference-of-two-convex-function (D.C.) programming technique is employed to solve the problems at optimality with significantly lower complexity compared with conventional method

Low Complexity Damped Gauss-Newton Algorithms for CANDECOMP/PARAFAC

The damped Gauss-Newton (dGN) algorithm for CANDECOMP/PARAFAC (CP) decomposition can handle the challenges of collinearity of factors and different magnitudes of factors; nevertheless, for factorization of an $N$-D tensor of size $I_1\times I_N$ with rank $R$, the algorithm is computationally demanding due to construction of large approximate Hessian of size $(RT \times RT)$ and its inversion where $T = \sum_n I_n$. In this paper, we propose a fast implementation of the dGN algorithm which is based on novel expressions of the inverse approximate Hessian in block form. The new implementation has lower computational complexity, besides computation of the gradient (this part is common to both methods), requiring the inversion of a matrix of size $NR^2\times NR^2$, which is much smaller than the whole approximate Hessian, if $T \gg NR$. In addition, the implementation has lower memory requirements, because neither the Hessian nor its inverse never need to be stored in their entirety. A variant of the algorithm working with complex valued data is proposed as well. Complexity and performance of the proposed algorithm is compared with those of dGN and ALS with line search on examples of difficult benchmark tensors