Sub-optimal and robust downlink beamforming in wireless networks with imperfect channel state information

The rapid growth in internet applications such as video streaming enforce the researcher to explore a new wireless technique to ensure high signal to interference noise ratio (SINR) at the end users, leading to high quality of service (QOS). The fourth generation (4G) wireless technologies introduced a with promising technique known as multiple-input-multiple-output (MIMO) paradigm. The MIMO offers spatial diversity of multiple signals between the source and the destination which can ensure high concentration of the desired power at the destination as well as combat the unwanted interference which can be done by the beamforming technique, implemented in two ways the up-link and the down-link. Two methods of beamforming have been addressed in MIMO wireless communications, the first consider the minimization of transmitted power for predefined SINR at the receiver and the second approach consider maximization of SINR at the recipient while maintaining the power at the sender to a small fixed value. Rigid beamforming is assured when the accurate channel state information (CSI) of the wireless system are acquired at the beamforming side. Because of some practical limitations in wireless systems such as feedback error, dynamic characteristics of wireless channel, etc., the ideal CSI cannot be obtained and thus the beamforming must consider the error in CSI. Three type of solutions have been developed to combat the effect of uncertain CSI these solutions are the non-robust, the sub-optimal and the robust solution. In this work the sub-optimal and the robust downlink beamforming in conventional wireless network are addressed. The solution considers a multicast, multi-group, multicell scenario. The uncertainty in CSI is modeled mathematically using Frobinius norm and the beamforming method used is the QOS method where the minimum SINR over all groups is maximized for small predefined transmitted power. Because the problem is difficult to be solved as a single optimization problem, it is divided into two problems. The first problem eliminates the effect of CSI uncertainty using the non-monotone spectral projected gradient (NMSPG) method, and the second problems use the successive convex approximation (SCA) method to extract the beamforming vectors for each group. The procedure goes through an iterate-alternative convex technique between the two methods until stopped by some predefined criteria. Wireless communication researchers have also achieved significant development in the area of spectrum scarcity by introducing the cognitive radio (CR) network. In a CR network the secondary users (SUs) can utilize the licensed frequency that is underutilized by the primary users (PUs). Two type of CR network were developed, the overlay and the underlay CR network. The beamforming in an overlay CR network follows the same procedure as in a conventional network while in an underlay CR network an extra constraint must be added to the beamforming problem which makes the problem more difficult to solved. In this thesis the beamforming problem in a CR network with multiple secondary transmitters that generate multiple beamforming vectors to multiple groups of secondary receivers under uncertain CSI are analyzed and solved. Two solutions were developed: the sub-optimal and the robust solutions. For the sub-optimal solution, the problem is split into two problems, the QOS and the interference power problem to combat the effect of CSI uncertainty then the two problems are combined to find the beamforming vectors using the SCA method. For the robust solution, the problem is also divided into two problems, the QOS and the interference power problem to eliminate the CSI uncertainty. The interference power problem is solved using the Lagrangian duality. The QOS problem is solved using the Lagrangian duality and the NMSPG method. after addressing CSI uncertainty, the beamforming vectors are extracted using the SCA method and, solved using the bisection search method.

Quadratic problems with two quadratic constraints: convex quadratic relaxation and strong Lagrangian duality

In this paper, we study a nonconvex quadratic minimization problem with two quadratic constraints, one of which being convex. We introduce two convex quadratic relaxations (CQRs) and discuss cases, where the problem is equivalent to exactly one of the CQRs. Particularly, we show that the global optimal solution can be recovered from an optimal solution of the CQRs. Through this equivalence, we introduce new conditions under which the problem enjoys strong Lagrangian duality, generalizing the recent condition in the literature. Finally, under the new conditions, we present necessary and sufficient conditions for global optimality of the problem.

Distributionally Robust XVA via Wasserstein Distance: Wrong Way Counterparty Credit and Funding Risk

This paper investigates calculations of robust X-Value adjustment (XVA), in particular, credit valuation adjustment (CVA) and funding valuation adjustment (FVA), for over-the-counter derivatives under distributional ambiguity using Wasserstein distance as the ambiguity measure. Wrong way counterparty credit risk and funding risk can be characterized (and indeed quantified) via the robust XVA formulations. The simpler dual formulations are derived using recent Lagrangian duality results. Next, some computational experiments are conducted to measure the additional XVA charges due to distributional ambiguity under a variety of portfolio and market configurations. Finally some suggestions for further work are discussed.