A Conic Quadratic Programming Approach to Physical Layer Multicasting for Large-Scale Antenna Arrays

We investigate the problem of downlink physical layer multicasting that aims at minimizing the transmit power with a massive antenna array installed at the transmitter site. We take a solution based on semidefinite relaxation (SDR) as our benchmark. It is shown that instead of working on the semidefinite program (SDP) naturally produced by the SDR, the dual counterpart of the same problem may provide a more efficient numerical implementation. Later, by using a successive convex approximation strategy, we arrive at a provably convergent iterative second-order cone programming (SOCP) solution. Our thorough numerical investigations report that the newly proposed SOCP solution offers improved power efficiency and a massively reduced computational complexity. Therefore, the SOCP solution is seen as a suitable candidate for obtaining beamformers that minimize transmit power, especially, when a very large number of antennas is used at the transmitter.

I. Introduction

The main benefits offered by the so called large-scale antenna systems include, but are not limited to, better spectral efficiency, enhanced link reliability and improved energy efficiency. For an overview of this topic, the interested reader is referred to [1]. The joint problem of multicasting the same message to a group of active users while also consuming minimum power with the aid of a massive array of antenna elements is intriguing at least from a couple of perspectives. Firstly, from the point of view of studying approximation techniques for this NP-hard problem. Secondly, due to the need of devising computationally efficient procedures to overcome the peculiar challenges that arise due to a very large number of transmit antenna elements. In their pioneering work, Sidiropoulos et al. [2], have studied the problem of the physical layer multicasting with a transmitter using a linear precoder. In addition to proving the NP-hardness of this problem, they have also devised an approximating scheme that relies on the semidefinite relaxation (SDR) and uses randomization techniques to recover the beamforming vector. The SDR approach has been extensively used in numerous problems under varying settings. The survey paper [3] provides a more complete overview of the relaxation technique and its signal processing applications.