Quadratically constrained quadratic programs (QC-QPs) have a wide range of applications in signal processing and wireless communications. Non-convex QCQPs are NP-hard in general. Existing approaches relax the non-convexity using semidefinite relaxation (SDR) or linearize the non-convex part and solve the resulting convex problem. However, these techniques are seldom successful in even obtaining a feasible solution when the QCQP matrices are indefinite. In this paper, a new feasible point pursuit successive convex approximation (FPP-SCA) algorithm is proposed for non-convex QCQPs. FPP-SCA linearizes the non-convex parts of the problem as conventional SCA does, but adds slack variables to sustain feasibility, and a penalty to ensure slacks are sparingly used. When FPP-SCA is successful in identifying a feasible point of the non-convex QCQP, convergence to a Karush-Kuhn-Tucker (KKT) point is thereafter ensured. Simulations show the effectiveness of our proposed algorithm in obtaining feasible and near-optimal solutions, significantly outperforming existing approaches. Index Terms-Non-convex QCQP, feasible point pursuit, successive convex approximation, semi-definite relaxation, linearization, multicast beamforming.

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This paper considers the (NP-)hard problem of joint multicast beamforming and antenna selection. Prior work has focused on using Semi-Definite relaxation (SDR) techniques in an attempt to obtain a high quality sub-optimal solution. However, SDR suffers from the drawback of having high computational complexity, as SDR lifts the problem to higher dimensional space, effectively squaring the number of variables. This paper proposes a high performance, low complexity Successive Convex Approximation (SCA) algorithm for max-min SNR "fair" joint multicast beamforming and antenna selection under a sum power constraint. The proposed approach relies on iteratively approximating the non-convex objective with a series of non-smooth convex subproblems, and then, a first order-based method called Saddle Point Mirror-Prox (SP-MP) is used to compute optimal solutions for each SCA subproblem. Simulations reveal that the SP-MP SCA algorithm provides a higher quality and lower complexity solution compared to the one obtained using SDR.

Massive MIMO is currently a leading physical layer technology candidate that can dramatically enhance throughput in 5G systems, for both unicast and multicast transmission modalities. As antenna elements are becoming smaller and cheaper in the mmW range compared to radio frequency (RF) chains, it is crucial to perform antenna selection at the transmitter, such that the available RF chains are switched to an appropriate subset of antennas. This paper considers the joint problem of multicast beamforming and antenna selection for a single multicast group in massive MIMO systems. The prior state-of-art for this problem relies on semi-definite relaxation (SDR), which cannot scale up to the massive MIMO regime. A successive convex approximation (SCA) based approach is proposed to tackle max-min fair joint multicast beamforming and antenna selection. The key idea of SCA is to successively approximate the non-convex problem by a class of non-smooth, convex optimization problems. Two fast and memory efficient first-order methods are proposed to solve each SCA subproblem. Simulations demonstrate that the proposed algorithms outperform the existing state-of-art approach in terms of solution quality and run time, in both traditional and especially in massive MIMO settings.