Multi-Group Multicast Beamforming by Superiorized Projections Onto Convex Sets

Abstract

In this paper, we propose an iterative algorithm to address the nonconvex multi-group multicast beamforming problem with quality-of-service constraints and per-antenna power constraints. We formulate a convex relaxation of the problem as a semidefinite program in a real Hilbert space, which allows us to approximate a point in the feasible set by iteratively applying a bounded perturbation resilient fixedpoint mapping. Inspired by the superiorization methodology, we use this mapping as a basic algorithm, and we add in each iteration a small perturbation with the intent to reduce the objective value and the distance to nonconvex rank-constraint sets. We prove that the sequence of perturbations is bounded, so the algorithm is guaranteed to converge to a feasible point of the relaxed semidefinite program. Simulations show that the proposed approach outperforms existing algorithms in terms of both computation time and approximation gap in many cases.

Superiorized Adaptive Projected Subgradient Method with Application to MIMO Detection

“…Nevertheless, we will consider nonconvex objective functions in the following. Moreover, as in [15], we slightly deviate from [8] and [7], by using proximal mappings instead of subgradients of the superiorization objective to define the perturbations. This allows for a simple trade-off between the perturbations' magnitude and their contribution to reducing the objective value.…”

“…The convergence of the superiorized APSM in ( 14) with perturbations according to (15) or ( 17) is investigated below.…”

“…Then the algorithm in (14) with (β n ) n∈N ∈ 1 + (N) and perturbations according to (15) or ( 17) is guaranteed to converge to a point x ∈ B minimizing all but finitely many functions of the sequence (Θ n ) n∈N .…”

“…The proposed iterative MIMO dectectors with perturbations according to (15) and (17), respectively, are summarized in Algorithm 1 below.…”

“…thereof follows automatically [7]. In recent years, bounded perturbation resilience has been proven for several classes of algorithms, including block-iterative projection methods [10], amalgamated projection methods [11], dynamic string averaging projection methods [12], [13], and fixed point iterations of averaged nonexpansive mappings [14] (which include POCS with relaxed projections [15]).…”