Grab-n-Pull: An optimization framework for fairness-achieving networks

Abstract

In this paper, we present an optimization framework for designing precoding (a.k.a. beamforming) signals that are instrumental in achieving a fair user performance through the networks. The precoding design problem in such scenarios can typically be formulated as a non-convex max-min fractional quadratic program. Using a penalized version of the original design problem, we derive a simplified quadratic reformulation of the problem in terms of the signal (to be designed). Each iteration of the proposed design framework consists of a combination of power method-like iterations and the Gram-Schmidt process, and as a result, enjoys a low computational cost. Moreover, the suggested approach can handle various types of signal constraints such as total-power, per-antenna power, unimodularity, or discrete-phase requirements-an advantage which is not shared by other existing approaches in the literature.

Max-min transmit beamforming via iterative regularization

“…To minimize w H Rw, we can equivalently maximize w H (−R)w. For total power constraint, this optimization problem can be solved efficiently using power-method [11]. The unimodular signal design can be handled by using powermethod like iterations introduced in [12].…”

“…It can be shown that the power constraint in (3) will be satisfied with equality, so we have w 2 = P . Following [11], the SINR expression can be written as…”

“…The unimodular signal design can be handled by using powermethod like iterations introduced in [12]. For more details on handling optimization w.r.t w please refer to [11].…”

“…where Ω is the feasible set of the problem governed by the constraint on w. Particularly, the total power constraint and unimodular signal design, Ω is defined respectively as [11], [12],…”

“…The proposed method is based on iterative paradigm involving the use of regularization based optimization approach. While the proposed framework can efficiently handle several constraints including total power and unimodularity constraints, it also improves on the complexity of related iterative Grab and Pull approach [11].…”