Sum-Rate Maximization via Convex Optimization Using Subgradient Projections Onto Nonlinear Spectral Radius Constraint Sets

TL;DR

This work solves the (weighted) sum-rate maximization problem over the set of achievable rates characterized by a nonlinear spectral radius function by exploiting subgradient projections onto the level sets of suitably reformulated spectral radius functions.

Abstract

We solve the (weighted) sum-rate maximization problem over the set of achievable rates characterized by a nonlinear spectral radius function. This set has been recently shown to be convex in some practically relevant settings in modern wireless networks, including cell-less networks. However, even under convexity, sum-rate maximization is challenging because the nonlinear spectral radius characterization of the achievable rate region is difficult to handle directly. We overcome this difficulty by exploiting subgradient projections onto the level sets of suitably reformulated spectral radius functions. Notably, the derived subgradient projection algorithm provably converges to the global optimum of the sum-rate maximization problem under the convexity condition. The efficacy of the proposed algorithm is illustrated in simulations for cell-less networks.

Figures (2)

• Figure 1: Behavior of algorithms for a simulation where $\bm{M}_l$ is an inverse Z-matrix for every $l \in \{1,\ldots,L\}$.
• Figure 2: Behavior of algorithms for a simulation where $\bm{M}_l$ is not an inverse Z-matrix for some $l \in \{1,\ldots,L\}$.

Theorems & Definitions (34)

• Definition 2.1
• Definition 2.2
• Definition 2.3
• Remark 2.1
• Remark 2.2
• Example 2.1
• Lemma 3.1
• proof
• Lemma 3.2
• Lemma 3.3