Proportional Fairness Beamforming Weight Design

• PFBWD is a family of wireless optimization formulations that blend proportional fairness with beam-domain control variables over diverse network architectures.
• It spans designs such as hierarchical beam scheduling, heterogeneous-rank subband beamforming, energy-efficient MISO NOMA, robust MISOME-SWIPT, and distributed HAPS-vHetNet methods.
• Research offers strategies from closed-form solutions and dynamic programming to SCA, ADMM, and SDR, highlighting open challenges in joint beamweight and resource-allocation design.

Searching arXiv for the cited papers and closely related PFBWD sources. Proportional Fairness Beamforming Weight Design (PFBWD) denotes a family of wireless optimization formulations in which proportional-fairness criteria are coupled to beam-domain control variables such as beam activation fractions, hybrid analog/digital beamformers, linear precoders, artificial-noise covariances, or distributed beamforming weights. The term does not refer to a single standardized problem. In the literature considered here, it spans hierarchical beam scheduling on a fixed beam tree, heterogeneous-rank hybrid beamforming across OFDMA subbands, energy-efficiency-fair beamforming for MISO NOMA, robust secrecy-energy-efficient beamforming for MISOME-SWIPT, and distributed proportional-fair beamforming for large-scale HAPS-empowered vertical heterogeneous networks (Floquet et al., 2018, Bedin et al., 2023, Al-Obiedollah et al., 2019, Dong et al., 2018, Shamsabadi et al., 11 Jul 2025).

1. Conceptual scope and formal variants

Across the cited formulations, proportional fairness appears through different utility layers. In hierarchical beamforming, proportional fairness is the $\alpha=1$ specialization of the $\alpha$-fair utility

$f_\alpha(r)= \begin{cases} \dfrac{r^{1-\alpha}}{1-\alpha}, & \alpha\neq 1,\[4pt] \ln r, & \alpha=1, \end{cases}$

applied to effective per-flow rates $R_k=r_k\gamma_{v_k}\delta_k$ on a predefined beam tree (Floquet et al., 2018). In heterogeneous-rank industrial beamforming, proportional fairness is operationalized through the per-RB metric $W=C_r/C_u$ with rate averaging $C_u\leftarrow \gamma C_u+(1-\gamma)\bar C_u$, and in joint scheduling-and-precoding form through weights $w_k=1/\bar R_k(t)$ in a weighted sum-rate maximization (Bedin et al., 2023). In MISO NOMA, the PF objective is the sum of logarithms of per-user energy efficiencies, $\max \sum_i \log(EE_i)$ (Al-Obiedollah et al., 2019). In MISOME-SWIPT, proportional fairness is imposed by secrecy-rate proportions $R_{s,n}\ge \alpha_n\tau$ with $\sum_n \alpha_n=1$, so the secrecy energy efficiency becomes $\alpha$0 at optimality (Dong et al., 2018). In HAPS-empowered vHetNets, the PF objective is cast as maximizing $\alpha$1 with slack variables satisfying $\alpha$2 (Shamsabadi et al., 11 Jul 2025).

Setting	Optimized variables	PF instantiation
Hierarchical beamforming	$\alpha$3 on fixed beams	$\alpha$4
Heterogeneous-rank beamforming	$\alpha$5	Per-RB $\alpha$6; weighted sum-rate
MISO NOMA	$\alpha$7	$\alpha$8
MISOME-SWIPT	$\alpha$9	$f_\alpha(r)= \begin{cases} \dfrac{r^{1-\alpha}}{1-\alpha}, & \alpha\neq 1,\[4pt] \ln r, & \alpha=1, \end{cases}$0
HAPS vHetNets	$f_\alpha(r)= \begin{cases} \dfrac{r^{1-\alpha}}{1-\alpha}, & \alpha\neq 1,\[4pt] \ln r, & \alpha=1, \end{cases}$1 and consensus variables	$f_\alpha(r)= \begin{cases} \dfrac{r^{1-\alpha}}{1-\alpha}, & \alpha\neq 1,\[4pt] \ln r, & \alpha=1, \end{cases}$2

This variation suggests that PFBWD is best understood as a class of proportional-fair beam management problems rather than a unique canonical formulation. What is held fixed in one paper, such as the beam codebook in hierarchical beamforming, becomes a design variable in another.

2. Hierarchical beamforming: proportional fairness without beam-vector optimization

In "Hierarchical Beamforming: Resource Allocation, Fairness and Flow Level Performance" the network is modeled by a hierarchical beam tree $f_\alpha(r)= \begin{cases} \dfrac{r^{1-\alpha}}{1-\alpha}, & \alpha\neq 1,\[4pt] \ln r, & \alpha=1, \end{cases}$3 in which each beam $f_\alpha(r)= \begin{cases} \dfrac{r^{1-\alpha}}{1-\alpha}, & \alpha\neq 1,\[4pt] \ln r, & \alpha=1, \end{cases}$4 covers a spatial region $f_\alpha(r)= \begin{cases} \dfrac{r^{1-\alpha}}{1-\alpha}, & \alpha\neq 1,\[4pt] \ln r, & \alpha=1, \end{cases}$5, parent-child edges encode nested coverage, siblings have disjoint coverage, and deeper beams have larger received power on their descendants’ regions. Flow $f_\alpha(r)= \begin{cases} \dfrac{r^{1-\alpha}}{1-\alpha}, & \alpha\neq 1,\[4pt] \ln r, & \alpha=1, \end{cases}$6 at location $f_\alpha(r)= \begin{cases} \dfrac{r^{1-\alpha}}{1-\alpha}, & \alpha\neq 1,\[4pt] \ln r, & \alpha=1, \end{cases}$7 is associated with the deepest beam covering that location, equivalently $f_\alpha(r)= \begin{cases} \dfrac{r^{1-\alpha}}{1-\alpha}, & \alpha\neq 1,\[4pt] \ln r, & \alpha=1, \end{cases}$8. A descent algorithm starting at the root and moving to a covering child finds $f_\alpha(r)= \begin{cases} \dfrac{r^{1-\alpha}}{1-\alpha}, & \alpha\neq 1,\[4pt] \ln r, & \alpha=1, \end{cases}$9 in $R_k=r_k\gamma_{v_k}\delta_k$0 time, which becomes $R_k=r_k\gamma_{v_k}\delta_k$1 for regular trees (Floquet et al., 2018).

The resource-allocation variables are beam activation fractions $R_k=r_k\gamma_{v_k}\delta_k$2 and within-beam sharing fractions $R_k=r_k\gamma_{v_k}\delta_k$3. Admissible activation satisfies the ancestor constraint

$R_k=r_k\gamma_{v_k}\delta_k$4

equivalently

$R_k=r_k\gamma_{v_k}\delta_k$5

Per-flow effective rate is

$R_k=r_k\gamma_{v_k}\delta_k$6

For proportional fairness, the paper gives a closed form: the optimal within-beam sharing is equal sharing,

$R_k=r_k\gamma_{v_k}\delta_k$7

where $R_k=r_k\gamma_{v_k}\delta_k$8, and the optimal conditional activation is

$R_k=r_k\gamma_{v_k}\delta_k$9

The resulting activation fractions follow from $W=C_r/C_u$0. More generally, the $W=C_r/C_u$1-fair allocator is computed in $W=C_r/C_u$2 through a non-iterative dynamic program with an ascending recursion for subtree values $W=C_r/C_u$3 and a descending pass for $W=C_r/C_u$4 (Floquet et al., 2018).

At flow level, the same paper derives closed-form PF performance for elastic traffic under a time-scale separation assumption. With Poisson arrivals $W=C_r/C_u$5, beam-level rates $W=C_r/C_u$6, and loads $W=C_r/C_u$7, stability holds iff

$W=C_r/C_u$8

The Markov process is reversible, its stationary distribution is explicit, and the results are insensitive to the flow-size distribution beyond the mean. Mean throughput on beam $W=C_r/C_u$9 is

$C_u\leftarrow \gamma C_u+(1-\gamma)\bar C_u$0

The paper also gives a practical randomized scheduler by sampling independent Bernoulli variables with means $C_u\leftarrow \gamma C_u+(1-\gamma)\bar C_u$1 and activating $C_u\leftarrow \gamma C_u+(1-\gamma)\bar C_u$2.

A central interpretive point is explicit in the paper: this formulation does not perform beamforming weight-vector design. The beam/codebook is predefined, coverage regions $C_u\leftarrow \gamma C_u+(1-\gamma)\bar C_u$3 are fixed, and there is no optimization of power allocation, weight orthogonality, or beam shapes. What is optimized is the proportional-fair allocation layer $C_u\leftarrow \gamma C_u+(1-\gamma)\bar C_u$4 over a fixed hierarchical beam structure. The paper therefore provides a resource-allocation foundation for PFBWD rather than a complete beamweight-design solution.

3. Heterogeneous-rank subband beamforming in industrial communications

"Heterogeneous Rank Beamforming for Industrial Communications" formulates a heterogeneous-rank architecture in which a UE with $C_u\leftarrow \gamma C_u+(1-\gamma)\bar C_u$5 antennas has one analog RF chain over the full bandwidth $C_u\leftarrow \gamma C_u+(1-\gamma)\bar C_u$6 and one or more tradable RF chains that can be repurposed for narrowband fully digital beamforming. If one ADC digitizes multiplexed per-antenna baseband branches, the effective digital beamforming bandwidth is

$C_u\leftarrow \gamma C_u+(1-\gamma)\bar C_u$7

while the analog chain remains operational on $C_u\leftarrow \gamma C_u+(1-\gamma)\bar C_u$8. More generally, the bandwidth is partitioned into subbands $C_u\leftarrow \gamma C_u+(1-\gamma)\bar C_u$9, each with effective rank $w_k=1/\bar R_k(t)$0, producing a heterogeneous-rank frequency map. The hybrid beamforming structure on subband $w_k=1/\bar R_k(t)$1 is

$w_k=1/\bar R_k(t)$2

with constant-modulus analog matrix entries $w_k=1/\bar R_k(t)$3 (Bedin et al., 2023).

The paper’s PF scheduler uses the per-RB metric

$w_k=1/\bar R_k(t)$4

where $w_k=1/\bar R_k(t)$5 is the spectral efficiency on RB $w_k=1/\bar R_k(t)$6 and $w_k=1/\bar R_k(t)$7 is the user’s running average rate updated as

$w_k=1/\bar R_k(t)$8

A user is eligible if its buffer is non-empty, and the RB is assigned to the user with the largest $w_k=1/\bar R_k(t)$9 subject to constraints. In the joint scheduling-and-precoding abstraction, with PF weights $\max \sum_i \log(EE_i)$0, the instantaneous problem is

$\max \sum_i \log(EE_i)$1

subject to $\max \sum_i \log(EE_i)$2, $\max \sum_i \log(EE_i)$3 or $\max \sum_i \log(EE_i)$4, total or per-subband power constraints, and the hybrid constraint $\max \sum_i \log(EE_i)$5. The per-subband SINR is

$\max \sum_i \log(EE_i)$6

The paper characterizes this problem as nonconvex because of log-SINR coupling, hybrid constraints, and combinatorial scheduling variables. It then describes a practical PF-BWD decomposition based on alternating optimization: PF-based user selection per subband, digital precoder design using weighted ZF, MMSE, or WMMSE, PF-aware power allocation, and analog precoder updates aligned with selected users’ subspaces. The inferred regularized-ZF form reported in the detailed exposition is

$\max \sum_i \log(EE_i)$7

with $\max \sum_i \log(EE_i)$8.

A distinctive analytical contribution is the set of spectral-efficiency bounds based on the analog, hybrid, and digital spectral efficiencies $\max \sum_i \log(EE_i)$9. If

$R_{s,n}\ge \alpha_n\tau$0

then the heterogeneous-mode upper bound is

$R_{s,n}\ge \alpha_n\tau$1

and heterogeneous mode outperforms classical hybrid when

$R_{s,n}\ge \alpha_n\tau$2

The overall piecewise maximum achievable average spectral efficiency is

$R_{s,n}\ge \alpha_n\tau$3

In the reported OFDMA setup with $R_{s,n}\ge \alpha_n\tau$4 RBs, $R_{s,n}\ge \alpha_n\tau$5 antennas per UE, $R_{s,n}\ge \alpha_n\tau$6 b/s/Hz, $R_{s,n}\ge \alpha_n\tau$7 b/s/Hz, and $R_{s,n}\ge \alpha_n\tau$8 b/s/Hz, the threshold becomes $R_{s,n}\ge \alpha_n\tau$9 users. The numerical results show that PF on heterogeneous mode closely tracks the upper bound beyond that threshold, that analog allocations under PF occur only when no digital-capable users have data, and that heterogeneous mode can save substantial RB resources at low traffic loads (Bedin et al., 2023).

4. Energy-efficiency proportional fairness in downlink MISO NOMA

In "Energy Efficiency Fairness Beamforming Designs for MISO NOMA Systems," PFBWD is formulated for a downlink MISO NOMA system with a BS with $\sum_n \alpha_n=1$0 antennas and $\sum_n \alpha_n=1$1 single-antenna users. The BS transmits

$\sum_n \alpha_n=1$2

and user $\sum_n \alpha_n=1$3 receives

$\sum_n \alpha_n=1$4

Users are ordered by channel strength, $\sum_n \alpha_n=1$5, and stronger users perform SIC on weaker users’ signals. For decoding user $\sum_n \alpha_n=1$6 at user $\sum_n \alpha_n=1$7,

$\sum_n \alpha_n=1$8

and the effective SINR for user $\sum_n \alpha_n=1$9 is

$\alpha$00

The achievable rate is $\alpha$01, subject to QoS constraints $\alpha$02 and SIC power-ordering constraints $\alpha$03 for all $\alpha$04 (Al-Obiedollah et al., 2019).

The paper defines per-user energy efficiency as

$\alpha$05

with total transmit power constraint $\alpha$06. The proportional-fair design, denoted OP$\alpha$07, is

$\alpha$08

subject to the SIC ordering, QoS, and power constraints. The companion MMEE design, OP$\alpha$09, maximizes $\alpha$10 under the same constraints. The PF objective therefore targets a balance between global energy efficiency and fairness between users in terms of achieved EE, whereas MMEE enforces strict worst-user equalization.

Because OP$\alpha$11 is nonconvex, the paper solves it through sequential convex approximation. The SINR/QoS constraints are converted to SOC form after phase alignment:

$\alpha$12

where $\alpha$13. The SIC received-power ordering is handled through linear minorization, and the fractional PF objective is reformulated with auxiliary variables $\alpha$14 so that each SCA iteration solves an SOCP. Feasible initialization is obtained from a power-minimization problem under QoS and SIC constraints, and the algorithm stops when the relative objective change falls below $\alpha$15 (Al-Obiedollah et al., 2019).

In the reported simulation setup with $\alpha$16, $\alpha$17, distances $\alpha$18 m, $\alpha$19, Rayleigh fading, $\alpha$20, $\alpha$21, $\alpha$22, and $\alpha$23 MHz, the PF design lies between GEE maximization and MMEE. At TX-SNR $\alpha$24 dB, MMEE gives approximately $\alpha$25 bits/J for the weakest user, about $\alpha$26 higher than GEE-max, while PF also significantly outperforms GEE-max and preserves more global EE than MMEE. The paper therefore positions PF-based beamweight design as an intermediate operating point between system-level efficiency and strict fairness.

5. Robust secrecy-energy-efficient PFBWD in MISOME-SWIPT

"Robust Secrecy Energy Efficient Beamforming in MISOME-SWIPT Systems With Proportional Fairness" studies a BST with $\alpha$27 antennas, single-antenna legitimate users, single-antenna eavesdroppers, and single-antenna energy-harvesting nodes. The transmitted signal is

$\alpha$28

where $\alpha$29 is artificial noise, $\alpha$30, and $\alpha$31. Legitimate CSI is perfect, whereas EVE and EHN channels are norm-bounded uncertain. The legitimate-user SINR is

$\alpha$32

and secrecy rates are controlled by robust leakage caps on the EVEs. Energy harvesting obeys the linear model

$\alpha$33

with minimum requirements $\alpha$34 (Dong et al., 2018).

Proportional fairness is imposed through weights $\alpha$35 satisfying $\alpha$36 and a common scaling variable $\alpha$37 such that

$\alpha$38

Since $\alpha$39, the fairness constraints are equivalent, for fixed $\alpha$40, to legitimate-user SINR thresholds

$\alpha$41

where

$\alpha$42

The secrecy energy efficiency is

$\alpha$43

and because $\alpha$44, at optimality $\alpha$45, yielding $\alpha$46.

Robustness to uncertain EVE and EHN CSI is enforced with S-procedure LMIs. For EVEs, with

$\alpha$47

the worst-case leakage constraint is transformed into an LMI involving slack $\alpha$48. For EHNs, the robust harvested-energy constraint is similarly turned into an LMI with slack $\alpha$49. The original SEE maximization is then handled by semidefinite relaxation and a two-stage procedure: an inner SDP minimizing radiated power for fixed $\alpha$50, and an outer one-dimensional search over feasible $\alpha$51. The exposition states that SDR is tight: for any feasible $\alpha$52, there exists an optimal solution with $\alpha$53 for all $\alpha$54, so beamforming vectors can be recovered from the principal eigenvectors (Dong et al., 2018).

The beamforming interpretation is structurally different from the NOMA case. Here, larger $\alpha$55 tightens $\alpha$56 and therefore pulls more power and alignment toward user $\alpha$57’s legitimate channel, while $\alpha$58 serves the dual role of raising the EVE denominator and contributing to energy harvesting. The paper reports that SEE is unimodal with respect to the allowed information leakage rate, that there is an optimal leakage cap around $\alpha$59 Knats/s for SDR-based designs in the reported setting, and that robustness to uncertainty incurs a moderate SEE loss relative to perfect CSI while preventing secrecy and EH outages.

6. Distributed PFBWD for large-scale HAPS-empowered vHetNets

"Two-Level Distributed Interference Management for Large-Scale HAPS-Empowered vHetNets" considers a harmonized-spectrum, cell-free architecture with $\alpha$60 terrestrial macro BSs and one HAPS, indexed as $\alpha$61, jointly serving all UEs. BS $\alpha$62 transmits

$\alpha$63

and UE $\alpha$64 receives

$\alpha$65

The SINR is

$\alpha$66

with per-BS power constraint $\alpha$67 (Shamsabadi et al., 11 Jul 2025).

The PF objective is introduced by slack variables $\alpha$68 and

$\alpha$69

subject also to $\alpha$70. The optimization is nonconvex because the beamformed SINRs globally couple all BS and UE variables. To handle this at scale, the paper proposes a two-level distributed PFBWD algorithm combining an outer augmented Lagrangian method and an inner three-block ADMM.

The distributed reformulation introduces global per-UE variables $\alpha$71 and local per-BS variables $\alpha$72. Here $\alpha$73, $\alpha$74 upper-bounds interference-plus-noise, $\alpha$75 approximates $\alpha$76, and $\alpha$77 approximates $\alpha$78. Consensus equalities $\alpha$79 and $\alpha$80 are relaxed with auxiliary variables $\alpha$81, which are driven to zero by the outer ALM. The inner three-block ADMM alternates among a global block, a local per-BS convex block solved in parallel, and an auxiliary block. The block-1 nonconvex constraint $\alpha$82 is treated by SCA, and the constraint $\alpha$83 is enforced through an SOCP-representable approximation to the exponential cone (Shamsabadi et al., 11 Jul 2025).

The paper gives convergence conditions and states that the proposed method has guaranteed convergence: the inner three-block ADMM converges to a point satisfying first-order optimality conditions for the inner problem, while the outer ALM iterates have a limit point where either consensus is achieved or a stationary point of the least-squares consensus problem is obtained; in practice, the outer loop rapidly drives the auxiliary variables to zero. Complexity is dominated by local SOCPs. The distributed per-iteration cost scales as

$\alpha$84

whereas the centralized baseline scales as

$\alpha$85

Message passing per ADMM iteration involves only consensus vectors of length approximately $\alpha$86 reals per BS.

In the reported urban setup with four MBSs, one HAPS, $\alpha$87 terrestrial arrays, an $\alpha$88 HAPS array, $\alpha$89 GHz, $\alpha$90 dBm, per-BS powers $\alpha$91 dBm for MBSs and $\alpha$92 dBm for the HAPS, and $\alpha$93 independent realizations, the outer PF objective saturates in about $\alpha$94 outer iterations. For $\alpha$95, the distributed method requires about $\alpha$96 real scalars per BS per iteration, compared with about $\alpha$97 reals per MBS and $\alpha$98 reals for the HAPS in the centralized scheme. The distributed PFBWD achieves SE and PF-utility CDFs close to centralized PF, and a vHetNet with four MBSs plus one HAPS matches the average SE per UE of a standalone terrestrial network with about twelve MBSs (Shamsabadi et al., 11 Jul 2025).

7. Unifying principles, misconceptions, and open directions

A persistent misconception is that PFBWD always means direct optimization of complex beamforming vectors under a sum-log rate objective. The hierarchical beamforming formulation is a counterexample: it assumes a fixed hierarchical codebook and optimizes only activation and time-sharing variables, explicitly stating that there is no beamforming weight-vector design, no power allocation, and no weight orthogonality optimization (Floquet et al., 2018). By contrast, the NOMA, MISOME-SWIPT, heterogeneous-rank, and HAPS-vHetNet papers do optimize beam-domain variables, but the fairness target itself changes: per-user rate, per-user energy efficiency, proportional secrecy rate, or distributed rate slack (Al-Obiedollah et al., 2019, Dong et al., 2018, Bedin et al., 2023, Shamsabadi et al., 11 Jul 2025).

A second misconception is that PF necessarily implies a single computational template. The literature here shows several distinct solver families. Hierarchical PF admits a closed form and a linear-time dynamic program; heterogeneous-rank PF relies on alternating scheduling and precoding heuristics; NOMA EE-fair beamforming uses SCA and SOCPs; MISOME-SWIPT uses robust LMIs, SDR, and a one-dimensional outer search; and large-scale HAPS-vHetNets require a two-level ALM plus three-block ADMM decomposition. This suggests that algorithmic structure is driven at least as much by the physical architecture and constraint set as by the fairness criterion itself.

A third point concerns what fairness is balancing. In hierarchical beamforming, PF yields positive throughput for all beams within the stability region but does not strictly equalize across the hierarchy, because activating shallow beams blocks deeper ones (Floquet et al., 2018). In heterogeneous-rank beamforming, PF naturally prioritizes high-rank subbands for underserved users, because larger $\alpha$99 raises instantaneous rates and therefore raises the PF metric on those subbands (Bedin et al., 2023). In MISO NOMA, PF is a deliberate compromise between weakest-user EE and global EE (Al-Obiedollah et al., 2019). In MISOME-SWIPT, PF is intertwined with secrecy and harvesting constraints, so beamweights and artificial noise jointly satisfy proportional secrecy-rate targets (Dong et al., 2018).

The open problems stated or implied by the cited works are correspondingly heterogeneous. The hierarchical paper explicitly notes that a full PFBWD would require SINR models linking weights to achievable rates, along with power and orthogonality constraints, since its own contribution is limited to allocation over a fixed codebook (Floquet et al., 2018). The industrial heterogeneous-rank paper lists robust PFBWD under CSI uncertainty, learning-based PF schedulers, cross-layer latency-aware PF, and uplink extensions as natural continuations (Bedin et al., 2023). The HAPS paper notes that robust per-BS constraints can be embedded in the local SOCPs, which suggests a direct path to uncertainty-aware distributed PFBWD (Shamsabadi et al., 11 Jul 2025). Taken together, these results indicate that proportional fairness is not a finished design doctrine but a transferable optimization principle whose concrete realization depends on whether the dominant bottleneck is hierarchy, rank heterogeneity, SIC structure, secrecy and energy harvesting, or network-scale coordination.