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arxiv: 2507.08299 · v3 · pith:DNNDAZEWnew · submitted 2025-07-11 · 📡 eess.SY · cs.SY

Two-Level Distributed Interference Management for Large-Scale HAPS-Empowered vHetNets

Afsoon Alidadi Shamsabadi , Animesh Yadav , Halim Yanikomeroglu This is my paper

Pith reviewed 2026-05-22 00:40 UTC · model grok-4.3

classification 📡 eess.SY cs.SY
keywords HAPSvHetNetsbeamformingdistributed optimizationinterference managementproportional fairnessADMM
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The pith

A two-level distributed algorithm designs beamforming weights for large HAPS vHetNets.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

This paper addresses interference in vertical heterogeneous networks where high-altitude platforms share spectrum with terrestrial stations. It adopts a cell-free MIMO setup in which multiple base stations serve users simultaneously through beamforming, but the resulting weight design forms a nonconvex high-dimensional problem that does not scale well. The authors introduce a two-level distributed proportional fairness beamforming weight design method that combines the augmented Lagrangian method with a three-block ADMM framework to decompose the optimization. A sympathetic reader would care because this approach promises to make large-scale integrated networks practical by cutting computation and signaling compared with centralized solvers. If the method works as claimed, HAPS integration can deliver higher capacity and coverage than ground-only systems while remaining feasible to implement.

Core claim

The paper claims that a two-level distributed proportional fairness beamforming weight design algorithm, formed by embedding the augmented Lagrangian method inside a three-block ADMM framework, solves the nonconvex beamforming problem in HAPS-empowered vHetNets and yields performance close to centralized solutions with substantially lower complexity and signaling overhead.

What carries the argument

Two-level distributed PFBWD algorithm that uses ALM inside a three-block ADMM framework to decompose nonconvex beamforming optimization.

If this is right

  • HAPS integration raises spectral efficiency and coverage relative to standalone terrestrial networks.
  • The distributed algorithm lowers computational complexity for large deployments.
  • Signaling overhead drops compared with fully centralized beamforming.
  • Proportional fairness among users is preserved under spectrum sharing.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The decomposition may allow faster updates in time-varying channels than fully centralized methods.
  • Similar two-level structures could be tested in satellite-terrestrial or UAV-assisted networks.
  • Real-world hardware trials would reveal whether quantization and synchronization errors preserve the claimed performance.

Load-bearing premise

The nonconvex beamforming optimization can be split into two levels via the three-block ADMM framework and still reach performance close to a centralized solution.

What would settle it

A simulation of a large-scale HAPS-terrestrial network that directly compares the distributed algorithm's achieved sum rate and fairness index against the centralized optimum and checks whether the gap stays small.

Figures

Figures reproduced from arXiv: 2507.08299 by Afsoon Alidadi Shamsabadi, Animesh Yadav, Halim Yanikomeroglu.

Figure 1
Figure 1. Figure 1: HAPS-empowered vHetNet system model. where hr,u represents the small-scale fading channel coefficient and follows Rayleigh distribution. ξ b u = 10ξ ′b u /10 denotes the log-normal shadowing gain, where ξ ′ b u is the Gaussian random variable with N (0, σξ dB). P Lb,u refers to the FSPL between MBS b and the UE u, and can me computed as P Lb,u = [PITH_FULL_IMAGE:figures/full_fig_p004_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: HAPS antenna architecture. where wb u ∈ C Nb refers to the beamforming vector for UE u at BS b, and σ 2 n is the variance of the additive white Gaussian noise (AWGN) at u. According to the above explanation, the preliminar PFBWD optimization problem can be expressed as maximize Wb X u∈U log(log2 (1 + γu)) (7a) s.t. γu ≥ γmin, ∀u, (7b) ∥Wb ∥ 2 F ≤ P max b , ∀b, (7c) where Wb = [wb 1 , . . . , wb U ] ∈ C Nb×… view at source ↗
Figure 3
Figure 3. Figure 3: Convergence behavior of the proposed PFBWD algorithm. 2) Convergence of the outer-level algorithm 3: Under assumptions 1-3, according to [ [32], Theorem 1], the outer-level Algorithm has a limit point (X ∗ , X ∗ , Z ∗ ) which is either feasible to the original problem, i.e., Z ∗ = 0, or (X ∗ , X ∗ ) is a stationary point of the problem minimize X ∈C, X ∈C 1 2 [PITH_FULL_IMAGE:figures/full_fig_p013_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Statistical behavior of SE per UE and the average PF objective function value in a HAPS-empowered vHetNet (4 MBSs + 1 HAPS) serving 16 UEs. VI. NUMERICAL RESULT AND DISCUSSION In this section, we evaluate the performance of the proposed distributed PFBWD algorithm in HAPS-empowered vHetNets and compare it against both standalone terrestrial networks and the centralized approach. The standalone terrestrial … view at source ↗
Figure 5
Figure 5. Figure 5: Performance comparison of centralized and distributed PFBWD algorithms for different numbers of UEs in a HAPS-empowered vHetNet (4 MBSs + 1 HAPS) [PITH_FULL_IMAGE:figures/full_fig_p014_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Average SE per UE: vHetNet vs. standalone terrestrial network across different numbers of MBSs. large values of δ may lead to numerical instability [24]. Figs. 4a and 4b plot the cumulative distribution function (CDF) of the average SE per UE and the average PF objective value, respectively. The results compare the proposed distributed PFBWD algorithm with the centralized algorithm in a vHetNet comprising … view at source ↗
Figure 7
Figure 7. Figure 7: Impact of HAPS antenna architecture on the average SE per UE in a vHetNet (4 MBSs + 1 HAPS) serving 16 UEs. VII. CONCLUSION This paper addressed the challenges of interference management in harmonized spectrum HAPS-empowered vertical het￾erogeneous networks (vHetNets). We highlighted the limitations of centralized interference management approaches, which face scalability issues due to the large number of … view at source ↗
read the original abstract

High altitude platform stations (HAPS) offer a promising solution for achieving ubiquitous connectivity in next-generation wireless networks (xG). Integrating HAPS with terrestrial networks, creating HAPS-empowered vertical heterogeneous networks (vHetNets), significantly improves coverage and capacity and supports emerging novel use cases. In HAPS-empowered vHetNets, HAPS and terrestrial network tiers can share the same spectrum, forming harmonized spectrum vHetNets that enhance spectral efficiency (SE). However, harmonized spectrum vHetNets face major challenges, including severe co-channel interference and scalability in large-scale deployments. To address the first challenge, we adopt a cell-free multiple-input multiple-output (MIMO) network architecture in which users are simultaneously served by multiple base stations using beamforming. However, beamforming weight design leads to a nonconvex, high-dimensional optimization problem, highlighting the scalability challenge. To address this second challenge, we develop a two-level distributed proportional fairness beamforming weight design (PFBWD) algorithm. This algorithm combines the augmented Lagrangian method (ALM) with a three-block ADMM framework. Simulation results demonstrate the performance improvements achieved by integrating HAPS with standalone terrestrial networks, as well as the reduced complexity and signaling overhead of the distributed algorithm compared to centralized algorithms.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The manuscript proposes a two-level distributed proportional fairness beamforming weight design (PFBWD) algorithm that integrates the augmented Lagrangian method (ALM) with a three-block ADMM framework. The approach targets co-channel interference and scalability challenges in large-scale HAPS-empowered vertical heterogeneous networks (vHetNets) that share spectrum between HAPS and terrestrial tiers. Simulations are reported to show performance gains from HAPS integration and lower complexity/signaling overhead relative to centralized beamforming solutions.

Significance. If the distributed algorithm reliably approaches centralized performance with provable convergence and reduced overhead, the contribution would be relevant for practical interference management in integrated aerial-terrestrial networks. The work addresses nonconvex high-dimensional beamforming in a distributed manner, which is pertinent to capacity and coverage goals in next-generation systems.

major comments (2)
  1. [algorithm development paragraph and associated method description] The description of the two-level PFBWD algorithm (combining ALM with three-block ADMM): the manuscript applies the three-block ADMM framework directly to the nonconvex beamforming subproblems without deriving or citing convergence conditions specific to the weighted sum-rate or proportional-fair objective and the interference coupling terms. Standard three-block ADMM lacks global convergence guarantees for general nonconvex problems absent additional structure such as strong convexity of blocks or Lipschitz continuity of gradients; this assumption is load-bearing for the central claim of performance close to the centralized solution in large-scale deployments.
  2. [simulation results paragraph] Simulation results paragraph: the reported performance improvements and overhead reductions are presented without accompanying convergence guarantees, error bounds, or analysis of sensitivity to the choice of penalty parameters and iteration counts. This weakens support for the scalability assertions when the underlying optimization is acknowledged to be nonconvex.
minor comments (2)
  1. [Abstract] The abstract refers to 'harmonized spectrum vHetNets' without a concise definition or reference to prior usage of the term; a brief clarifying sentence would improve readability.
  2. [system model section] Notation for the beamforming weights and interference terms could be introduced more explicitly before the algorithm description to aid readers unfamiliar with cell-free MIMO formulations.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the constructive and detailed comments. We address each major comment below and indicate the planned revisions to strengthen the manuscript.

read point-by-point responses

  1. Referee: [algorithm development paragraph and associated method description] The description of the two-level PFBWD algorithm (combining ALM with three-block ADMM): the manuscript applies the three-block ADMM framework directly to the nonconvex beamforming subproblems without deriving or citing convergence conditions specific to the weighted sum-rate or proportional-fair objective and the interference coupling terms. Standard three-block ADMM lacks global convergence guarantees for general nonconvex problems absent additional structure such as strong convexity of blocks or Lipschitz continuity of gradients; this assumption is load-bearing for the central claim of performance close to the centralized solution in large-scale deployments.

    Authors: We agree that the lack of specific convergence conditions for the nonconvex setting is a limitation in the current presentation. In the revised manuscript we will add a dedicated discussion subsection that cites relevant literature on ADMM variants applied to nonconvex beamforming and weighted sum-rate problems. We will also report empirical convergence behavior observed across the simulated scenarios and note the parameter regimes in which stable performance is obtained, while explicitly acknowledging that global convergence guarantees do not hold for arbitrary nonconvex instances. revision: yes

  2. Referee: [simulation results paragraph] Simulation results paragraph: the reported performance improvements and overhead reductions are presented without accompanying convergence guarantees, error bounds, or analysis of sensitivity to the choice of penalty parameters and iteration counts. This weakens support for the scalability assertions when the underlying optimization is acknowledged to be nonconvex.

    Authors: We accept this observation. The revised version will include additional simulation figures that plot the evolution of the objective and constraint violation over iterations for different network sizes. We will also add a sensitivity study showing the effect of penalty-parameter values and iteration budgets on both achieved proportional fairness and signaling overhead. These additions will provide concrete support for the scalability claims while remaining transparent about the nonconvex nature of the problem. revision: yes

Circularity Check

0 steps flagged

No circularity: algorithm derivation is self-contained via ALM+ADMM framework and simulation validation

full rationale

The paper presents a two-level distributed PFBWD algorithm that combines the augmented Lagrangian method with a three-block ADMM framework to solve the nonconvex beamforming weight design problem in HAPS-empowered vHetNets. This is introduced as a novel decomposition to address scalability and interference, with performance claims supported by simulation results comparing to centralized solutions and standalone terrestrial networks. No load-bearing step reduces by construction to fitted parameters, self-citations, or renamed inputs; the central claims rest on the proposed algorithmic structure and empirical outcomes rather than tautological definitions or unverified self-referential premises. The derivation chain is independent of the target performance metrics.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claim rests on standard assumptions of cell-free MIMO channel models and convergence properties of ADMM for nonconvex problems; no explicit free parameters or invented entities are named in the abstract.

axioms (2)
  • domain assumption Cell-free MIMO architecture can be applied to HAPS-empowered vHetNets with shared spectrum
    Invoked when stating that users are simultaneously served by multiple base stations using beamforming.
  • domain assumption Three-block ADMM framework converges for the formulated nonconvex beamforming problem
    Implicit in the claim that the distributed algorithm addresses the scalability challenge.

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