Multi-Beam HAPS Optimization

• The paper introduces a joint design of beamforming, user association, and platform trajectory to maximize throughput, fairness, and energy efficiency under strict power and coverage constraints.
• It employs advanced techniques such as metaheuristic algorithms, successive convex approximation, and distributed optimization to solve high-dimensional, non-convex mixed-integer problems.
• The study highlights critical trade-offs, balancing max-min fairness with sum-rate maximization, and demonstrates improvements in spatial multiplexing and energy-resilient operations.

A high-altitude platform station (HAPS) equipped with beamforming arrays is a transformative architecture for wide-area wireless coverage, vertical heterogeneous networking, and integrated sensing-and-communication (ISAC) missions in 6G and beyond. Multi-beam HAPS optimization refers to the joint design of beamforming, user/resource allocation, and—where relevant—platform deployment and trajectory, with the goal of maximizing network performance metrics (throughput, fairness, beampattern, cost, and energy efficiency) subject to stringent constraints on power, backhaul, coverage, and user quality-of-service. Optimization in this context is fundamentally high-dimensional, non-convex, and frequently mixed-integer, and underpins the ability of new HAPS infrastructure to simultaneously deliver equitable communication, robust sensing, and energy-resilient operation over large geographic areas.

1. System Models for Multi-Beam HAPS Architectures

A typical system comprises a quasi-stationary or mobile HAPS at 18–24 km altitude, equipped with a uniform planar array (UPA), hemispherical array, or custom multi-beam FSO banks, serving ground users via multi-antenna downlink and, in some cases, uplink or ISAC signaling. Users may be single-antenna communication users (CUs), ground-based sensors, or targets for SAR imaging or radar missions. The propagation model is generally a Rician fading channel for air-to-ground links, often incorporating LoS-dominance for HAPS–UE channels and NLoS components for terrestrial links (Kanani et al., 24 Jul 2025, Javed et al., 2024, Abbasi et al., 2024).

Beamforming architectures may include:

• UPA or vertical/horizontal arrays with digital/analog (hybrid) beamforming (Kanani et al., 24 Jul 2025, Javed et al., 2024).
• Hemispherical arrays for hemisphere-shaped coverage, supporting uniform capacity footprint by strategic placement of elements (Abbasi et al., 2024).
• Multi-beam free-space-optical (FSO) banks for optical links to ground terminals, where coverage is determined by principal plus supplementary beams, each with specific divergence and steering (Truong et al., 2023).
• Hybrid satellite-HAPS-ground or HAPS-UAV networks, incorporating terrestrial BSs or UAVs as sub-tiers, with HAPS additionally managing backhaul or as a computation hub (Liu et al., 2022, Kanani et al., 18 Jul 2025).

The multi-beam nature arises as the array forms spatially distinct, high-gain boresight beams aimed at user clusters, sensing targets, or probe points, with the number and geometry of beams determined subject to hardware, coverage, and interference trade-offs (Javed et al., 2024, Abbasi et al., 2024, Truong et al., 2023).

2. Joint Beamforming, User Association, and Sensing Design

Central to multi-beam HAPS optimization is the problem of selecting and shaping downlink (and, in ISAC, sensing) beams, while associating users to beams and allocating transmit power in order to optimize the target metric. The problem is formalized as a joint decision over:

• Digital/analog beamformer vectors for downlink to each user or user group (Kanani et al., 24 Jul 2025, Abbasi et al., 2024).
• Sensing beamformers (ISAC): spatially selective transmission toward ground targets using pilot signals and additional array weightings (Kanani et al., 24 Jul 2025, Kanani et al., 18 Jul 2025).
• User-grouping: partitioning the user set into clusters, each served by a distinct beam or spot, with grouping determined by geometric disk cover techniques or association optimization (Javed et al., 2024).
• User association: assignment of each user to a unique or shared beam/platform, leveraging binary association variables in mixed-integer programming formulations (Liu et al., 9 Nov 2025, Liu et al., 2022, Shamsabadi et al., 2023).
• Platform trajectory: for dynamic HAPS, 3D flight trajectory or quasi-stationary positioning is optimized jointly with the beamformers and schedule (Zhang et al., 12 Jun 2025, Javed et al., 2022).

For ISAC-enabled platforms, both communication and sensing objectives/constraints are addressed. For example, the transmit signal may be

$$x[n] = \sum_{k=1}^K w_k[n] s_k[n] + \sum_{j=1}^J r_j[n] s'_j[n]$$

where $\{w_k\}$ are communication beamformers and $\{r_j\}$ are sensing beamformers; constraints are imposed both on SINR to all CUs and sensing beampattern gain to designated ground locations (Kanani et al., 24 Jul 2025).

3. Optimization Problems and Solution Methods

The underlying optimization problems are highly non-convex, involving quadratic or fractional quadratic forms (e.g., SINRs), mixed-integer association variables, and, for ISAC, max-min or weighted sum-of-log objectives for fairness and sum-rate (Kanani et al., 24 Jul 2025, Liu et al., 9 Nov 2025, Shamsabadi et al., 2023).

Canonical formulations include:

• Max-min fairness: maximize the minimum SINR across users or the minimum beampattern gain across targets, under power and QoS constraints (Kanani et al., 24 Jul 2025, Liu et al., 9 Nov 2025, Abbasi et al., 2024).
• Weighted sum-rate maximization: maximize total throughput with per-user minimum rate constraints and power budgets (Liu et al., 2022, Liu et al., 9 Nov 2025).
• Multi-objective optimization (ISAC): jointly maximize worst-case SINR, minimum beampattern, and/or echo reception from targets, using Pareto trade-off parameterization (Kanani et al., 18 Jul 2025).
• Resource-constrained multi-beam coverage: minimize platform or network cost subject to solar budget, energy, and hardware constraints (Truong et al., 2023).

Solution techniques include:

• Metaheuristic algorithms: genetic algorithm (GA) optimizes beamformer coefficients directly, especially in non-convex ISAC fairness formulations (Kanani et al., 24 Jul 2025, Kanani et al., 18 Jul 2025).
• Successive Convex Approximation (SCA): handles non-convex quadratic–fractional constraints iteratively, replacing them with convex surrogates at each iteration (Liu et al., 9 Nov 2025, Shamsabadi et al., 2023).
• Semidefinite Relaxation (SDR): relaxes rank-one constraints on beamforming weight matrices to convexify the problem (Zhang et al., 12 Jun 2025).
• Block-coordinate methods: alternating between user association (solved as a generalized assignment problem or ILP) and beamforming (via quadratic programming, WMMSE, or SCA/SDR) (Liu et al., 2022, Liu et al., 9 Nov 2025).
• Distributed optimization: augmented Lagrangian and multi-block ADMM decompositions for cell-free and large-scale settings, facilitating scalable and provably convergent distributed solutions (Shamsabadi et al., 11 Jul 2025).
• Closed-form analysis and sequential geometric algorithms for FSO coverage, disk cover, and beam center optimization (Truong et al., 2023, Javed et al., 2024).

The selection of method depends on the structure and dimensionality of the constraints, the presence of binary/integer variables, and the system scalability requirements.

4. Key Performance Results and Trade-offs

Extensive simulation studies across these works provide the following validated findings for multi-beam HAPS optimization:

• Multi-beam HAPS with large UPAs (>64 elements) substantially improves both minimum user rates and worst-case sensing gain, yielding near-flat beampattern coverage over footprints and uniform per-user throughput (Kanani et al., 24 Jul 2025, Abbasi et al., 2024).
• HAPS-based ISAC achieves simultaneous fairness in communication (max-min SINR) and strong, uniform sensing (via beampattern optimization), outclassing UAV-based or non-joint schemes (Kanani et al., 24 Jul 2025, Kanani et al., 18 Jul 2025, Zhang et al., 12 Jun 2025).
• Fairness-oriented optimization (max-min) ensures equitable distribution but trades off some aggregate throughput; sum-rate maximization can increase total rates by up to 20% but decreases fairness (Jain’s index from ~0.9 to ~0.7) (Liu et al., 9 Nov 2025).
• More beams and larger arrays increase both spatial multiplexing and uniformity but risk higher sidelobe/inter-beam interference if not controlled by judicious design (e.g., antenna selection, phase steering) (Abbasi et al., 2024, Javed et al., 2024).
• HAA architectures outperform RAA/CAA in area-uniformity, coverage, and sum-rate; e.g., HAA achieves up to 14 Gbps throughput at 50 dBm, 60%+ higher than RAA at the same power (Abbasi et al., 2024).
• In hybrid and cell-free deployments, adding a HAPS augments capacity, matches the performance of multiple terrestrial BSs, and enables both "super-connecting" urban users and "connecting the unconnected" rural users (Liu et al., 2022, Shamsabadi et al., 11 Jul 2025).
• Energy-constrained multi-beam FSO design demonstrates that increasing the number of beams can up to double coverage and halve network cost, provided solar and hardware budgets permit (Truong et al., 2023).
• NOMA-based HAPS schemes, jointly optimizing user grouping, beam width, and power offer 20–30% energy and throughput gains over OFDMA/OMA, with outage probability falling by more than an order of magnitude at high SNR (Javed et al., 2024, Javed et al., 2022).

5. Hardware and Architectural Innovations

Recent advances in multi-beam HAPS optimization are enabled by novel platform and array architectures:

• Hemispherical Antenna Array (HAA): Provides hemispheric placements of antennas aligned with ground coverage, leveraging selection algorithms for analog phase-shifted beamforming and achieving near-uniform user rates across wide areas (Abbasi et al., 2024).
• Flexible multi-FSO transceiver bundles: Optimal placement and divergence of supplementary FSO beams for wide ground footprint, analytically deriving coverage and cost-minimization formulas, with 14-beam bundles found optimal with current platform parameters (Truong et al., 2023).
• Dynamic HAPS deployment strategies: Alternating between "stop-and-go" (quasi-stationary) and continuous circular trajectories, optimizing 3D positioning and beamforming to maximize both sum-rate and SAR imaging fidelity (Zhang et al., 12 Jun 2025).
• HAPS-UAV ISAC integration: Divides computation (with HAPS as CPU) and analog beamforming (UAV APs), balancing energy efficiency, fairness, and beam pattern performance through joint optimization (Kanani et al., 18 Jul 2025).

6. Complexity, Scalability, and Distributed Implementation

Given the scale and dimensionality of multi-beam HAPS systems, special attention is required to computational complexity:

• Mixed-integer and non-convex QCQP problems scale exponentially in the number of beams/users, requiring decomposition and distributed methods for tractability (e.g., distributed SOCP, ALM-ADMM) (Liu et al., 9 Nov 2025, Shamsabadi et al., 11 Jul 2025).
• User association solved via GAP admits decentralized implementation, with only interference summaries exchanged, supporting hundreds of users per HAPS (Liu et al., 9 Nov 2025, Shamsabadi et al., 11 Jul 2025).
• Distributed PF beamforming frameworks achieve near-centralized performance with order-of-magnitude reductions in signaling overhead and local computation (Shamsabadi et al., 11 Jul 2025).
• Genetic algorithms are preferred for global, highly non-convex fairness/ISAC formulations, exploring millions of candidate solutions under high-dimensional constraints (Kanani et al., 24 Jul 2025).

7. Design Principles and Insights

Empirical and theoretical studies establish the following principles for multi-beam HAPS design:

• Maximize spatial degrees-of-freedom (using large and/or flexible arrays) to support both user multiplexing and sharp beam patterning.
• Jointly optimize beamforming—both at the analog and digital level—with resource assignment (user association, power allocation) and deployment (position/trajectory).
• For ISAC systems, control trade-offs between communication SINR and beampattern gain by either max-min objectives or scalarization (Pareto weighting), with explicit auxiliary variables (Kanani et al., 24 Jul 2025, Kanani et al., 18 Jul 2025).
• When FSO is used, balance between coverage extension and platform energy budget to select optimal number and divergence of beams (Truong et al., 2023).
• Leverage fairness-driven optimization (e.g., max-min SINR) to ensure digital equity, particularly in rural/underserved regions, while switching to sum-rate maximization as capacity permits (Liu et al., 9 Nov 2025).
• Implement distributed solution architectures in multi-access and cell-free HAPS–terrestrial networks to permit scalable computation and near-optimal performance (Shamsabadi et al., 11 Jul 2025, Shamsabadi et al., 2023).

Each of these operational facets is substantiated with analytically specified algorithms and simulation studies in the referenced works, and together they define the rigorous framework for multi-beam HAPS optimization in current and future wide-area wireless, ISAC, and hybrid network deployments.