Max-Min Rate Problem in RA-Enhanced Wireless

• Max-Min Rate Problem is a formulation that seeks to maximize the minimum SINR or rate in multi-user wireless communications through joint beamforming and rotatable antenna orientation control.
• Alternating optimization methods using SOCP and SCA decouple the complex design challenges, yielding up to 6 dB SINR improvement for under-served users.
• Integrated RA design enhances system performance by dynamically adjusting antenna orientations, offering robust fairness and cost-effective hardware solutions in distributed networks.

The max-min rate problem is a central optimization formulation in multi-user wireless communications and array processing that aims to maximize the worst-case (minimum) SINR or achievable rate among all users, thereby ensuring fairness and coverage guarantees even for users with unfavorable channel conditions. Its relevance is heightened in advanced architectures such as systems with rotatable antennas (RAs), which introduce additional spatial degrees of freedom through dynamic orientation control of antenna elements or arrays. The ensuing coupling between digital beamforming and mechanical/electronic steering renders the max-min rate problem highly nontrivial, requiring specialized algorithmic treatments. This article provides a comprehensive treatment, emphasizing the integration of RA technology, mathematical formulations, solution algorithms, and practical system design.

1. Mathematical Formulation of the Max-Min Rate Problem

Max-min fairness in multi-user downlink or uplink systems is typically characterized by the objective

$$\max_{\mathcal{W},\mathcal{\Theta}}\,\min_{k=1,\dots,K}\;R_k(\mathcal{W},\mathcal{\Theta}),$$

where $R_k$ is the achievable rate (or SINR) for user $k$, which depends on both the digital beamforming variables $\mathcal{W}$ (e.g., precoder or combiner weights) and, in RA-enhanced systems, the antenna orientation parameters $\mathcal{\Theta}$ (e.g., per-element or per-panel 3D boresight angles) (Zheng et al., 5 Jan 2025, Wu et al., 2024, Peng et al., 23 Jan 2026). Constraints may include per-antenna power limits, mechanical tilt bounds, and unit-norm orientation vectors: \begin{align*} &|\mathcal{W}b|^2 \le P{\max}, \quad \forall b\ &\theta_{b,m} \in [0,\theta_{\max}],\; \phi_{b,m} \in [0,2\pi),\; |\mathbf{f}_{b,m}| = 1 \end{align*} where $\mathbf{f}_{b,m}$ specifies the 3D orientation of antenna $m$ at AP $b$ (Peng et al., 23 Jan 2026). The problem is inherently non-convex due to the bilinear coupling between beamforming and array orientations and the non-convex structure of physical-layer rate functions.

2. Rotatable Antenna Modeling and Channel Characteristics

The deployment of RAs affects both array response and directional gain patterns. An RA element is parameterized by zenith/elevation ($\theta$) and azimuth ($\phi$), forming a unit boresight vector $\mathbf{f}(\theta,\phi)$. The element gain relative to a signal direction $\mathbf{s}$ is typically modeled as

$G(\epsilon) = \kappa_{\max} \cos^{2p}(\epsilon)$

where $\epsilon = \arccos(\mathbf{f}^T \mathbf{s})$, $\kappa_{\max}=2(2p+1)$, and $p$ sets directivity (Zheng et al., 5 Jan 2025, Peng et al., 23 Jan 2026). The full array response to user $k$ is a composition of phase shifts and rotation-dependent gains: $$h_k = \left[ \sqrt{G(\epsilon_{b,m,k})} e^{-j\frac{2\pi}{\lambda} r_{k,b,m}}\right]_{b,m}$$ with $r_{k,b,m}$ the propagation distance and the spatial term determined by orientation (Peng et al., 23 Jan 2026).

In the presence of multipath and distributed APs (cell-free MIMO), the channel stacking incorporates RA orientation for every AP element, creating a high-dimensional coupling between macro-diversity and local directivity optimization (Pan et al., 4 Dec 2025, Peng et al., 23 Jan 2026).

3. Solution Algorithms

The prevailing approach utilizes alternating optimization (AO) between beamforming and orientation blocks:

A. SOCP for Beamformer Update:

For fixed RA orientations, $R_k(\mathcal{W})$ is convex in $\mathcal{W}$ and per-user SINR constraints can be cast as second-order cone constraints. Feasibility is checked for a candidate target $\gamma$, yielding a bisection algorithm over $\gamma$ for global optimality: $$\frac{|\mathbf{h}_k^H \mathbf{w}_k|^2}{\sum_{j\neq k} |\mathbf{h}_k^H \mathbf{w}_j|^2 + \sigma^2} \ge \gamma$$ can be reformulated and solved using standard SOCP solvers (Peng et al., 23 Jan 2026).

B. Successive Convex Approximation (SCA) for Orientation Update:

With fixed beamforming, the SINR as a function of $\{\theta,\phi\}$ is fractional and highly non-convex. Fractional programming and SCA are deployed:

• Taylor/Young’s inequality yields convex quadratic surrogates for desired and interference power terms: $$S_k(\mathbf{f}) \geq \underline{S}_k^{[t]}(\mathbf{f}), \ I_k(\mathbf{f}) \leq \overline{I}_k^{[t]}(\mathbf{f})$$ where gradients and curvature constants are computed for local approximations (Peng et al., 23 Jan 2026).
• The unit-norm and mechanical bounds on orientations are relaxed to convex sets and enforced via projection after each update.
• AO proceeds by alternating SOCP and SCA, with guaranteed monotonic convergence to a stationary point (local optimum).

C. Two-Stage Low-Complexity Schemes:

A simplified alternative decouples geometry from digital design. Stage 1 optimizes RA orientations for aggregate channel gain or a proportional-fair utility: $$U(\mathbf{f}) = \sum_k \ln (\eta_k(\mathbf{f}) + \varepsilon),\quad \eta_k = \mathbf{h}_k^H \mathbf{h}_k$$ using manifold-aware Frank-Wolfe retraction (Peng et al., 23 Jan 2026). Stage 2 solves the beamformer SOCP once for the resultant array configuration. Performance is competitive, especially for moderate $p$ and actuator limits.

4. Special Cases and Analytical Insights

• Single-user, Free-space:

Optimal RA orientation aligns each element directly toward the user, maximizing array gain: $$\gamma^* = \bar{P} \sum_{n=1}^N \frac{G_0 \cos^{2p}(\epsilon_n)}{r_n^2}$$ Closed-form solutions exist for the independent steering of each element (Zheng et al., 5 Jan 2025, Wu et al., 2024).

• Asymptotic Analysis:

With $N\to\infty$ and wide steering range, the max-min SINR approaches

$$\lim_{N\to\infty} \gamma_{\rm RA} / \gamma_{\rm fixed} = \theta_{\max} + \cos \theta_{\max}$$

demonstrating the provable gain of RA versus fixed arrays (Zheng et al., 5 Jan 2025, Wu et al., 2024).

• Multi-user, Multi-path:

AO and SCA provide up to 3–6 dB improvement in min-SINR compared to fixed or random orientation schemes, with higher robustness to user geometry and path diversity (Zheng et al., 5 Jan 2025).

5. System-Level Impact, Hardware, and Algorithmic Considerations

RA Hardware Costs and Control:

• Per-element 3D orientation actuators scale hardware cost linearly; cross-linked and panel-level rotation schemes (CL-RA) reduce motor count to $O(M+N)$, with a negligible loss in max-min rate compared to fully flexible designs (Zheng et al., 8 Jan 2026).
• Mechanical tilt limits $\theta_{\max}$ need only be moderate ($\pi/10$ to $2\pi/10$) to capture the majority of max-min-SINR gains (Peng et al., 23 Jan 2026).
• Fine orientation granularity (sub-degree) is ideal; codebooks and genetic/CEM algorithms handle coarser discrete angular controls (Zheng et al., 8 Jan 2026, Peng et al., 24 Sep 2025).

Algorithmic Trade-offs:

• Joint beamforming/orientation optimization is computationally demanding ($O(L(N^{3.5}+KN^3))$), but two-stage methods and subarray grouping significantly reduce complexity (Peng et al., 23 Jan 2026, Zheng et al., 5 Jan 2025).
• Proper initialization (e.g., geometry-based orientation) aids convergence.
• AO and Frank-Wolfe algorithms preserve monotonic increase in worst-user rate or SINR.

Performance and Fairness Enhancements:

• In cell-free networks, RA-enhanced max-min rate designs equalize the rates across users despite blockages and spatial disparity, by leveraging macro-diversity with orientation-aware directivity control (Pan et al., 4 Dec 2025, Peng et al., 23 Jan 2026).
• Larger directivity factor $p$ increases gains for optimized RA designs, but can degrade fairness for non-optimized or randomly oriented arrays.

6. Numerical Results and Design Guidelines

Representative numerical findings across systems with RA include:

Scheme	SINR Gain (dB, min-user)	Hardware Complexity	Rotation DoFs
Per-element RA + AO	4–6	High	$N$
CL-RA element-level	+25% over panel-level	Moderate	$M+N$ actuators
Fixed orientation	0	Low	None
Panel-level rotation	-17% vs CL-element	Lowest	$O(M+N)/\sqrt{Q_b}$
Two-stage AO/Frank-Wolfe	≈24–49% over random	Low–Moderate	$N$ (but one-shot)

Performance saturates after moderate tilt angles; gains are robust to localization errors and persist across user densities.

7. Extensions and Research Directions

• Advanced materials (meta-surfaces, fluid antennas) may further lower actuation latency and broaden feasible orientation ranges (Zheng et al., 22 May 2025).
• Integration of real-time ML for orientation prediction under mobility and partial CSI is an active area (Zheng et al., 22 May 2025, Peng et al., 23 Jan 2026).
• Multi-cell RA coordination and joint optimization for distributed massive MIMO are open problems.
• Application to secure communication (secrecy max-min), cognitive spectrum sharing (with interference constraints), and ISAC (integrated sensing and communication) are active extensions (Dai et al., 14 Apr 2025, Tan et al., 30 Sep 2025, Zhou et al., 13 Mar 2025).

Rotatable antennas thus fundamentally shift the design space for max-min rate and fairness optimization by introducing orientation as an explicit degree of freedom, demanding new analytical and algorithmic methods to fully exploit their capabilities in next-generation distributed and cell-free wireless networks (Peng et al., 23 Jan 2026, Zheng et al., 5 Jan 2025, Wu et al., 2024, Pan et al., 4 Dec 2025).