Alternating Optimization Algorithm

Updated 30 January 2026

Alternating-optimization-based algorithms are iterative methods that decompose nonconvex problems with coupled variables into simpler, manageable subproblems.
They employ techniques like convex programming, successive convex approximation, and block coordinate descent to optimize interdependent parameters such as beamforming and antenna orientation.
These algorithms are pivotal for advanced wireless systems, enhancing performance in applications like multi-user MIMO, integrated sensing and communications, and secure transmissions.

An alternating-optimization-based algorithm is a metaheuristic and iterative procedure used to solve non-convex optimization problems with multiple coupled variable blocks. In wireless communications, this approach is deployed when key system variables (e.g., beamformers and antenna orientations) are interdependent, and direct joint optimization is computationally prohibitive or analytically intractable. Alternating optimization decomposes the high-dimensional joint problem into a sequence of subproblems, each of which is often more tractable and solvable using dedicated techniques—such as convex programming, successive convex approximation (SCA), block coordinate descent (BCD), or gradient projection—applied while fixing all other variable blocks. This framework is central to much contemporary array and multi-antenna system design involving new physical degrees of freedom such as rotatable antennas, movable arrays, and fluid-antenna systems.

1. Formalism and Problem Structure

Alternating optimization (AO) begins with the joint optimization problem: $\max_{\mathbf{x},\,\mathbf{y}}\,F(\mathbf{x},\mathbf{y}),\qquad \mathbf{x}\in\mathcal{X},\ \mathbf{y}\in\mathcal{Y}$ where $F(\mathbf{x},\mathbf{y})$ is often non-convex in both blocks $\mathbf{x}$ (e.g., beamforming weights) and $\mathbf{y}$ (e.g., antenna orientation angles). AO alternates between:

$\displaystyle \mathbf{x}^{(t+1)} = \arg\max_{\mathbf{x}\in\mathcal X} F(\mathbf{x},\mathbf{y}^{(t)})$
$\displaystyle \mathbf{y}^{(t+1)} = \arg\max_{\mathbf{y}\in\mathcal Y} F(\mathbf{x}^{(t+1)},\mathbf{y})$

Convergence is guaranteed if each subproblem incrementally improves the objective and the feasible sets are compact (Zheng et al., 5 Jan 2025, Wu et al., 2024, Peng et al., 23 Jan 2026). In rotatable antenna systems, this paradigm is necessary due to the coupling between beamforming vectors and orientation-dependent channel gains.

2. Application in Rotatable Antenna Array Optimization

Alternating-optimization-based algorithms are extensively used to solve non-convex design problems in systems leveraging the spatial degree of freedom from rotatable antenna elements or arrays. Typical applications include:

Uplink and Downlink Multi-user MIMO: Max-min SINR or sum-rate maximization via AO between receive (or transmit) beamforming and per-element orientation (zenith/azimuth deflection), subject to physical rotation constraints (Zheng et al., 5 Jan 2025, Wu et al., 2024, Peng et al., 23 Jan 2026).
Integrated Sensing and Communications (ISAC): Joint radar/communication performance, iteratively updating beamforming for communication/sensing and antenna orientations for improved spatial aperture (Zhou et al., 13 Mar 2025, Wang et al., 10 Sep 2025).
Cell-free and Distributed MIMO: Alternating between beamformer updates (via convex optimization or SOCP) and orientation updates (via SCA or Frank-Wolfe) to achieve max-min fairness across distributed APs (Pan et al., 4 Dec 2025, Peng et al., 23 Jan 2026).
Secure Communications and Cognitive Radio: Secrecy-rate maximization by alternating between generalized Rayleigh-quotient beamforming and per-element SCA-based orientation, incorporating interference or power constraints (Dai et al., 14 Apr 2025, Tan et al., 30 Sep 2025).

3. Algorithmic Components and Techniques

Subproblem solvers in AO-based architectures for rotatable or movable antenna arrays include:

Beamforming Update: For fixed orientations, classical MMSE/ZF/MRC or WMMSE closed-form solvers are utilized (Zheng et al., 5 Jan 2025). In more complex interference/channel models, second-order cone programming (SOCP) is deployed (Peng et al., 23 Jan 2026).
Orientation Update: For fixed beamformers, orientation variables often appear in non-convex forms (through gain patterns, steering vectors, array responses). Successive convex approximation (SCA), first-order Taylor linearization, or Frank-Wolfe methods on spherical caps are used (Zheng et al., 5 Jan 2025, Wu et al., 2024, Peng et al., 23 Jan 2026, Peng et al., 24 Sep 2025).
Fractional Programming and Quadratic Transforms: In hybrid beamforming regimes (ISAC), quadratic-transforms and auxiliary-variable-based surrogates decompose fractional SINR/SCNR objectives, making each block tractable (Wang et al., 10 Sep 2025, Peng et al., 24 Sep 2025, Pan et al., 4 Dec 2025).
Metaheuristics for Angle Selection: When physical limitations dictate discrete rotation steps (finite-resolution motors), genetic algorithms, particle swarm optimization (PSO), or cross-entropy methods are embedded within the AO loop to select optimal angular settings (Zheng et al., 8 Jan 2026, Zhang et al., 5 Sep 2025, Peng et al., 24 Sep 2025).

Table 1: Core Subproblems and Methods in AO Frameworks (Representative)

Variable Block	Typical Method	Context
Beamforming weights	MMSE/ZF, SOCP, WMSE	Multi-user MIMO/CF-MIMO
Orientations	SCA, Frank-Wolfe, PSO	Element/panel orientation update
Power Allocation	SCA, Closed-form	Interference/joint optimization

4. Representative AO Algorithmic Workflow

A canonical AO-based algorithm for rotatable antenna arrays follows:

Initialization: Set initial orientation vector $\Theta^{(0)}$ and beamformers $W^{(0)}$ .
Beamformer Update: Given $\Theta^{(t)}$ , compute $W^{(t+1)}$ to maximize SINR, sum-rate, secrecy, etc.
Orientation Update: Given $W^{(t+1)}$ , optimize $\Theta$ by maximizing the current objective, via SCA or projection.
Convergence: Repeat steps 2–3 until the relative improvement is below $\varepsilon$ .

for t in range(max_iter):
    W_new = solve_beamforming(SINR, Theta_current)
    Theta_new = optimize_orientation(W_new, constraints)
    if objective(Theta_new, W_new) - objective(Theta_current, W_current) < tol:
        break
    W_current, Theta_current = W_new, Theta_new

This structure ensures monotonic improvement and tractable per-iteration complexity (often

\mathcal{O}(KN^3 + N^{3.5}\ln(1/\varepsilon))

for

K

users,

N

antennas) (Wu et al., 2024).

5. Case Studies and Performance Characteristics

a) Max-Min SINR in Uplink RAs

In (Wu et al., 2024), AO alternates between MMSE beamforming (fixed orientations) and per-element orientation (via SCA and subsequent projection). This achieves near-maximum array gain for arbitrary user geometries and rapidly outperforms both fixed-antenna and random-orientation baselines.

b) Secure Communications with Rotatable Arrays

In (Dai et al., 14 Apr 2025), the secrecy-rate maximization is solved by AO, switching between Rayleigh-quotient-based beamformer updates and SCA for element orientations. Nulls are steered toward eavesdroppers while maintaining high main-lobe gain for legitimate users.

c) Joint Power and Orientation in Mixed Near-/Far-Field

For modular rotatable arrays with subarray-wise rotation (power plus angle vectors), a double-layer AO is used: SCA for power, metaheuristic PSO for orientation. This structure enables near interference-free operation in challenging upper mid-band deployments (Zhang et al., 5 Sep 2025).

d) Integrated Sensing and Communication (ISAC)

Fractional programming combined with AO—alternately optimizing beamforming (quadratic transforms) and rotation angles (closed-form or gradient)—achieves substantial improvement in joint radar-communication utility (Wang et al., 10 Sep 2025).

6. Design Guidelines, Advantages, and Complexity

Key advantages of AO-based algorithms in rotatable/movable antenna optimization include:

Decomposition of non-convex coupled objectives, enabling practical solution throughput
Flexibility to incorporate a wide range of physical constraints (discrete rotations, collision avoidance, limited actuator ranges)
Provable monotonicity and convergence to local optima in most cases (unless stochastic solvers are embedded)
Compatibility with continuous, discrete, and hybrid physical architectures

Complexity is determined by subproblem dimensionality and the degree of coupling, and can be reduced via non-iterative suboptimal variants with modest performance loss (Zheng et al., 5 Jan 2025, Peng et al., 23 Jan 2026).

7. Broader Impacts and Theoretical Insights

Alternating-optimization-based algorithms are pivotal for enabling new spatial degrees of freedom in wireless systems without increasing hardware count, power, or complexity. They allow element-wise orientation adaptation, maximize spatial coverage, enhance interference mitigation, and enable new modes such as joint sensing/communication (Zheng et al., 22 May 2025, Wang et al., 10 Sep 2025, Wu et al., 2024). They have also facilitated analytical breakthroughs, such as the proof of higher asymptotic array gain for RA versus fixed arrays (Zheng et al., 5 Jan 2025), and closed-form full-gain nulling conditions in RAA-enhanced zero-forcing arrays (Wen et al., 13 Dec 2025).

AO frameworks will remain essential as future systems incorporate flexible arrays, hybrid architectures, physical meta-materials, and distributed optimization under dynamic spatial constraints.