Tri-Loop Alternating Optimization (TLAO)

• TLAO is a tri-level framework that decomposes the joint design of digital, analog, and EM beamformers into three nested loops, addressing non-convex and combinatorial challenges.
• The method employs PDD, WMMSE, and coordinate descent in its inner, middle, and outer loops respectively, iteratively enhancing spectral efficiency under realistic hardware constraints.
• By integrating dual quadratic and Lagrange dual fractional programming, TLAO offers a practical trade-off between computational accuracy and speed for achieving energy-efficient beamforming.

Searching arXiv for the cited paper and closely related beamforming/RCRAA context. Tri-Loop Alternating Optimization (TLAO) is a tri-level alternating optimization framework for the joint design of digital beamforming, analog beamforming, and electromagnetic (EM) beamforming in Radiation-Center Reconfigurable Antenna Array (RCRAA)-based Tri-Hybrid Beamforming (THBF). In the formulation introduced in "Tri-Hybrid Beamforming for Radiation-Center Reconfigurable Antenna Array: Spectral Efficiency and Energy Efficiency" (Li et al., 21 Aug 2025), the three beamforming layers are represented by the digital precoder $\mathbf{F}_{\rm BB}$, the RF-domain phase-shifter network $\mathbf{F}_{\rm RF}$, and radiation-center (RC) selection $\mathbf{r}$, respectively. TLAO decomposes a highly non-convex and combinatorial joint optimization problem into three nested loops—inner, middle, and outer—so that each block can be updated tractably while iteratively improving spectral efficiency (SE) or, when embedded in a fractional-programming framework, energy efficiency (EE).

1. Architectural setting and design objective

TLAO is defined within a THBF architecture that combines three distinct beamforming domains. The digital beamformer operates at baseband through $\mathbf{F}_{\rm BB}$, the analog beamformer operates in the RF domain through $\mathbf{F}_{\rm RF}$, and the EM beamformer is modeled as RC selection through the discrete variable $\mathbf{r}$. The paper characterizes this as a joint design problem in which EM beamforming is not an auxiliary refinement of conventional hybrid beamforming, but a third optimization layer coupled to the other two.

The central motivation is that RC configurability introduces additional degrees of freedom beyond those of conventional hybrid beamforming architectures. The paper states that the TLAO scheme is designed to “jointly optimize the three layers of beamforming” in RCRAA-based THBF and that this tri-loop structure allows tractable alternating optimization of each block while iteratively improving the overall objective. This suggests that TLAO should be understood as a block-structured framework for coordinated optimization across baseband, RF, and aperture-domain decisions rather than as a single closed-form beamformer update.

The three-loop decomposition is fixed as follows: the inner loop optimizes the digital beamformer, the middle loop optimizes the analog beamformer, and the outer loop optimizes EM beamforming through RC selection. Because RC selection is discrete and subject to hardware and mutual-coupling restrictions, the outer loop is combinatorial, whereas the inner and middle loops address a continuous but still non-convex digital-analog coupling.

2. Spectral-efficiency formulation

For SE maximization, the core problem is given as \begin{align} \max_{\mathbf{r},\; \mathbf{F}{\rm RF},\; \mathbf{F}{\rm BB}}~ & R_{\rm SE} = \sum_{k=1}^K \log_2(1+\gamma_k) \tag{18a}\ \text{s.t.}~\quad & \mathbf{r} \in \mathcal{A}r \qquad\quad\,~ \text{(RC selection constraints)} \tag{18b}\ & \mathbf{F}{\rm RF} \in \mathcal{F}{\rm FC}\text{ or } \mathcal{F}{\rm PC} \qquad \text{(analog beamformer constraints)} \tag{18c}\ & |\mathbf{F}{\rm RF}\mathbf{F}{\rm BB}|F^2 \leq P{\rm max} \qquad \text{(power constraint)} \tag{18d} \end{align} with $\gamma_k$ denoting the SINR for user $k$ and depending on all three beamformers (Li et al., 21 Aug 2025).

The SINR is expressed as

$$\gamma_k = \frac{|\mathbf{h}_k^H(\mathbf{r}) \mathbf{F}_{\rm RF}[\mathbf{F}_{\rm BB}]_{:, k}|^2} {\sum_{i \neq k} |\mathbf{h}_k^H(\mathbf{r}) \mathbf{F}_{\rm RF}[\mathbf{F}_{\rm BB}]_{:, i}|^2 + \sigma_n^2},$$

where $\mathbf{h}_k(\mathbf{r})$ is the user-channel vector induced by the current RC selection. In this formulation, $\mathbf{F}_{\rm RF}$0 denotes the feasible RC-selection set, which the paper describes as combinatorial because of mutual coupling and selection rules, while $\mathbf{F}_{\rm RF}$1 and $\mathbf{F}_{\rm RF}$2 denote the feasible sets for fully connected and partially connected analog beamformers under constant-modulus constraints.

The RC selection constraints include binary selection, a constraint on the number of selected RCs, and a mutual-coupling condition. The paper writes these as a binary condition $\mathbf{F}_{\rm RF}$3 and a mutual-coupling inequality $\mathbf{F}_{\rm RF}$4. The significance of these constraints is that the EM beamformer is not a continuous aperture weighting but a discrete hardware selection problem embedded directly into the beamforming design.

3. Tri-loop mechanics

The defining feature of TLAO is its nested-loop decomposition. With $\mathbf{F}_{\rm RF}$5 fixed, the digital-analog subproblem remains non-convex because of the coupling constraint $\mathbf{F}_{\rm RF}$6, where $\mathbf{F}_{\rm RF}$7 is an auxiliary variable. The paper addresses this through penalty dual decomposition (PDD) alternating minimization in the inner and middle loops, while the outer loop uses coordinate descent (COD) over RC selections.

Loop	Variable(s) optimized	Method
Inner	$\mathbf{F}_{\rm RF}$8, $\mathbf{F}_{\rm RF}$9, auxiliary variables	PDD, WMMSE/DQTFP/LDTFP
Middle	$\mathbf{r}$0	PDD, block update
Outer	$\mathbf{r}$1	Coordinate descent

Within the PDD framework, the relaxed objective includes the penalty term

$\mathbf{r}$2

Here, $\mathbf{r}$3 is the weighted MSE for user $\mathbf{r}$4, $\mathbf{r}$5 and $\mathbf{r}$6 are auxiliary variables from the WMMSE transformation, and $\mathbf{r}$7 is a penalty parameter decreased during iterations to enforce the coupling. The update sequence given in the paper is: MMSE closed-form updates for $\mathbf{r}$8, closed-form updates for $\mathbf{r}$9, a convex quadratic update for $\mathbf{F}_{\rm BB}$0 under the power constraint via KKT conditions, a non-convex update for $\mathbf{F}_{\rm BB}$1 using closed-form or approximations for the FC and PC cases, and a least-squares or SVD-based update for $\mathbf{F}_{\rm BB}$2 depending on FC or PC.

The outer loop performs RC selection by coordinate descent. The initialization starts from $\mathbf{F}_{\rm BB}$3 by selecting the best RCs based on effective channel gain. Then, for each RC element, the method tries replacing it with all other possible RC candidates while respecting hardware and mutual-coupling constraints, re-runs the inner and middle loop optimizations for each candidate, and selects the configuration that yields maximal SE. The stopping criterion is that no further improvement occurs after one full round. Because each step is accepted only when it is non-decreasing in the objective, the COD stage is monotonic by construction.

4. Energy-efficiency extensions

For EE maximization, the paper retains the TLAO structure but embeds it in a fractional-programming framework. The EE objective is written as

$\mathbf{F}_{\rm BB}$4

with $\mathbf{F}_{\rm BB}$5 detailed in eq. (16), and the performance metric summarized as $\mathbf{F}_{\rm BB}$6 (Li et al., 21 Aug 2025).

Two fractional-programming variants are integrated with TLAO. The first is the dual quadratic transform-based fractional programming (DQTFP) scheme, and the second is the Lagrange dual transform-based fractional programming (LDTFP) scheme. The paper states that both combine with TLAO for the THBF design, with DQTFP being more accurate and LDTFP being much faster. More specifically, DQTFP solves a convex problem via CVX, whereas LDTFP yields closed-form updates at each iteration except for a simple bisection associated with the power constraint.

The relation between the SE and EE formulations is structurally important. In the SE problem, TLAO directly optimizes the tri-hybrid beamformer under power and hardware constraints. In the EE problem, the same nested logic is preserved, but the inner and middle loops are wrapped by fractional transforms. This suggests that TLAO functions as a reusable optimization scaffold, while DQTFP and LDTFP determine the specific continuous-update mechanism for the energy-efficiency objective.

5. Convergence and computational profile

The paper attributes monotonic convergence properties to all three loops. In the inner and middle loops, each alternating step either strictly reduces the objective value in minimization form or increases it in maximization form; lower-boundedness under engineering constraints then guarantees convergence to at least a local optimum. In the outer loop, every RC update is non-decreasing in SE or EE because the coordinate-descent step explicitly selects the best candidate among the tested RC replacements. The paper is explicit that the overall method converges to local optima rather than global optima because the joint problem is non-convex and combinatorial (Li et al., 21 Aug 2025).

The computational burden is dominated by the outer loop because each RC update requires solving the full digital-analog beamformer subproblem. The paper states that this scales with $\mathbf{F}_{\rm BB}$7, the number of RCs per array, and $\mathbf{F}_{\rm BB}$8, the RC candidate pool. Within the inner and middle loops, PDD-HBF/WMMSE requires matrix inversions and, for the FC case, SVD. DQTFP carries higher complexity because it solves convex problems via CVX. By contrast, LDTFP has closed-form updates per iteration except for a simple bisection for the power constraint.

A concrete runtime comparison is provided for a specific scenario: the paper notes “7.7 ms vs. 12.2 s for LDTFP vs. DQTFP.” The stated interpretation is that LDTFP significantly reduces computational complexity with only minor performance loss. This trade-off is especially consequential in the outer loop, where many candidate RC evaluations may be required.

6. Significance, comparison, and interpretive issues

The paper states that the TLAO approach “substantially improves both SE and EE” relative to conventional HBF architectures without RC/EM beamforming and to fully digital approaches with fixed-position antenna arrays. The reasons listed are that additional RC selection allows finer electromagnetic control for maximizing channel gain and minimizing interference, that it can reduce the required active hardware for a given performance level, and that partial connections can closely approach ideal SE and EE with much lower complexity and cost (Li et al., 21 Aug 2025).

The claimed empirical comparison is that simulation results confirm that RCRAA/THBF architectures with TLAO outperform FPA and even high-dimensional FDBF solutions, “especially in scalability and cost.” Because those findings are reported at the level of the paper’s simulation study, a cautious reading is that TLAO is positioned not merely as an optimization routine but as a mechanism for exploiting a broader hardware-design space created by radiation-center reconfigurability.

Several interpretive clarifications follow directly from the formulation. A common misconception would be to regard TLAO as a conventional two-block hybrid beamforming solver with an added discrete heuristic. The paper’s definition is stricter: the EM beamformer is itself part of the THBF architecture and is modeled as RC selection, so the outer loop is not peripheral but constitutive. Another possible misconception is that monotonic improvement implies global optimality. The paper explicitly rejects that conclusion, stating that the non-convex and combinatorial structure yields local optima rather than global ones. A plausible implication is that TLAO should be understood as a principled local-optimization framework whose practical value derives from coordinated updates across three coupled beamforming domains.

At the level of the paper’s internal organization, the main components are identified as follows: problem definition in Eq. (18) for SE and Eq. (42) for EE; PDD-HBF/WMMSE in Eq. (21)–(35) and Algorithm 1; RC selection by COD in Section III.C and Algorithm 2; the overall TLAO framework in Section III.D and Algorithm 3; and DQTFP/LDTFP integrations in Sections IV.B and IV.C with Algorithms 4 and 5. Taken together, these components define TLAO as the organizing optimization structure for joint digital, analog, and EM beamformer design in RCRAA-based THBF.