WMMSE Algorithm for Wireless Resource Allocation

• WMMSE algorithm is a technique that reformulates complex weighted sum-rate problems into tractable MSE minimization using auxiliary variables and closed-form updates.
• It employs a block-coordinate descent scheme to iteratively update MMSE receivers, weights, and transmit strategies, ensuring convergence to a stationary point.
• Recent variants integrate deep learning and matrix-free methods, reducing computational complexity and enabling efficient use in MU-MIMO, OFDM, and modern wireless systems.

The Weighted Minimum Mean Square Error (WMMSE) algorithm is a cornerstone technique for resource allocation and transceiver design in modern wireless communications, particularly for solving the nonconvex weighted sum-rate (WSR) maximization problem. By transforming the WSR objective through auxiliary variables and block-coordinate descent, WMMSE admits iterative closed-form updates for transmit and receive strategies. Its fundamental principles, mathematical equivalence, algorithmic generalizations, and recent integration into deep learning architectures enable efficient, scalable, and near-optimal solutions for MU-MIMO, OFDM, and emerging network paradigms.

1. Mathematical Foundation and Equivalence to WSR

WMMSE reformulates the original WSR maximization—such as $$\max_{\mathbf{p}} \sum_i \lambda_i \log_2(1 + \mathrm{SINR}_i(\mathbf{p}))$$ under explicit power constraints—into a weighted sum-MSE minimization via auxiliary variables: transmit scalars/beamformers %%%%1%%%%, MMSE filters $u_i$, and positive weights $w_i$ (Yang et al., 2023). For multiuser MIMO,

$$\min_{u, v, w}\; \sum_{i=1}^N \lambda_i \left[w_i e_i(u,v) - \log w_i \right], \quad 0 \leq v_i^2 \leq p_{\max}$$

where the per-user MSE is

$$e_i(u, v) = (1 - u_i h_{ii} v_i)^2 + \sum_{j \neq i} (u_i h_{ij} v_j)^2 + \sigma^2 u_i^2$$

and $v_i$ typically denotes the square root of transmit power for user $i$.

The transformation holds generally for both SISO and MIMO setups through block-convex surrogates; stationary points of the WMMSE minimization are stationary points of the original WSR maximization (Pellaco et al., 2022, Gao et al., 23 Oct 2025).

2. Block-Coordinate Descent Iterative Scheme

The WMMSE algorithm iterates over three blocks:

• MMSE receiver update: For each user, compute

$$u_i^{(k)} = \frac{h_{ii} v_i^{(k-1)}}{\sum_{j=1}^N |h_{ij}|^2 (v_j^{(k-1)})^2 + \sigma^2}$$

• Weight update: Set

$$w_i^{(k)} = \frac{1}{e_i(u^{(k)}, v^{(k-1)})} = [1 - u_i^{(k)} h_{ii} v_i^{(k-1)}]^{-1}$$

• Transmit power/beamformer update: Solve (via KKT stationarity)

$$v_i^{(k)} = \left[ \frac{\lambda_i u_i^{(k)} h_{ii} w_i^{(k)}}{\sum_{j=1}^N |h_{ji}|^2 (u_j^{(k)})^2 w_j^{(k)}} \right]_{0}^{\sqrt{p_{\max}}}$$

or, for MIMO/beamforming, use matrix equations for transmit and receive filters (Pellaco et al., 2022, Gao et al., 23 Oct 2025).

The algorithm proceeds until the sum-rate increment or sum-MSE decrement is less than a threshold or up to a fixed iteration count. Block convexity ensures monotonic objective improvement and convergence to a stationary point (Zhang et al., 2023).

3. Computational Complexity and Scalable Variants

Classical WMMSE in MIMO contexts incurs $\mathcal{O}(M^3)$ complexity due to matrix inversions/bisections in each transmit update (for $M$ base-station antennas). Several recent advances reduce this cost:

• Matrix-inverse-free WMMSE: Employs gradient descent for transmit/receive update and Schulz iteration for weight matrix inverses (quadratic convergence), replacing hard-to-parallelize matrix operations with GEMMs and enabling real-time hardware mapping (Pellaco et al., 2022).
• Accelerated Mixed Weighted-Unweighted MMSE (A-MMMSE): Uses projected block gradient descent (BCGD) with extrapolation and two-stage warm-start. Per-iteration complexity is downgraded to $\mathcal{O}(KM^2d)$ and highly parallelizable, offering significant speedup on GPU/FPGA (Gao et al., 23 Oct 2025).
• Reduced-Complexity Algorithms: R-WMMSE and PAPC-WMMSE leverage low-dimensional subspace structures and recursive per-antenna updates, achieving linear scaling in $M$ and making massive MU-MIMO tractable (Zhao et al., 2022, Yoo et al., 2024).
• Functional WMMSE: For continuous aperture arrays, functions replace finite-dimensional vectors/matrices, with all integrations discretized into weighted matrix products via quadrature, preserving closed-form update structure (Chen, 21 Sep 2025).

Algorithm Variant	Complexity per Iteration	Parallelizability
Classical WMMSE	$O(M^3)$	Low
Matrix-inverse-free WMMSE	$O(M^2)$	High
A-MMMSE	$O(KM^2d)$	Very high (GPU)
R-WMMSE, PAPC-WMMSE	$O(M)$	High
Functional WMMSE (CAPA)	$O((Kn_A)^3)$	Medium

4. Deep Unrolling and Graph Neural Networks

Modern deployments seek millisecond-level latency and topology-generalization, motivating hybrid deep learning approaches:

• Deep Unrolled WMMSE: Each iteration maps to a neural-network layer, typically using GNNs for D2D/graph-topology scenarios. Aggregation mirrors channel-aware summation, and update blocks encode the local power calculation steps. This "knowledge injection" substantially reduces sample complexity, training epochs, and inference latency (Yang et al., 2023, Pellaco et al., 2020, Wang et al., 19 Jun 2025).
• RL-driven Deep Unfolding (RLDDU-Net): In wideband MU-MIMO-OFDM, SWMMSE updates are implemented as learnable DU layers, featuring compensation matrix adaptation via reinforcement learning. This exploits beam-domain sparsity and subcarrier correlations for accelerated convergence and robust power allocation under imperfect CSI (Wang et al., 19 Jun 2025).

Empirical results indicate unrolled WMMSE architectures can achieve or surpass traditional WMMSE with only a few layers, generalize seamlessly across graphs or user counts, and require significantly less computation in dynamic environments.

5. Connections to Quadratic Transforms and MM Algorithms

The WMMSE algorithm is subsumed within the quadratic transform (QT) and minorization-maximization (MM) frameworks:

• Fractional Programming View: QT decouples nonconvex log-SINR ratios with auxiliary variables; block-coordinate MM steps yield the standard WMMSE update rules (Shen et al., 2023).
• Accelerated WMMSE: By recognizing the WMMSE block-coordinate step as an implicit gradient projection, Nesterov's extrapolation yields $O(1/k^2)$ convergence (iteration error), improving on the $O(1/k)$ rate of vanilla WMMSE (Shen et al., 2023).
• WSR-FP, WSR-MM Connections: WMMSE, WSR-FP (fractional programming), and WSR-MM (minorization-maximization) are algorithmically equivalent under specific transforms and surrogate constructions. Enhanced variants (WSR-MM+, WSR-FP+) eliminate all matrix inversions via isotropic quadratic surrogates, producing efficient gradient-projection updates (Zhang et al., 2023).

6. Practical Applications and Recent Extensions

The WMMSE paradigm underpins a spectrum of wireless optimization problems:

• MU-MIMO, OFDM, V2X, D2D: Resource allocation, joint beamforming, and power-control in both centralized and distributed settings (Gao et al., 2019, Yang et al., 2023, Feng et al., 2024, Gao et al., 23 Oct 2025).
• RIS-assisted and cell-free networks: Joint transceiver and phase-shift design with embedded WMMSE; operator-valued free probability enables deterministic equivalent AO steps for cell-free massive MIMO with stringent fronthaul constraints (Choi et al., 2024, Pan et al., 2024).
• ISAC (Integrated Sensing and Communication): MI-based WMMSE approaches tune beamforming for trade-offs between communication rate and sensing performance (Peng et al., 2023, Shen et al., 2023).
• Symbol-level precoding: WMMSE naturally arises as the special case of constructive interference region-based formulations with closed-form updates (Wang et al., 2022).

Application Area	WMMSE Role	Key Outcome
MU-MIMO/OFDM	Sum-rate maximization, robust precoding	Near-optimal rate, scalable
Graph-based D2D/GNN	Knowledge injection, rapid inference	Generalization, low-latency
RIS/CF networks	Joint transceiver/phase, statistical AO	Fronthaul reduction, SE gain
ISAC	Trade-off optimization, MMSE design	Sensing-communication trade
V2X/resource allocation	BCD WMMSE, DNN training targets	Real-time throughput boost

7. Convergence Theory and Limitations

Convergence to a stationary point is guaranteed under mild regularity: objective convexity in each block, continuity, and compact feasible sets (Pellaco et al., 2022, Zhang et al., 2023). Empirical convergence is rapid—often in $\sim$10–20 iterations—but the solution is in general a local optimum due to nonconvex WSR structure.

Open issues include:

• Non-global optimality in strongly coupled, multiuser interference channels
• Cubic complexity without adaptation for extremely large-scale arrays (addressed by R-WMMSE, deep unfolding, beam-domain approximation)
• Dependence on channel model and statistical stationarity for robustness under mobility, aging, or pilot contamination

• (Yang et al., 2023): Knowledge-Driven Resource Allocation for D2D Networks: A WMMSE Unrolled Graph Neural Network Approach
• (Pellaco et al., 2022): A matrix-inverse-free implementation of the MU-MIMO WMMSE beamforming algorithm
• (Gao et al., 23 Oct 2025): An Accelerated Mixed Weighted-Unweighted MMSE Approach for MU-MIMO Beamforming
• (Feng et al., 2024, Zhao et al., 2022, Pan et al., 2024, Zhang et al., 2023, Choi et al., 2024, Peng et al., 2023, Yoo et al., 2024, Gao et al., 2019, Wang et al., 2022, Pellaco et al., 2020, Chen, 21 Sep 2025, Shen et al., 2023, Wang et al., 2022, Wang et al., 19 Jun 2025)

The WMMSE algorithm, through algebraic transformations, scalable iterative updates, and domain-aligned learning architectures, remains one of the most versatile and effective tools for wireless system optimization and resource management.