Utility-Oriented Precoder Design

• The paper introduces a framework that maximizes a system’s utility by optimizing beamforming matrices subject to power, hardware, and fairness constraints.
• It employs diverse algorithmic paradigms—such as Riemannian manifold, symplectic, and weighted MMSE optimization—to handle non-convex, high-dimensional precoding problems.
• Simulations demonstrate significant performance gains, reduced computational complexity, and robust scalability in massive MIMO and FD-MIMO applications.

A utility-oriented precoder construction method refers to the design and optimization of linear precoders (beamforming matrices) to directly maximize a system-level performance metric—termed the “utility function”—which is often a function of the users' achievable rates, error probabilities, or application-specific goals, subject to resource constraints such as power, fairness, or hardware limitations. Rather than relying on myopic, channel-centric, or heuristic designs (e.g., ZF/MRT precoding), the utility-oriented methodology formulates and solves the precoder synthesis problem with the goal of global or multi-objective optimality under the selected utility and operational constraints.

1. Mathematical Principles and General Formulation

The utility-oriented approach first identifies a suitable utility function $U(\cdot)$ mapping system variables to a scalar (or vectors in multi-objective settings), e.g.,

• weighted sum-rate $\sum_k \alpha_k \log_2(1+\gamma_k)$,
• minimum rate $\min_k R_k$,
• proportional fairness $\sum_k \log R_k$,
• application-specific objectives such as frame error rate (FER), mean squared error (MSE), or sensing-comm trade-off surfaces.

The generic utility-optimized precoder design can be written as: $\max_{W \in \mathcal{W}} \; U(\{R_k(W)\}_{k=1}^K)$ subject to structural (e.g., per-antenna/UE/BS power), hardware, or fairness constraints, where $W$ encodes the beamforming/precoding matrices, and $R_k(W)$ is the rate or relevant metric for user $k$ (Björnson et al., 2021, Sun et al., 2023, Lin et al., 31 Jul 2025). When $U$ is differentiable and problem structure permits, this admits convex, quasi-convex, or monotonic programming formulations; but in general, the presence of interference, hardware or coupling constraints renders the problem non-convex and often high-dimensional.

2. Algorithmic Frameworks

Several algorithmic paradigms have been advanced for utility-oriented precoder construction:

• Matrix Manifold Optimization: For weighted sum-rate maximization under total, per-user, or per-antenna power constraints, the feasible set is cast as a Riemannian manifold (sphere, product sphere, oblique), and the problem is solved via Riemannian gradient, conjugate-gradient, or trust-region methods. Closed-form projections, gradients, Hessians, and retractions are derived for each constraint type to enable efficient manifold optimization without large matrix inverses (Sun et al., 2023).
• Symplectic Optimization: In massive MIMO networks (including user-centric clusterings), the sum-utility precoder problem is formulated as a dissipative Hamiltonian dynamical system. By transforming all variables to the real field, the utility (e.g., negative sum-rate) acts as the system potential, and per-antenna/BS constraints are enforced via Lagrange multipliers. Symplectic integrators such as RATTLE, plus explicit damping and adaptive step size rules, yield matrix-inversion-free, parallelizable, and structure-preserving updates that achieve high efficiency and rapid convergence (Lin et al., 31 Jul 2025).
• Weighted MMSE and Dual Optimization: For cognitive radio or SWIPT (simultaneous wireless information and power transfer), and in multicell/MIMO-NOMA, non-convex sum-utility maximization is often reformulated as a weighted MMSE problem (when possible), enabling equivalence between certain utilities (rate, error) and convex MSE surrogates. The zero-duality gap is exploited, and dual-based or ellipsoid subgradient methods are utilized to solve the dual problem efficiently, with closed-form water-filling in special cases (Song et al., 2018, Nguyen et al., 2017).
• Pareto-Optimal and Multiobjective Methods: For vector-valued utilities (e.g., SINR tuples in per-antenna power-constrained MU-MIMO), the Pareto boundary is parameterized with a weight vector $\boldsymbol\nu$, linking each setting to a unique SINR trade-off point. KKT conditions yield explicit mappings from the weight vector to the globally Pareto-optimal precoder, and iterative scaling solves the per-antenna-power constraint with $O(n^3)$ complexity (Petrov et al., 13 Aug 2025).
• Unsupervised Learning and Product Quantization: In high-dimensional FD-MIMO, especially for codebook-based precoding, the utility-oriented codebook construction problem (e.g., minimizing expected mutual information loss) connects to unsupervised clustering on product Grassmann manifolds. Alternating k-means on Cartesian-product structures, with distortion metrics induced by the utility, yields practical codebooks with near-optimal performance but orders of magnitude lower complexity (Bhogi et al., 2021).

3. Utility Function Classes and Problem Families

Utility-oriented precoding is fundamentally shaped by the selected utility:

Utility Function	System Objective	Common Algorithms/Notes
Sum-rate, weighted sum-rate	Network throughput, spectral efficiency	WMMSE, manifold/symplectic, dual-gradient
Proportional fairness	Throughput-fairness trade-off	Water-filling, log-domain splitting, dual, Riemannian
Min-rate / Max-min	User-rate balancing	Linear programming, monotonic programming
Frame/bit error rate	Reliability, URLLC, ISAC	Deep learning, convex surrogate + dual updates
Mutual information bound	Codebook quantization, limited feedback	Model-free clustering on matrix/tensor manifolds
Multiobjective SINR-front	Pareto-efficient trade-off	Explicit KKT-based weight parameterization

Specialized architectures (e.g., product codebooks exploiting array separability (Bhogi et al., 2021), DD-domain CNN+LSTM predictive precoding for FER minimization (Liu et al., 2023), BER-CRB dual utility for OTFS-based ISAC (Wu et al., 2023)) emerge depending on system structure, channel uncertainty, and application context.

4. Complexity, Scalability, and Structural Insights

The principal challenge in utility-oriented precoder construction is computational tractability, especially as system scale or constraint granularity increases. Scalability is achieved by:

• Dimensionality Reduction: Exploiting array separability (e.g., Kronecker structures, Tucker decomposition (Bhogi et al., 2021)), user-centric clustering (Lin et al., 31 Jul 2025), or per-antenna/BS decoupling.
• Product/Manifold Codebooks: Decomposing the high-dimensional search into lower-dimensional factor learning (product codebooks (Bhogi et al., 2021)).
• Closed-form and Iterative Updates: Achieving per-iteration cost near-order optimal (e.g., $O(n^3)$, matching ZF) (Petrov et al., 13 Aug 2025). Manifold and symplectic methods further avoid large matrix inverses.
• Surrogate Optimization: Deploying matrix surrogates (e.g., KKT stationarity with explicit power/SINR balancing, convex under-approximations, etc.) (Nguyen et al., 2017, Song et al., 2018).

Simulations consistently demonstrate that utility-based methods yield:

• Strict Pareto-optimality (for multiobjective problems with explicit weight parametrization) (Petrov et al., 13 Aug 2025).
• Order-of-magnitude faster convergence and reduced CPU burden vs. legacy WMMSE or VQ designs, with negligible loss in utility (Bhogi et al., 2021, Sun et al., 2023, Lin et al., 31 Jul 2025).
• Robustness to system size as $N_t, K$ increase; e.g., symplectic and manifold methods scale efficiently in massive MIMO (Sun et al., 2023, Lin et al., 31 Jul 2025).

5. Generalizations and Adaptability

The utility-oriented paradigm is highly adaptable:

• Any unitarily invariant utility (e.g., trace, determinant, sum-log-eigenvalue) or differentiable function of subspace/projected channel output can replace the loss in the formulation (Bhogi et al., 2021, Wu et al., 2023, Björnson et al., 2021).
• System extensions to hybrid or conformal arrays, joint sensing-communication, and MIMO-OTFS/ISAC domains are tractable by suitable reformulation in tensor, delay-Doppler, or block-diagonal forms (Bhogi et al., 2021, Wu et al., 2023, Liu et al., 2023).
• Jointly optimized designs beyond precoding—e.g., joint training/precoding (Pastore et al., 2012), joint rate-energy (SWIPT) (Song et al., 2018), or integrated pilot+precoder beamforming—fit the same meta-framework.
• Deep learning architectures can be trained end-to-end on utility-derived losses for model-free adaptation under channel prediction or uncertainty (Liu et al., 2023).

6. Key Algorithmic Procedures

Matrix Manifold Riemannian Optimization (WSR Example) (Sun et al., 2023):

1. Initialize precoder $W^0$ on constraint manifold (sphere, product sphere, oblique).
2. Iteratively compute Riemannian gradient, project onto tangent space.
3. Update along Riemannian gradient, with retraction to maintain feasibility.
4. Optionally, use trust-region subproblems or conjugate gradient acceleration.
5. Terminate when utility increment or gradient norm falls below a threshold.

Symplectic Dynamical System (WSR Example) (Lin et al., 31 Jul 2025):

1. Map all complex variables to real field.
2. Construct augmented symplectic Hamiltonian with utility as potential and introduce conjugate momenta.
3. Iterate with symplectic integrator (RATTLE, etc.), alternately updating state and enforcing per-BS constraints.
4. Add friction term for dissipation, adapt step size via normed increments.
5. Stop when relative potential (negative utility) improvement is negligible; recover complex-domain precoder.

Product Codebook k-means (Mutual Information Example) (Bhogi et al., 2021):

1. Project channel tensor onto separate factor Grassmannians via Tucker decomposition.
2. Alternate between assignment (nearest centroid under induced utility-based distance) and centroid update (top-r eigenvectors).
3. Construct aggregate codebook as Cartesian product of factor codebooks.
4. Quantize/unquantize via nearest codewords; evaluate mutual information loss.

7. Representative Case Studies and Benchmarks

• FD-MIMO product codebook design achieves $>$95% of ideal vector-quantization performance using only ~10% of CPU time at typical antenna and bit configurations (Bhogi et al., 2021).
• Symplectic optimization in user-centric networks yields up to +29% WSR gains over WMMSE in large-scale deployments, without large-matrix inversion (Lin et al., 31 Jul 2025).
• Dual-based utility optimization in SWIPT/CR admits global optimality under single-user configuration with convexity and strong duality, while in multiuser cases, local optima with practical convergence rates are typical (Song et al., 2018).
• Deep learning-based utility minimization (FER, BER) can match the performance of ideal (perfect-CSI) oracle mechanisms, outperforming classical linear and MMSE/ZF alternatives under channel uncertainty (Liu et al., 2023, Wu et al., 2023).

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References:

• (Bhogi et al., 2021) Tensor Learning-based Precoder Codebooks for FD-MIMO Systems
• (Björnson et al., 2021) Utility-Based Precoding Optimization Framework for Large Intelligent Surfaces
• (Sun et al., 2023) Precoder Design for Massive MIMO Downlink with Matrix Manifold Optimization
• (Lin et al., 31 Jul 2025) Precoder Design for User-Centric Network Massive MIMO: A Symplectic Optimization Approach
• (Petrov et al., 13 Aug 2025) Per-antenna power constraints: constructing Pareto-optimal precoders with cubic complexity under non-negligible noise conditions
• (Song et al., 2018) Optimal Precoder Designs for Sum-utility Maximization in SWIPT-enabled Multi-user MIMO Cognitive Radio Networks
• (Nguyen et al., 2017) Precoder Design for Signal Superposition in MIMO-NOMA Multicell Networks
• (Wu et al., 2023) Optimal BER Minimum Precoder Design for OTFS-Based ISAC Systems
• (Liu et al., 2023) Deep Learning-empowered Predictive Precoder Design for OTFS Transmission in URLLC
• (Pastore et al., 2012) A Framework for Joint Design of Pilot Sequence and Linear Precoder