Sum-Rate Maximization in Wireless Systems

• Sum rate maximization is the process of optimizing multi-user wireless network performance by strategically allocating power, beamforming, and scheduling.
• It employs iterative non-convex optimization techniques such as SCA, SDR, and alternating optimization to tackle SINR coupling and hardware constraints.
• Practical implementations demonstrate significant throughput gains and improved spectral efficiency in MISO/MIMO, RSMA, RIS-assisted, and multi-hop networks.

The sum rate maximization problem is central to the design and optimization of multi-user wireless communication networks. It concerns the optimal allocation of physical-layer resources—such as transmit power, beamforming vectors, scheduling, precoding, and the configuration of auxiliary devices (e.g., RIS, relay, pinching antennas)—to maximize the aggregate throughput summed over all users, typically under practical constraints (power, quality-of-service, hardware, and topology limits). Addressing the sum-rate maximization problem is essential for achieving high spectral efficiency and robust system operation in multi-antenna, multi-cell, and next-generation wireless systems (Li et al., 2023, Gadamsetty et al., 2024, Wang et al., 2019).

1. Fundamental Problem Formulation

The canonical sum-rate maximization problem is typically posed as

$$\max_{\boldsymbol{x}} \sum_{k=1}^K R_k(\boldsymbol{x})$$

where $R_k$ denotes the instantaneous or ergodic achievable rate of user $k$, and $\boldsymbol{x}$ collects all design variables: transmit powers, beamforming vectors, codebooks, auxiliary device parameters (e.g., phase shifts for RIS), and scheduling decisions. The rates $R_k$ are generally of the form

$$R_k = \log_2\left(1 + \frac{S_k(\boldsymbol{x})}{I_k(\boldsymbol{x}) + N_k}\right)$$

where $S_k$ is the received signal power, $I_k$ is the (possibly user-coupled) interference, and $N_k$ is noise. This basic structure is common across MISO/MIMO downlink and uplink (Wang et al., 2019, Tran et al., 2012), MISO interference channels (Li et al., 2010), networks with RIS-assisted channels (Li et al., 2023, Wang et al., 2019), relay networks (Dayarathna et al., 2022, Khabbazibasmenj et al., 2012), and advanced access schemes such as RSMA (Li et al., 2023, Yang et al., 2019).

Sum-rate maximization is almost always non-convex: the SINR expressions are nonlinear and variables are coupled (notably, interference depends on global variable allocations), and design spaces frequently combine continuous and discrete variables (e.g., relay selection, RIS phase quantization).

2. Representative System Models and Sum-Rate Metrics

Several prominent system configurations and their sum-rate objectives are:

• Multiuser MISO/MIMO Downlink and Uplink: Users are served by one or more base stations using beamforming. The sum-rate function is coupled via interference across users (Wang et al., 2019, Tran et al., 2012).
• Rate-Splitting Multiple Access (RSMA): The sum-rate includes both a common part (decoded by all users) and private parts (user-specific), with rate allocation variables subject to common-decoding and QoS constraints (Li et al., 2023, Yang et al., 2019).
• IRS/RIS-Assisted Systems: The sum-rate depends on both conventional beamforming and the (active or passive) RIS phase and amplitude configuration, modeling cascaded channels and often subject to power or hardware constraints (Li et al., 2023, Wang et al., 2019, Hu et al., 2020).
• Relay- and Multihop Networks: Achievable rates are bottlenecked by weakest-hop SINRs; joint relay-path selection and hopwise power allocation is required (Dayarathna et al., 2022).
• Interference and Outage Settings: Weighted or instantaneous sum-rate is maximized given SINR or outage constraints, e.g., in MISO interference channels with channel uncertainty (Li et al., 2010, 0806.2860).
• Novel topologies: Pinching antenna systems (Zhang et al., 23 Apr 2025, Zhou et al., 21 Apr 2025), D2D underlay (Cao et al., 2020), digital-holographic RIS architectures (Gadamsetty et al., 2024).

3. Algorithmic Paradigms and Solution Techniques

Given the non-convexity and high dimensionality, current methods for sum-rate maximization employ iterative, block-coordinate, and surrogate optimization schemes. Major algorithmic paradigms include:

• Alternating Optimization and Block Coordinate Descent: Decompose the variable set into blocks (e.g., BS beamforming, RIS configuration, relay selection) and optimize each block in turn while fixing others. Convergence to stationary/KKT points is typically guaranteed if each subproblem is optimally (or sufficiently) solved (Li et al., 2023, Wang et al., 2019, Gadamsetty et al., 2024, Al-Shatri et al., 2015).
• Sequential Convex Approximation (SCA): Non-convex fractional or bilinear SINR constraints are linearized via Taylor expansions or convex surrogates. Each iteration solves a convex surrogate problem, e.g., QCQP or SOCP, guaranteeing monotonic improvement of the sum-rate objective (Li et al., 2023, Tran et al., 2012, Al-Shatri et al., 2015, Wang et al., 2019, Gadamsetty et al., 2024).
• Semidefinite Relaxation (SDR): Rank relaxation is applied to matrix-valued beamforming variables. Convex SDPs are solved at each step, and rank-reduction or randomization recovers feasible rank-one (precoder) solutions (Hu et al., 2020, Li et al., 2010, Khabbazibasmenj et al., 2012).
• Fractional Programming and Quadratic Transform: SINR-coupled objectives are tackled via quadratic transforms or fractional programming, introducing auxiliary variables to expose convexity or to decouple variables (Li et al., 2023, Zhang et al., 23 Apr 2025, Li et al., 2010).
• Rate–MMSE Equivalence: By expressing the rate in terms of MSE-minimization variables, problem nonconvexity is handled through the alternating closed-form updates of the WMMSE framework (Wang et al., 2019, Al-Shatri et al., 2015, Hashmi et al., 30 Sep 2025).
• Metaheuristics and Discrete Optimization: For problems with large discrete decision sets (e.g., relay selection, antenna/phase configuration), heuristic search methods such as Ant Systems or greedy algorithms provide tractable approximations to otherwise intractable enumerations (Othman et al., 2020, Xiu et al., 2021, Zhou et al., 21 Apr 2025).
• Advanced Learning-Based Approaches: Deep learning (e.g., graph neural networks) is utilized for policy approximation under fading uncertainty and user-rate variability with risk-averse or ultra-reliable objectives (Hashmi et al., 30 Sep 2025).

4. Structure of Constraints and Nonconvexity

Key constraint classes which render sum-rate maximization challenging include:

• Power Constraints: Total (sum), per-antenna, per-branch, and per-relay power budgets; possibly including hardware limits as in active RIS (Li et al., 2023).
• Quality of Service / Rate Constraints: User-specific minimum rates, proportional fairness (rate-splitting), or reliability via outage/CVaR constraints (Li et al., 2023, Li et al., 2010, Hashmi et al., 30 Sep 2025).
• Hardware Constraints: Unit-modulus or discrete phase restrictions on RIS/RHS, binary on/off indicators on relay/IRS (Li et al., 2023, Gadamsetty et al., 2024, Xiu et al., 2021).
• Decoding Constraints: SIC ordering, polymatroid region consistency, and rate allocation coupling in RSMA and multiuser MAC (Li et al., 2023, Yang et al., 2019, Yang et al., 2019).

Nonconvexity arises due to the multi-affine coupling of variables in SINR, the combinatorial nature of scheduling/discrete selections, and constraints which are typically not jointly convex.

5. Illustrative Algorithmic Structure (Active RIS-aided RSMA Example)

The sum-rate maximization for an active RIS-assisted RSMA system (Li et al., 2023) features all the complexities above. The approach is:

1. System model: BS with $N_T$ antennas, $K$ users, active RIS of $L$ elements; RIS precoder is $\Psi=P\Theta$ (phase $\Theta$, amplitude $P$); the effective channel is $$\mathbf h_k^H = \mathbf g_k^H + \mathbf f_k^H\,\Psi\,\mathbf G$$.
2. Optimization Problem $\mathcal P_0$: Maximize $R_{sum} = \sum_{k=1}^K(r_{c,k} + R_{p,k})$ over $\{\mathbf w_0,...,\mathbf w_K\}$, $\mathbf r_c$, and $\Psi$, subject to coupled SINR constraints, rate-allocation, BS and RIS power limits, and minimum rates.
3. Algorithmic Solution:
   • Block 1: For fixed $\Psi$, jointly optimize beamformers and rate split. SCA is applied to linearize (bi)linear terms (e.g., $|\mathbf h_k^H \mathbf w_k|^2$), and the resulting convex QCQP is solved for the BS variables.
   • Block 2: For fixed beamformers and rate split, optimize $\Psi$. This subproblem reduces to a nonconvex QCQP over the diagonal RIS parameters after Lagrangian dual and fractional programming transforms.
   • Alternating steps: Repeat block optimizations until convergence.
   • Convergence: Monotone sum-rate improvement, typically within 10–20 iterations.

6. Practical Performance, Trade-Offs, and Design Guidelines

Empirical and simulation findings across architectures and solution paradigms consistently show:

• RIS/active IRS: Joint optimization of BS beamforming and RIS phase/amplitude control yields sum-rate gains up to 45% over passive RIS, and the benefit amplifies with the number of reflecting elements (Li et al., 2023). A few tens of RIS elements suffice to approach passive-beamforming limits; phase quantization with only 1–3 bits approaches continuous-phase performance (Wang et al., 2019, Hu et al., 2020).
• Multiuser MISO/MIMO and Interference: Fast SCA-based SOCP and SDR algorithms can achieve near-optimal weighted sum-rates rapidly, converging in few iterations and scaling to moderate network size (Tran et al., 2012, Al-Shatri et al., 2015, Li et al., 2010).
• Relay/Multihop Networks: Alternating max-min relay selection with SCA power control increases sum-rate by up to 80% over naive or greedy schemes, especially as the number of hops and relays grows (Dayarathna et al., 2022).
• RSMA/Advanced Access: RSMA architectures enable rate splitting with closed-form or low-complexity allocation, improving sum-rate by 10–21% over NOMA/OFDMA (Li et al., 2023, Yang et al., 2019).
• Antenna- and Placement-Adaptation: Adaptive placement and configuration (PASS, pinching antennas, digital/RHS, or RIS) offers further degrees of freedom, yielding significant throughput gains over conventional static architectures (Zhang et al., 23 Apr 2025, Gadamsetty et al., 2024, Zhou et al., 21 Apr 2025).
• Reliability-Driven Optimization: Sum-rate maximization with risk-aware (CVaR) criteria suppresses deep-fade outages and improves per-user reliability at a controllable sacrificed mean sum-rate (Hashmi et al., 30 Sep 2025).
• Complexity: Per-iteration complexity is typically $O(K^{3.5}N_T^{3.5})$ for SCA-based beamforming, $O(L^{4.5})$ for RIS optimization, and polynomial for block coordinate schemes. Metaheuristic approaches are used for intractable discrete subproblems (Li et al., 2023, Othman et al., 2020, Zhou et al., 21 Apr 2025).

7. Extensions and Open Directions

Sum-rate maximization frameworks have been extended to accommodate:

• Outage probability and stochastic channel models: Convex approximation and sequential SDP approaches efficiently handle outage-constrained sum-rate maximization (Li et al., 2010).
• Self-sustainable, energy-harvesting IRS: Joint mode and phase scheduling with beamformer adaptation achieves near-ideal sum-rates even with low-resolution IRS elements (Hu et al., 2020).
• Large systems and model-based learning: Deep (unfolded) GNNs trained on model-augmented WMMSE loss achieve near-optimal and ultra-reliable (risk-aggregated) sum-rate, facilitating scalable online adaptation (Hashmi et al., 30 Sep 2025).
• Multi-cell, multi-hop, and distributed architectures: Scalable clustering, distributed algorithms, and multi-convex surrogates enable tractable sum-rate maximization in realistic, interference-limited cellular and relay networks (Dayarathna et al., 2022, Al-Shatri et al., 2015, 0806.2860).
• Practical design guidelines: Placement of smart surfaces and relays should balance proximity to transmitters (energy harvesting/coverage) and receivers (signal intensity), with phase quantization, device activation/switching, and user pairing further optimizing performance (Hu et al., 2020, Xiu et al., 2021).

The sum-rate maximization problem remains a central, rich, and evolving area of research driving fundamental advances in network information theory, optimization, and practical wireless system design. The state of the art leverages a diverse spectrum of mathematical, algorithmic, and learning-based approaches tailored to topology and hardware, under increasingly tight performance, reliability, and scalability requirements (Li et al., 2023, Wang et al., 2019, Gadamsetty et al., 2024, Hashmi et al., 30 Sep 2025).