Max-Min Beamforming (MMB)

• MMB is an optimization framework that maximizes the minimum user performance to ensure fairness across rate, SINR, and energy efficiency in multi-user systems.
• It adapts to various architectures—including hybrid mmWave, cell-free massive MIMO, and RIS-assisted systems—by tailoring constraints and optimization variables.
• Algorithmic solutions transform the nonconvex max-min problem into convex subproblems using soft surrogates and alternating updates for efficient fairness enforcement.

Searching arXiv for recent and foundational work on max-min beamforming to ground the article in the cited literature. Max-Min Beamforming (MMB) denotes a class of beamforming designs in which the beamformer is chosen to maximize a worst-case utility—most commonly the minimum user rate or minimum user SINR, but also the minimum received power, minimum user energy efficiency, or a weighted minimum communication-and-sensing utility, depending on the system model. Its defining feature is fairness: the optimization target is not total throughput alone, but the protection of the weakest user, group, target, or service dimension. In the literature represented here, MMB appears in hybrid mmWave/sub-THz downlinks, cell-free massive MIMO, multicell multicast, RIS- and SIM-assisted systems, integrated sensing and communications, visible light communications, SWIPT, and two-way relaying (Zhu et al., 2023).

1. Core definition and fairness criterion

In its strictest form, MMB is the problem of choosing beamforming variables so that the minimum user performance is as large as possible. In the hybrid multi-user mmWave/sub-THz downlink formulation, the objective is

$$\max_{\theta,v^B}\; \min_{k\in\{1,\dots,K\}} r_k(\theta,v^B),$$

with the explicit interpretation that the hybrid beamformer is designed so that the worst-user downlink rate is as large as possible (Zhu et al., 2023). In cell-free massive MIMO, the corresponding fairness target is the worst-user SINR,

$$\underset{W}{\text{maximize}} \quad \underset{k\in\{1,\dots,K\}}{\min}\ \gamma_k,$$

under per-AP power constraints (Zhou et al., 2020). In coordinated multicell multicast, the fairness variable is the minimum received SINR over all users and cells,

$$\max_{\{\mathbf w_i\}_{i=1}^N}\ \min_{\forall i,\forall k} \mathrm{SINR}_{i,k},$$

again under individual power budgets (Xiang et al., 2012).

The same max-min structure persists when the performance metric is not rate or SINR. In RIS fair beam allocation, the objective is to choose RIS phase shifts so that the minimum received power across users or observation points is maximized,

$$\max_{\omega_1,\dots,\omega_N\in[0,2\pi)} \min_k \{\mathbf w^H\mathbf H_k\mathbf w\},$$

with $\mathbf w=[e^{j\omega_1},\dots,e^{j\omega_N}]^H$ constrained to the unit-modulus torus (Xiong et al., 2023). In multicell multiuser joint transmission, the fairness metric becomes user energy efficiency,

$$\max_{\mathbf W}\ \min_n \frac{\log(1+\operatorname{SINR}_n)}{\|\mathbf w_n\|^2+\frac{MP_c+KP_0}{N}},$$

which makes MMB an energy-efficiency analogue of max-min SINR or max-min rate beamforming (He et al., 2013).

A central misconception addressed repeatedly in this literature is that fairness-oriented beamforming merely redistributes loss. The explicit rationale for max-min design is that conventional sum-rate maximization can give some users very low or even zero rates, whereas max-min optimization attempts to make all users’ rates as uniformly high as possible (Zhu et al., 2023). In multicast systems the same logic appears through group decoding: because all users in a multicast group must decode the same stream, the effective group rate is limited by the worst user in that group (Joudeh et al., 2016).

2. Feasible sets, architectures, and constraint classes

MMB is defined not only by its objective but also by the physical structure of the beamforming variables. In hybrid beamforming for mmWave/sub-THz downlink, the transmitter has $N$ antennas, $N_{\mathrm{RF}}$ RF chains, one analog beamformer $V_{RF}$, and one digital/baseband beamformer $v^B$. Under the array-of-subarrays (AOSA) architecture, the analog beamformer is block diagonal and each phase entry is quantized to $$\underset{W}{\text{maximize}} \quad \underset{k\in\{1,\dots,K\}}{\min}\ \gamma_k,$$0-bit resolution, which produces unit-modulus, discrete-phase constraints as the main nonconvexity. Because of the AOSA structure, the transmit-power constraint simplifies to

$$\underset{W}{\text{maximize}} \quad \underset{k\in\{1,\dots,K\}}{\min}\ \gamma_k,$$1

with $$\underset{W}{\text{maximize}} \quad \underset{k\in\{1,\dots,K\}}{\min}\ \gamma_k,$$2 (Zhu et al., 2023).

In IRS- and RIS-assisted formulations, the constraints are phase-only rather than power-allocation-only. In the multicell IRS-aided MISO setting, the reflective beamforming vector satisfies $$\underset{W}{\text{maximize}} \quad \underset{k\in\{1,\dots,K\}}{\min}\ \gamma_k,$$3, while the BS precoders satisfy $$\underset{W}{\text{maximize}} \quad \underset{k\in\{1,\dots,K\}}{\min}\ \gamma_k,$$4 (Xie et al., 2019). In the geometrical-optics RIS beam allocation model, the beamforming vector is explicitly restricted to

$$\underset{W}{\text{maximize}} \quad \underset{k\in\{1,\dots,K\}}{\min}\ \gamma_k,$$5

so the problem is intrinsically nonconvex even before the max-min operator is considered (Xiong et al., 2023). In SIM-aided wave-domain beamforming, the feasible set is defined by layered unit-modulus phase-shift matrices $$\underset{W}{\text{maximize}} \quad \underset{k\in\{1,\dots,K\}}{\min}\ \gamma_k,$$6 and total power $$\underset{W}{\text{maximize}} \quad \underset{k\in\{1,\dots,K\}}{\min}\ \gamma_k,$$7, so the max-min objective is coupled to a multi-layer electromagnetic transformation rather than a conventional digital precoder (Quran et al., 13 May 2025).

Other system classes introduce different constraint geometries. Cell-free massive MIMO uses per-AP power constraints $$\underset{W}{\text{maximize}} \quad \underset{k\in\{1,\dots,K\}}{\min}\ \gamma_k,$$8 or $$\underset{W}{\text{maximize}} \quad \underset{k\in\{1,\dots,K\}}{\min}\ \gamma_k,$$9 (Zhou et al., 2020, Wang et al., 25 Jul 2025). ISAC and JCAS formulations use per-antenna power constraints of the form

$$\max_{\{\mathbf w_i\}_{i=1}^N}\ \min_{\forall i,\forall k} \mathrm{SINR}_{i,k},$$0

while jointly optimizing transmit and receive beamformers (Fang et al., 2024, Ma et al., 2024). RSMA-aided VLC adds nonnegativity, current-range, and electrical-power constraints because the transmitted signal must satisfy optical intensity constraints, making its robust MMF design structurally different from RF beamforming (Qiu et al., 2024). In green Cloud-RAN SWIPT, the max-min rate design is further coupled to ER harvesting constraints, fronthaul-capacity constraints modeled through an $$\max_{\{\mathbf w_i\}_{i=1}^N}\ \min_{\forall i,\forall k} \mathrm{SINR}_{i,k},$$1-type sparsity term, and per-RRH green-energy budgets (Chen et al., 2018).

These formulations show that MMB is not a single optimization template with interchangeable notation. The common element is the max-min fairness criterion; the feasible set is architecture-specific and often determines the dominant source of nonconvexity.

3. Reformulations and algorithmic solution methods

A recurring methodological pattern is to convert the non-smooth, non-convex max-min problem into a sequence of convex or nearly convex subproblems. In the hybrid mmWave/sub-THz setting, discrete analog phases are handled through a penalized formulation with a continuous surrogate $$\max_{\{\mathbf w_i\}_{i=1}^N}\ \min_{\forall i,\forall k} \mathrm{SINR}_{i,k},$$2,

$$\max_{\{\mathbf w_i\}_{i=1}^N}\ \min_{\forall i,\forall k} \mathrm{SINR}_{i,k},$$3

followed by alternating optimization over the digital beamformer, the continuous analog variable, and the quantized phases. Each block update is based on a tight concave minorant of the user-rate expression, and the quantized phase update is closed-form rounding to the nearest $$\max_{\{\mathbf w_i\}_{i=1}^N}\ \min_{\forall i,\forall k} \mathrm{SINR}_{i,k},$$4-bit phase level (Zhu et al., 2023).

When the fairness objective can be written through SINR threshold feasibility, bisection combined with conic programming is standard. In cell-free massive MIMO, the max-min SINR problem is shown to be quasi-concave because the upper-level set $$\max_{\{\mathbf w_i\}_{i=1}^N}\ \min_{\forall i,\forall k} \mathrm{SINR}_{i,k},$$5 is convex after second-order-cone reformulation. This yields a bisection-plus-SOCP solution in which each fixed-$$\max_{\{\mathbf w_i\}_{i=1}^N}\ \min_{\forall i,\forall k} \mathrm{SINR}_{i,k},$$6 feasibility problem is convex (Zhou et al., 2020). A closely related principle appears in coordinated multicast beamforming, where the max-min SINR problem and weighted peak-power minimization are inverse problems; the practical algorithm performs one-dimensional search over $$\max_{\{\mathbf w_i\}_{i=1}^N}\ \min_{\forall i,\forall k} \mathrm{SINR}_{i,k},$$7 and solves a convex SDP feasibility subproblem at each step (Xiang et al., 2012).

For rate-splitting max-min formulations, WMMSE is the dominant tool. In multigroup multicast RS, common and private rates are mapped to MMSE quantities through augmented weighted MSEs, producing an equivalent alternating optimization in which equalizers and weights have closed-form MMSE updates and precoders are optimized via convex programming (Joudeh et al., 2016). The same WMMSE principle underlies the overloaded multigroup multicast RS design, where the common stream and private streams are jointly optimized under a max-min group-rate objective (Joudeh et al., 2017).

Other problem classes motivate different transforms. Fair beam allocation through RIS uses the Moreau-Yosida approximation of the maximum operator to smooth the non-smooth max-min quadratic problem, giving a differentiable surrogate with gradient

$$\max_{\{\mathbf w_i\}_{i=1}^N}\ \min_{\forall i,\forall k} \mathrm{SINR}_{i,k},$$8

and an outer loop that adaptively tightens the approximation parameter $$\max_{\{\mathbf w_i\}_{i=1}^N}\ \min_{\forall i,\forall k} \mathrm{SINR}_{i,k},$$9 (Xiong et al., 2023). Multiuser two-way relaying uses vectorization, semidefinite relaxation, and a primal Dinkelbach-type algorithm to solve the relaxed max-min fractional SDP with Q-superlinear convergence (Fang et al., 2013). Large-scale cell-free MIMO replaces deterministic feasibility solvers by a randomized ADMM that updates only a random subset of beamformer blocks each iteration while preserving an $$\max_{\omega_1,\dots,\omega_N\in[0,2\pi)} \min_k \{\mathbf w^H\mathbf H_k\mathbf w\},$$0 convergence rate (Wang et al., 25 Jul 2025).

Across these methods, convergence claims are typically local or stationary-point-based for alternating schemes, exact only for relaxed problems in SDR-based methods, and globally optimal only in cases where quasi-concavity or inverse convex feasibility is available.

4. Soft max-min surrogates and fairness proxies

Strict max-min optimization is frequently too expensive or too brittle to optimize directly, so several papers replace it with smooth surrogates. In hybrid mmWave/sub-THz beamforming, a soft max-min objective minimizes

$$\max_{\omega_1,\dots,\omega_N\in[0,2\pi)} \min_k \{\mathbf w^H\mathbf H_k\mathbf w\},$$1

where

$$\max_{\omega_1,\dots,\omega_N\in[0,2\pi)} \min_k \{\mathbf w^H\mathbf H_k\mathbf w\},$$2

with small $$\max_{\omega_1,\dots,\omega_N\in[0,2\pi)} \min_k \{\mathbf w^H\mathbf H_k\mathbf w\},$$3. This is described as a smooth surrogate for max-min fairness that approximates the hard min-rate objective while enabling closed-form alternating updates and per-iteration complexity about $$\max_{\omega_1,\dots,\omega_N\in[0,2\pi)} \min_k \{\mathbf w^H\mathbf H_k\mathbf w\},$$4 (Zhu et al., 2023).

A more cautionary case appears in long-term statistical beamforming for full-dimensional massive MIMO. There, the intractable max-min ergodic-rate objective is replaced by max-min SLNR, GM-SLNR, and soft max-min SLNR objectives. The important empirical finding is not merely that these are cheaper, but that fairness proxy choice matters: maximizing the minimum SLNR does not necessarily maximize the minimum ergodic rate, and the simulations show that the SLNR-max-min solution can have very balanced SLNRs but still produce poor ergodic-rate fairness. In the reported results, soft max-min SLNR yields the best fairness and the highest minimum ergodic rate among the proposed methods (Zhu et al., 2023).

ISAC and JCAS formulations use related smoothing ideas for the min operator itself. In monostatic ISAC, simplex variables and softmax updates approximate the worst-user and worst-target operators, preventing oscillation at exact simplex vertices while preserving the intended fairness ordering (Fang et al., 2024). In model-based machine learning for JCAS max-min fairness, the min over users and targets is rewritten through simplex weights and then smoothed by exponential weights $$\max_{\omega_1,\dots,\omega_N\in[0,2\pi)} \min_k \{\mathbf w^H\mathbf H_k\mathbf w\},$$5, after which quadratic transforms yield alternating closed-form updates and a projected transmit-beam update (Ma et al., 2024).

These results support a broader interpretation: max-min fairness is often implemented through a continuum of surrogates, and the choice of surrogate may change both complexity and the operational meaning of fairness. A plausible implication is that in large systems, the tractability of the fairness proxy can be as important as the exactness of the hard min operator.

5. System-specific expansions of the MMB paradigm

One major branch of MMB research concerns overloaded interference-limited systems. In multigroup multicast beamforming, conventional designated-stream MMF beamforming saturates in overloaded regimes because inter-group interference dominates at high SNR when the number of antennas is insufficient for interference nulling. Rate-splitting addresses this by splitting each group message into a common part and a designated part, encoding the common parts into a super common stream that all users decode and cancel before decoding the designated streams. The resulting RS MMF design achieves significant performance gains over the conventional multigroup multicast beamforming strategy (Joudeh et al., 2016). A later overloaded-system treatment sharpens the picture through DoF analysis: classical beamforming can have MMF-DoF $$\max_{\omega_1,\dots,\omega_N\in[0,2\pi)} \min_k \{\mathbf w^H\mathbf H_k\mathbf w\},$$6 in fully overloaded regimes, degraded beamforming guarantees $$\max_{\omega_1,\dots,\omega_N\in[0,2\pi)} \min_k \{\mathbf w^H\mathbf H_k\mathbf w\},$$7, and RS interpolates between designated and degraded transmission with a strictly better DoF profile (Joudeh et al., 2017).

Another branch replaces digital beamforming with programmable surfaces. In RIS fair beam allocation, MMB means choosing RIS phase shifts so that the minimum received power is maximized, enabling explicit beam-splitting, fair beam allocation, and wide-beam generation under a geometrical-optics model (Xiong et al., 2023). In SIM-aided RSMA, the fairness objective becomes the minimum user rate in a fully analog wave-domain beamforming architecture, with alternating optimization between SCA-based power allocation and Riemannian conjugate-gradient optimization of the SIM phases (Quran et al., 13 May 2025). In multicell IRS-aided MISO systems, the max-min metric is the minimum weighted received SINR, jointly optimized over BS beamformers and IRS coefficients via alternating SOCP and SDR/SCA updates (Xie et al., 2019).

A third branch extends max-min design beyond communications-only metrics. In downlink monostatic ISAC, the objective is a weighted combination of the worst communication-user SINR and the worst sensing-target SCNR, so fairness is imposed simultaneously across communication and sensing functionalities (Fang et al., 2024). The related JCAS formulation uses the weighted sum of the minimum communication rate and the minimum sensing term under per-antenna power constraints and then accelerates the optimization by algorithm unfolding (Ma et al., 2024). In RSMA-aided VLC, robust max-min beamforming maximizes the worst-user achievable rate under optical, electrical, and imperfect-CSIT constraints, using CCCP, SDR, and a rank-one penalty (Qiu et al., 2024).

MMB also appears in systems with heterogeneous service constraints. In green Cloud-RAN SWIPT, the objective is the minimum data rate across data receivers while simultaneously satisfying minimum harvested RF energy at energy receivers, fronthaul-capacity limits, and green-energy budgets (Chen et al., 2018). In multicell multiuser joint transmission, the fairness variable is minimum user energy efficiency rather than rate, which directly changes the numerator-denominator structure of the optimization problem (He et al., 2013). These variants show that the max-min principle is portable across performance criteria, but the associated optimization geometry can change substantially.

6. Theory, benchmarks, and empirical regularities

Several theoretical results clarify what MMB can and cannot guarantee. In cell-free massive MIMO, the max-min SINR design solved by SOCP is explicitly presented as an optimum beamforming benchmark, identifying the best achievable beamforming performance under the assumed CSI and power constraints (Zhou et al., 2020). In large-system coordinated cellular beamforming, the KKT analysis reveals a nested zero-forcing structure: some cells behave selfishly with generalized MMSE-like beamformers, while others become altruistic and zero-force interference toward the selfish group, with a recursive reduced-dimension structure in the large-system limit (Zakhour et al., 2012). In coordinated multicell TDD beamforming without data sharing, the objective is not only max-min fairness but a point on the Pareto boundary with max-min fairness, and the proposed two-step centralized algorithm is provably max-min Pareto optimal for the two-BS, two-user case (Huang et al., 2012).

Empirical results also complicate the simple fairness-versus-throughput dichotomy. In hybrid mmWave/sub-THz beamforming, soft max-min optimization not only approaches the minimum user rate of strict max-min optimization but also achieves a sum rate similar to that of sum-rate maximization; the same paper reports that sum-rate maximization alone can produce zero-rate users, that 3-bit phase quantization performs close to infinite-resolution in many cases, and that AOSA often performs well under the same total power budget (Zhu et al., 2023). In RIS fair beam allocation, the Moreau-Yosida method outperforms Fminimax and QuantRand and is substantially faster in the reported comparisons (Xiong et al., 2023). In ISAC, the low-complexity first-order design is reported to be about $$\max_{\omega_1,\dots,\omega_N\in[0,2\pi)} \min_k \{\mathbf w^H\mathbf H_k\mathbf w\},$$8 faster when $$\max_{\omega_1,\dots,\omega_N\in[0,2\pi)} \min_k \{\mathbf w^H\mathbf H_k\mathbf w\},$$9 and $\mathbf w=[e^{j\omega_1},\dots,e^{j\omega_N}]^H$0 faster when $\mathbf w=[e^{j\omega_1},\dots,e^{j\omega_N}]^H$1, while remaining close to the standard FP benchmark in fairness performance (Fang et al., 2024). In large-scale cell-free MIMO, randomized ADMM reduces per-iteration cost from $\mathbf w=[e^{j\omega_1},\dots,e^{j\omega_N}]^H$2 to $\mathbf w=[e^{j\omega_1},\dots,e^{j\omega_N}]^H$3 and retains the same $\mathbf w=[e^{j\omega_1},\dots,e^{j\omega_N}]^H$4 convergence order as deterministic ADMM (Wang et al., 25 Jul 2025).

A final point concerns fairness proxies and benchmark interpretation. The long-term massive-MIMO study shows that a design with the highest minimum SLNR can still yield the lowest minimum ergodic rate, whereas soft max-min SLNR produces the most balanced ergodic-rate distribution (Zhu et al., 2023). This suggests that MMB should be interpreted at the level of the optimized metric actually used in the objective. Fairness in minimum SINR, fairness in minimum rate, fairness in minimum received power, and fairness in minimum energy efficiency are related but not interchangeable, and the beamforming literature repeatedly treats the distinction as structurally important rather than cosmetic.