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Optimizations with Intelligent Reflecting Surfaces (IRSs) in 6G Wireless Networks: Power Control, Quality of Service, Max-Min Fair Beamforming for Unicast, Broadcast, and Multicast with Multi-antenna Mobile Users and Multiple IRSs

T0 review · 4 major / 5 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read One alternating recipe solves IRS power and fairness beamforming.

desk verdict A genuinely useful IRS problem-formulation paper whose 'efficient algorithms' claim outruns the analysis; referee it, but push for validation or softer claims. read the letter →

arxiv 1908.03965 v2 pith:ZGNDSUXC submitted 2019-08-11 cs.IT eess.SPmath.IT

Jun Zhao This is my paper · ORCID
classification cs.ITeess.SPmath.IT
keywords intelligentreflectingsurfaces6Gwirelessnetworkspowercontrolunderqualityofservicemax-minfairbeamformingsemidefiniterelaxationalternatingoptimizationmulticastphaseshiftdesign
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

Intelligent reflecting surfaces (IRSs) are thin arrays of reconfigurable elements that reflect base-station signals so that they arrive stronger at mobile users, and they are expected to be a part of 6G networks. This paper claims that a single optimization template—alternate between designing the base station's transmit beamformers and the IRS's phase-shift matrix, relaxing the hard rank-one constraints, then recovering feasible solutions by Gaussian randomization—can handle two core problems: minimizing transmit power while meeting per-user signal-to-interference-plus-noise ratio targets, and maximizing the minimum fair SINR. The template is shown to cover unicast, broadcast, and multicast traffic, and to extend to mobile users with several antennas and to systems with several IRSs. Existing IRS studies had addressed only the unicast power-control case, so the paper's contribution is a unified formulation and a concrete algorithmic recipe for the broader design space. If the algorithms deliver what the formulations promise, network designers can use one toolbox to configure IRS-aided downlinks under power, fairness, and traffic-pattern constraints.

What carries the argument

The load-bearing object is the lifted phase-shift matrix V := vv^H, built from v = t[φ; 1] where φ collects the IRS reflection coefficients β_n $e^{{jθ_n}}$ and t is a unit-modulus auxiliary variable. This lifting turns the nonconvex phase-shift feasibility constraint—a product of phase coefficients inside SINR terms—into a linear constraint on V plus a rank-one constraint, which is then relaxed by dropping rank-one and solved as a semidefinite program. The same lifting is reused for the beamformer side with X_k := w_k w_k^H, so that every SINR inequality becomes a trace inequality. Alternating optimization cycles between the W-side SDP and the Φ-side SDP, with Gaussian randomization post-processing to produce rank-one candidates.

What would settle it

Take a small system small enough for brute force—say one base station with two antennas, one IRS with two elements whose phase shifts are discrete, and two single-antenna users—and enumerate every phase configuration to compute the true least transmit power meeting the SINR targets; then run the paper's Algorithm 1 from several random initial phases. If the algorithm's returned power is consistently higher than the brute-force optimum by more than the randomization tolerance, or if it stops with no feasible phase vector when one exists, the central claim of an efficient near-optimal solution is contradicted.

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Extended reading notes

Core claim

The paper's central claim is that joint design of the BS beamforming matrix W and the IRS reflection coefficient matrix Φ can be reduced, for all six problem combinations (power control or max-min fairness, times unicast, broadcast, or multicast), to two alternating semidefinite programs. On the beamforming side, the rank-one beamformers X_k = w_k w_k^H are relaxed to positive semidefinite matrices; on the phase side, the phase vector φ is lifted through v = t[φ; 1] into V = vv^H, whose diagonal entries encode the unit-modulus or amplitude constraints of the IRS. Dropping rank-one constraints makes both subproblems convex, and a final Gaussian-randomization step is used to project the relaxed solutions back to rank-one beamformers and phase shifts. The paper asserts that this recipe works for multi-antenna mobile users, where the channel term becomes a sum over receive antennas, and for multiple IRSs, where all phase vectors are stacked into one lifted variable, and that the max-min fairness problem is solved by introducing the auxiliary fairness level t and using the same alternating machinery. No optimality or convergence proof is given; the contribution is the unified problem formulation and the efficient SDR-based algorithms.

Load-bearing premise

The load-bearing premise is that after dropping the rank-one constraints, the semidefinite relaxation plus Gaussian randomization reliably returns a feasible and near-optimal beamformer and phase shift at each alternating step, even though the paper gives no proof that such a solution exists or that the alternating objective improves monotonically.

Editorial extensions

If this is right

  • The max-min fair QoS problems are solved by the same alternating SDR machinery as power control: introducing the fairness level t turns them into feasibility checks with scaled SINR targets t·γ_i, so no new algorithmic structure is needed.
  • Multi-antenna mobile users reduce to replacing each channel outer product |h_i(Φ)^H w_k|^2 with w_k^H H_i(Φ) w_k, where H_i(Φ) sums over the user's antennas, and all subsequent lifting steps carry through unchanged.
  • Multiple IRSs are handled by stacking the per-surface phase vectors into a single φ and lifting to a larger V, so the number of IRSs changes only the dimension of the semidefinite programs.
  • Discrete phase-shift constraints are accommodated at the final mapping step after Gaussian randomization, because the SDP relaxation itself only fixes the diagonal magnitudes of V.
  • If the SDR solutions are near-optimal, the paper's formulations give a common framework for configuring IRS-aided 6G downlinks under power limits, QoS targets, and fairness criteria.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • A natural, testable extension is to replace the hard stopping rule in Algorithms 1 and 2, which breaks when no randomized phase vector satisfies the SINR constraints, with an adaptive search that scales the number of Gaussian samples Z with the channel dimensions; the paper gives no guidance on how Z should scale, so success probability versus Z and N is an open empirical question.
  • Because the max-min fair problem contains the power-control problem as a limiting case through bisection on t, any approximation guarantee proven for one would transfer to the other; the paper leaves this transfer implicit.
  • The same lifted V formulation should carry over to related IRS objectives such as weighted sum-rate or energy efficiency, since those objectives also enter the SDP only through the same trace inequalities; the paper lists weighted sum-rate as future work without developing the connection.
  • If channel estimation errors are introduced, the alternating SDR template would likely need a robust reformulation; the paper states this as a future direction, so a concrete next step is to replace the exact channel matrices by their estimated versions and measure SINR violation rates.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

4 major / 5 minor

Summary. The manuscript considers an IRS-aided downlink system with a multi-antenna base station and mobile users, and formulates total-power minimization under SINR constraints (power control under QoS) and max-min fair SINR maximization for unicast, broadcast, and multicast traffic. It extends these formulations to multi-antenna users, multiple IRSs, and their combination. The solution approach in Section IV is alternating optimization: for fixed phase shifts, the beamforming subproblem is relaxed by dropping rank-one constraints and post-processed by Gaussian randomization; for fixed beamformers, the phase-shift subproblem is transformed into a semidefinite program in V = vv^H via the auxiliary variable v = t[φ;1], then relaxed and randomized. Algorithms 1-4 instantiate this scheme. The paper claims these are efficient algorithms, with the stated novelty that prior IRS work considered only unicast power control and no IRS work considered max-min fair QoS.

Significance. If the algorithmic claims were supported, the paper would provide a useful unified treatment of IRS beamforming across a broad set of system models; the problem formulations in Tables I-IV are clearly organized and the SDR steps from SINR constraints to trace constraints are mostly correct. The paper also honestly lists approximation bounds and extensive experiments as future work. The central limitation is that the proposed algorithms are heuristics with no convergence, feasibility, or approximation guarantees and no simulations, so the principal claim of 'efficient algorithms' is not yet established. The contribution is therefore best viewed as a formulation-plus-heuristic paper pending substantial additional support.

major comments (4)
  1. [Section IV-A, Algorithm 1 lines 19-20 (and Algorithms 2-4)] The alternating algorithm stops without returning a feasible point when none of the Z randomized phase vectors satisfies the SINR condition (34) (or (47), (65), (80)). No argument is given that such a φ exists whenever the SDR (P1d) is feasible, nor that the objective f^(r) is monotone, nor that the iterates converge; the stopping criterion in line 7 is also not quantified. Since feasible output is the central promise of solving Problems (P1)-(P6), this gap undermines the 'efficient algorithms' claim and must be closed by either a proof under stated assumptions or a clearly labeled heuristic with numerical evidence.
  2. [Section IV-A, after Eq. (39) and after Eq. (48)] The statement that 'appropriate post-processing such as Gaussian randomization ... is applied to convert the candidate solution into a solution which satisfies the rank constraint' is an assertion, not a consequence of the cited references [13]-[14]. For SINR-constrained feasibility and for the multicast W-subproblem, whose hardness the paper itself leaves as future work, Gaussian randomization does not generally produce a feasible rank-one point from a feasible SDR solution. The authors should either prove a feasibility or approximation result for these specific problems or explicitly present the methods as heuristics and support them with simulations.
  3. [Section IV-B, Algorithm 2 lines 4-19; Algorithm 4] In the max-min fair case, t is obtained from the SDR (P4b) before rank-one recovery. After Gaussian randomization on W, the randomized beamformers are not shown to achieve the same t, yet the same t is used in the phase-shift feasibility check (47)/(80). If the randomized W achieves a smaller achievable t, the phase-shift step can fail and the loop breaks at line 19 even when the original problem is feasible. The algorithm should recompute t from the feasible W (e.g., via bisection) or otherwise ensure that the t used in the φ-update is achievable.
  4. [Section VI (Conclusion)] The abstract and the contribution list say 'efficient algorithms' are proposed, but no convergence rate, per-iteration complexity, or numerical demonstration is provided. The conclusion explicitly defers 'proving approximation bounds' and 'conducting extensive experiments' to future work. At minimum, the claims should be scaled back to 'heuristic algorithms' unless complexity and empirical behavior are added.
minor comments (5)
  1. [Section IV-A, after Eq. (21)] The sentence 'Replacing k by j in Eq. (21)' should yield |h_i^H(Φ)w_j|^2 = trace(X_j H_i(Φ)), not trace(X_k H_j(Φ)); the resulting inequality (23) is correct, so this appears to be a typo.
  2. [Section IV-A, text near Eq. (23)] The phrase 'we express the the constraint' contains a duplicated 'the'; similarly, Algorithm 1 line 5 and Algorithm 3 line 5 use 'object function' where 'objective function' is intended.
  3. [Algorithm 1 line 7 and Algorithm 2 line 6] The stopping rule 'the relative difference ... is small' is not a testable condition; a specific tolerance or a normalized convergence criterion should be given.
  4. [Table II, row (P2-MA)] The displayed SINR constraint contains a stray vertical bar and an unmatched absolute-value bracket; the denominator should be a sum of nonnegative terms, not |w_j^H H_k(Φ)w_j + σ_k^2|.
  5. [Section IV-A and IV-B] The paper states that the SDPs can be solved efficiently via [15], but it does not give the dimensions or per-iteration complexity of the SDPs solved in Algorithms 1-4; adding these would help substantiate the 'efficient' claim.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity; the paper's formulations and algorithms are assembled from external cited methods and no output is equivalent to an input by construction.

full rationale

The paper's central derivation chain is: formulate SINR expressions (Eq. (3)-(4)), define power-control and max-min-fair problems (P1)-(P6), then solve each by alternating optimization between the W-subproblem (SDR after dropping rank constraints in Eq. (24d)/(45e)) and the Phi-subproblem (SDR after dropping rank constraints in Eq. (39f)/(48e)), with Gaussian randomization to recover rank-one solutions. These subproblems are solved with methods from Karipidis et al. [12] and Wu & Zhang [4], which are external, citable results, not results derived from the paper's own outputs. The max-min-fair problems are reduced to the power-control feasibility check via the auxiliary variable t in (P4a) and (P4d); this is a standard epigraph reformulation, not a circular equivalence. No parameter is fitted to data, no prediction is statistically forced by a fitted input, and no uniqueness claim is imported from the authors' prior work. The only self-citation is reference [27] (the author's own IRS survey) in the related-work section, used merely as a survey pointer; it is not load-bearing for any derivation. The lack of a formal feasibility or monotonicity guarantee for the alternating SDR-plus-randomization loops in Algorithms 1-4 is a correctness/rigor limitation (the paper itself lists 'proving approximation bounds' as future work), but it is not circularity: the algorithms' outputs are not equal to their inputs by construction. Therefore no circular step is identifiable in this manuscript.

Assumptions & free parameters 3 free parameters · 6 assumptions · 0 invented entities

The central algorithms rest on standard convex optimization (SDR/SDP) plus two unproved heuristics: Gaussian randomization recovers rank-one feasible solutions, and alternating optimization converges. There are no fitted physical parameters, but the algorithm depends on hand-chosen hyperparameters: randomization count Z, convergence threshold, and initialization.

free parameters (3)
  • Gaussian randomization sample count Z = unspecified ('sufficiently large')
    Number of Gaussian randomization trials in Algorithms 1-4; the paper gives no value or scaling rule, so the outcome depends on the implementer's choice.
  • Convergence tolerance = unspecified ('small')
    Stopping criterion in Algorithms 1-4; the relative difference threshold is not quantified, which makes termination depend on an arbitrary choice.
  • Initial IRS phase matrix = random
    Algorithms 1-4 initialize Phi randomly; the alternating scheme may land in different local solutions depending on this choice.
assumptions (6)
  • domain assumption The BS has perfect downlink channel state information for all links (BS-IRS, IRS-MU, BS-MU).
    Section II states the BS obtains h_r,i, H_b,r, and h_b,i at the channel estimation stage via a TDD reciprocity protocol; no estimation error is modeled.
  • domain assumption The IRS is modeled as a diagonal phase-shift or reflection-coefficient matrix with independently controllable elements.
    Equations (1) and (2) define Phi = diag(...); this ignores coupling between IRS elements and mutual impedance effects.
  • domain assumption Transmitted signals are normalized to unit power and noise is additive white Gaussian.
    Section II defines the SINR in Eq. (4) and states signals are normalized to unit power.
  • ad hoc to paper Semidefinite relaxation followed by Gaussian randomization yields a feasible rank-one solution to the beamforming and phase-shift subproblems.
    Section IV-A after Problem (P1c): 'appropriate post-processing such as Gaussian randomization... is applied to convert the candidate solution into a solution which satisfies the rank constraint.' No guarantee is given for the nonconvex multicast SINR constraints.
  • ad hoc to paper Alternating optimization over W and Phi converges to a good stationary or locally optimal solution.
    Algorithms 1-4 alternate without a convergence proof; Section VI lists proving approximation bounds as future work.
  • standard math SDP problems of the form (P1d), (P1-MA-MR-d), etc., can be solved efficiently and the methods of Karipidis et al. [12] apply to the relaxed subproblems.
    Section IV-A: 'Problem (P1d) belongs to semidefinite programming and can be solved efficiently [15]' and 'we can apply methods of [12]'.

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Cite this review

Pith. "Pith review of Optimizations with Intelligent Reflecting Surfaces (IRSs) in 6G Wireless Networks: Power Control, Quality of Service, Max-Min Fair Beamforming for Unicast, Broadcast, and Multicast with Multi-antenna Mobile Users and Multiple IRSs." pith.science (2026). https://pith.science/paper/ZGNDSUXC

@misc{pith2026190803965,
  author       = {Pith},
  title        = {Pith review of: Optimizations with Intelligent Reflecting Surfaces (IRSs) in 6G Wireless Networks: Power Control, Quality of Service, Max-Min Fair Beamforming for Unicast, Broadcast, and Multicast with Multi-antenna Mobile Users and Multiple IRSs},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/ZGNDSUXC}},
  note         = {Machine review of arXiv:1908.03965}
}
read the original abstract

Intelligent reflecting surfaces (IRSs) have received much attention recently and are envisioned to promote 6G communication networks. In this paper, for wireless communications aided by IRS units, we formulate optimization problems for power control under quality of service (QoS) and max-min fair QoS under three kinds of traffic patterns from a base station (BS) to mobile users (MUs): unicast, broadcast, and multicast. The optimizations are achieved by jointly designing the transmit beamforming of the BS and the phase shift matrix of the IRS. For power control under QoS, existing IRS studies in the literature address only the unicast setting, whereas no IRS work has considered max-min fair QoS. Furthermore, we extend our above optimization studies to the novel settings of multi-antenna mobile users or/and multiple intelligent reflecting surfaces. For all the above optimizations, we provide detailed analyses to propose efficient algorithms. To summarize, our paper presents a comprehensive study of optimization problems involving power control, QoS, and fairness in wireless networks enhanced by IRSs.

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Pith tools

Reviewed August 14, 2026 · model on record in the stance chip above.