Optimal User and Target Scheduling, User-Target Pairing, and Low-Resolution Phase-Only Beamforming for ISAC Systems

Luis F. Abanto-Leon; Setareh Maghsudi

REVIEW 2 major objections 3 minor 23 references

Optimal User and Target Scheduling, User-Target Pairing, and Low-Resolution Phase-Only Beamforming for ISAC Systems

T0 review · 2 major / 3 minor · reviewed 2026-08-10 · deepseek-v4-flash

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Pith's one-line read Joint scheduling, pairing, and low-resolution beamforming in ISAC exactly reduces to a MILP.

desk verdict The joint ISAC formulation is new, but the claimed exact MILP reformulation uses mis-scaled big-M constants and is a strict restriction of the original problem. read the letter →

arxiv 2501.11593 v1 pith:BLUQJZ4N submitted 2025-01-20 eess.SP cs.ITcs.NImath.IT

Luis F. Abanto-Leon , Setareh Maghsudi This is my paper · ORCID

classification eess.SPcs.ITcs.NImath.IT

keywords ISACintegratedsensingandcommunicationsresourceallocationbeamformingdiscretephaseshiftsschedulinguser-targetpairingMILPreformulation

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

This paper studies a joint resource-allocation problem for an integrated sensing and communications (ISAC) base station: which users to schedule, which targets to sense, how to pair each sensed target with exactly one served user, and what few-bit constant-modulus phase-only beamforming vectors to use. The resulting optimization problem is a nonconvex mixed-integer nonlinear program (MINLP). The paper's central claim is that this MINLP can be transformed step by step into an equivalent mixed-integer linear program (MILP), so that any globally optimal solution of the MILP is also globally optimal for the original problem. This gives an exact, non-heuristic route to optimal joint design in a regime where existing approaches rely on approximations or decoupled stages. Simulations indicate that the optimal joint design consistently outperforms four heuristic stage-wise baselines, whose relative ordering changes with scenario characteristics.

What carries the argument

The load-bearing device is the reformulation chain itself, and its main components are: binary phase-selection vectors $x_{u,n}$ that pick one of the $L$ constant-modulus phases per antenna element; lifted matrices $Y_{u,n,m} = x_{u,n}x_{u,m}^{T}$, whose integrality is guaranteed by total unimodularity after relaxing entries to $[0,1]$; big-M constants $D_t = (P_{\mathrm{tx}}/K)\mathrm{Tr}(G_t)$ and $B_u = P_{\mathrm{tx}}\mathrm{Tr}(\tilde{H}_u)+1$ that switch off sensing and SINR constraints when scheduling or pairing indicators are zero; and auxiliary binary products $\pi_{t,q}$ that linearize $\lambda_t\lambda_q$. Together these variables and inequalities eliminate every nonconvexity and nonlinear coupling, leaving a linear objective and linear constraints.

What would settle it

Solve the original problem P and the reformulated problem Q on a minimal instance with a single user, a single weak target, and $N=1$ antenna. If the optimal $\tau$ in P can exceed $D_t = (P_{\mathrm{tx}}/K)\mathrm{Tr}(G_t)$ while Q's constraint J1 with $\lambda_t=0$, $\rho_{u,t}=0$ forces $\tau \le 2D_t$, the claimed solution-space preservation fails. Similarly, with two targets where $D_q > D_t$, check whether an optimal solution that illuminates target q while pairing the user with target t is excluded by K5; if so, Q is not equivalent to P.

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Extended reading notes

Core claim

The core discovery is a chain of eight equivalent reformulations (Propositions 1-8) that map the nonconvex MINLP P onto the MILP Q. Discrete-phase beamforming is encoded with binary phase-selection vectors; SINR constraints are linearized by a big-M bound and a rank-one lifting that is exact; products of binary variables are replaced by auxiliary matrices whose integrality follows from total unimodularity; and sensing power-gain and cross-interference constraints are rewritten with big-M constants derived from a trace inequality. At each step the original solution space is preserved, so an optimal solution of Q is claimed to be optimal for P as well. Solving Q by branch-and-cut yields the globally optimal schedules, pairings, and low-resolution beamformers.

Load-bearing premise

The reformulation is exact only if the big-M constants $D_t$ are large enough to deactivate the sensing constraints when a target is unscheduled or a pair is inactive; if a desired sensing level $\tau$ exceeds $D_t$ for a weak target, or the needed bound for a cross-interference term is $D_q$ rather than $D_t$, the MILP becomes a strict restriction of the original problem.

Editorial extensions

If this is right

A practitioner can now compute provably optimal schedules, pairings, and low-resolution beams for ISAC systems of moderate size, replacing heuristic stage-wise designs.
The advantage of joint optimization is scenario-dependent: the best heuristic baseline changes between correlated and uncorrelated user channels, while the optimal design is consistently best.
Higher phase resolution improves attainable sensing accuracy, as shown by the Cramér-Rao bound on angle-of-arrival estimation, and the optimal design exploits this automatically.
The linear structure of the reformulated problem lets branch-and-cut solvers reach the global optimum in a small fraction of the exhaustive-search complexity.

Reading between the lines

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

The exactness claim implicitly requires the big-M constants to be upper bounds in all inactive cases; a natural stress test is to verify on random instances whether Q's optimum ever violates a constraint of P.
Because the reformulation is exact, the MILP optimum can serve as a certificate against which any heuristic can be measured, allowing one to quantify the suboptimality gap of stage-wise designs in a given scenario.
The lifting-and-total-unimodularity technique is transferable to other discrete-phase ISAC resource-allocation problems, such as many-to-one pairing or multiple users per target, which the paper notes as possible extensions.
The Cramér-Rao analysis suggests that phase resolution, scheduling, and pairing interact; an extension could treat phase resolution as an optimizable variable rather than a fixed parameter.

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Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

Straight to it: the paper's central claim is wrong. The MILP Q is not an exact reformulation of P. The big-M constants in Proposition 7 (J1) and Proposition 8 (K5) are per-target DPG bounds, but the inactive cases require bounds on tau and on other targets' DPGs. In J1, with lambda_t=0 and rho_{u,t}=0, the constraint becomes Tr(G_t W_u) >= tau - 2D_t. D_t bounds the left side, not tau, and tau can be much larger than D_t when target t is weak and unscheduled. The single-antenna example is decisive: Ptx=10, alpha1=1, alpha2=0.1, true optimum tau=10, MILP caps tau at 3 via the weak target's constraint. K5 has the same scaling problem with D_t instead of D_q. So Q is a strict restriction of P, and the claimed global optimality for P does not follow.

What is genuinely new is the joint problem. The paper is the first, as far as it cites, to put user scheduling, target scheduling, pairing, and few-bit phase-only beamforming in one optimization. That is a legitimate modeling contribution, and the qualitative result that the best heuristic depends on the scenario (BL2 wins with correlated users, BL1 wins when users are separated) is believable and useful. The reformulation machinery (big-M, RLT, total unimodularity) is applied competently and the proofs are mostly complete; Proposition 6 is a sketch, but the TU argument is plausible.

The simulation section has no code, no data, no error bars, and the measured gains are all relative to heuristics whose suboptimality is expected. The real issue is that the 'optimal' numbers are not optimal for the stated problem. That kills the paper's main claim, though the framework might be salvageable as a heuristic or a relaxation.

I'd send this to a serious referee because the problem is timely and the error is specific and fixable in revision. But the exactness claim as written should not be published. Not a paper I'd cite for optimality; the formulation might be worth citing for the problem statement if the authors fix the scaling.

Referee Report

2 major / 3 minor

Summary. The paper studies joint user/target scheduling, user-target pairing, and low-resolution phase-only beamforming for an ISAC downlink. The problem is formulated as a nonconvex mixed-integer nonlinear program (MINLP), and the authors propose an exact mixed-integer linear program (MILP) reformulation through a chain of eight equivalence propositions. The reformulation is intended to preserve the original solution space and thus yield a globally optimal solution to the MINLP. The paper also provides simulations comparing the proposed optimal design against heuristic scheduling/pairing baselines.

Significance. If correct, the exact MILP reformulation would be a valuable contribution: it would allow global optimality for a practically relevant ISAC resource-allocation problem and provide a benchmark for heuristic designs. The paper is clearly written and the reformulation machinery (big-M linearizations, total unimodularity arguments) is standard. However, the central claim of exactness is load-bearing, and it is false as stated. The simulation results are therefore not evidence for the optimality of the proposed method.

major comments (2)

[Proposition 7, constraint J1] The big-M constant D_t = (Ptx/K)Tr(G_t) is an upper bound on Tr(G_t W_u), but the linearized constraint J1 also needs a bound on the auxiliary variable τ. In the inactive case λ_t=0 (and hence ρ_{u,t}=0 by C6), J1 reduces to Tr(G_t W_u) ≥ τ − 2D_t. Since τ can be much larger than D_t when a strong target is scheduled while a weak target t is unscheduled, this inequality is not automatically satisfied and cuts feasible solutions. Concrete counterexample: N=1, K=1, Ptx=10, U=1, T=2, J=1, α1=1, α2=0.1, with the single user paired with target 1. The original problem P achieves τ=10, but J1 for the unscheduled target 2 gives 1 ≥ τ − 2, so τ ≤ 3 in Q. Thus Q is a strict restriction of P, and the statement that an optimal solution to Q is also optimal to P is false.
[Proposition 8, constraint K5] The same mis-scaled big-M constant appears in K5: Tr(G_q W_u) ≤ ξ_th + (2 − π_{t,q} − ρ_{u,t}) D_t. The left-hand side is bounded by D_q = (Ptx/K)Tr(G_q), not by D_t. When D_q > D_t, the constraint for an unpaired or unscheduled strong target q can become active, excluding beams that are perfectly feasible in the original problem P. For example, if λ_q=1 but ρ_{u,t}=0 or π_{t,q}=0, the right-hand side may be smaller than the maximum possible value of Tr(G_q W_u), so the reformulation further restricts the feasible set. This confirms that the error is not isolated to J1 but affects the full reformulation.

minor comments (3)

[Abstract and Section I] There is a typo: 'commmunications' appears in the abstract and again in the first line of Section I.
[Fig. 5 and Scenario IV] The paper reports specific Cramér-Rao bound values for different phase resolutions but does not describe how these bounds are computed for the given beampatterns; a brief derivation or reference would help reproducibility.
[Appendix, proof of Proposition 6] The proof invokes total unimodularity to justify relaxing H3 to I3, but the argument that the row/column sums force the outer product can be made directly; the current wording is slightly misleading.

Circularity Check

0 steps flagged · score 1.0 of 10

No significant circularity: the MILP reformulation is a self-contained transformation chain, and the only self-citation is a non-load-bearing reference to a standard big-M technique.

full rationale

The paper's central claim is that the nonconvex MINLP P is equivalently recast as the MILP Q via Propositions 1 through 8. Each proposition transforms a constraint using binary encodings, matrix lifting, or big-M linearization, and none of these steps defines the target quantity (e.g., the optimal value tau or the scheduling/pairing decisions) in terms of the quantity it is supposed to predict. There is no data fitting, no calibrated parameter that is later called a prediction, and no benchmark result that is used to infer the model parameters. The only self-citation is reference [10], which is cited in the proofs of Propositions 2, 7, and 8 as a source for the big-M method. That citation is not load-bearing: the big-M technique is a standard textbook transformation, the paper states the transformation explicitly, and the equivalence claimed in the propositions does not depend on the correctness or content of the authors' prior work. The potential flaw noted by the skeptic, namely that the big-M constant D_t in Proposition 7 may not dominate tau when a target is unscheduled, is a mathematical correctness issue in the claimed equivalence; it is not a circular argument in which the output reduces to the input by construction. Accordingly, no circular step is identified.

Assumptions & free parameters 0 free parameters · 7 assumptions · 0 invented entities

The reformulation relies on standard mathematical tools (trace inequalities, cyclic trace property, binary-product linearization, total unimodularity) and on the physical system model assumptions. No new entities or fitted parameters are introduced. The critical flaw identified in this review is not in these axioms but in the magnitude of the big-M constants, which are derived from the model but used incorrectly.

assumptions (7)

standard math Trace inequality Tr(AB) ≤ Tr(A)Tr(B) for PSD matrices (Lemma 1)
Used to derive the big-M bounds B_u and D_t in Propositions 2 and 7; cited to [23].
standard math Cyclic property of trace Tr(ABCD)=Tr(BCDA) (Lemma 2)
Used to convert quadratic forms into linear trace expressions in Propositions 2, 7, and 8.
standard math Linearization of a product of binary variables via c≤a, c≤b, c≥a+b−1, c∈[0,1] (Lemma 3)
Used in Proposition 8 to replace π_{t,q}=λ_t λ_q with constraints K1-K4.
standard math Total unimodularity of the multiple-choice and transportation constraint matrix
Used in Proposition 6 to relax binary Y to continuous [0,1] without changing the solution space; cited to [22, ch.13].
standard math Validity of the big-M method for linearizing products of binary and continuous variables
Used in Propositions 2, 7, and 8; the method is standard, but the paper's choice of constants is flawed.
domain assumption Monostatic co-located radar with identical AoD/AoA and point-target model
The sensing model in Section II assumes G_t = α_t a(θ_t) a^H(θ_t) and equal departure and arrival angles.
domain assumption Rician fading channel and 3GPP UMa path-loss model
Simulation channel model; LoS angles are randomized with a fixed separation Δ, which is a modeling choice.

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Cite this review

Pith. "Pith review of Optimal User and Target Scheduling, User-Target Pairing, and Low-Resolution Phase-Only Beamforming for ISAC Systems." pith.science (2026). https://pith.science/paper/BLUQJZ4N

@misc{pith2026250111593,
  author       = {Pith},
  title        = {Pith review of: Optimal User and Target Scheduling, User-Target Pairing, and Low-Resolution Phase-Only Beamforming for ISAC Systems},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/BLUQJZ4N}},
  note         = {Machine review of arXiv:2501.11593}
}

read the original abstract

We investigate the joint user and target scheduling, user-target pairing, and low-resolution phase-only beamforming design for integrated sensing and communications (ISAC). Scheduling determines which users and targets are served, while pairing specifies which users and targets are grouped into pairs. Additionally, the beamformers are designed using few-bit constant-modulus phase shifts. This resource allocation problem is a nonconvex mixed-integer nonlinear program (MINLP) and challenging to solve. To address it, we propose an exact mixed-integer linear program (MILP) reformulation, which leads to a globally optimal solution. Our results demonstrate the superiority of an optimal joint design compared to heuristic stage-wise approaches, which are highly sensitive to scenario characteristics.

Figures

Figures reproduced from arXiv: 2501.11593 by the authors.

**Figure 2.** Performance as a function of transmit power. [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗

**Figure 3.** Performance as a function of SINR threshold in correlated channels. [PITH_FULL_IMAGE:figures/full_fig_p004_3.png] view at source ↗

Figures from the paper (2 more)

**Figure 4.** Figure 4: Performance as a function of SINR threshold in uncorrelated channels. [PITH_FULL_IMAGE:figures/full_fig_p004_4.png]

**Figure 5.** Figure 5: Beampatterns for scheduled and paired users and targets. [PITH_FULL_IMAGE:figures/full_fig_p004_5.png]

Discussion (0). Continue with ORCID to comment.

Reference graph

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Pith tools

Reviewed August 10, 2026 · model on record in the stance chip above.

[1] [1]

Joint radar and communication design: Applications, state-of-the-art, and the road ahead,

F. Liu, C. Masouros, A. P. Petropulu, H. Griffiths, and L. Hanzo, “Joint radar and communication design: Applications, state-of-the-art, and the road ahead,” IEEE Trans. Commun., vol. 68, no. 6, pp. 3834–3862, 2020

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T. Mao, J. Chen, Q. Wang, C. Han, Z. Wang, and G. K. Karagiannidis, “Waveform design for joint sensing and communications in millimeter- wave and low terahertz bands,” IEEE Trans. Commun., vol. 70, no. 10, pp. 7023–7039, 2022

work page 2022

[3] [3]

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S. Dutta, C. N. Barati, D. Ramirez, A. Dhananjay, J. F. Buckwalter, and S. Rangan, “A case for digital beamforming at mmWave,” IEEE Trans. Wireless Commun., vol. 19, no. 2, pp. 756–770, 2020

work page 2020

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W. Lyu, S. Yang, Y . Xiu, Y . Li, H. He, C. Yuen, and Z. Zhang, “CRB minimization for RIS-aided mmWave integrated sensing and communications,” IEEE Internet Things J. , vol. 11, no. 10, pp. 18 381– 18 393, 2024

work page 2024

[5] [5]

Waveform design for joint radar-communications with low com- plexity analog components,

A. Kaushik, A. Arora, C. Tsinos, C. Masouros, F. Liu, and S. Chatzino- tas, “Waveform design for joint radar-communications with low com- plexity analog components,” in Proc. of IEEE JC&S , 2022, pp. 1–5

work page 2022

[6] [6]

Dual-function radar-communication systems with constant-modulus and similarity con- straints,

C. G. Tsinos, A. Arora, S. Chatzinotas, and B. Ottersten, “Dual-function radar-communication systems with constant-modulus and similarity con- straints,” in Proc. of IEEE SAM , 2022, pp. 231–235

work page 2022

[7] [7]

Simultaneous multi-beam sweeping for mmWave massive MIMO integrated sensing and communication,

Z. Xiao, S. Chen, and Y . Zeng, “Simultaneous multi-beam sweeping for mmWave massive MIMO integrated sensing and communication,” IEEE Trans. Veh. Technol., vol. 73, no. 6, pp. 8141–8152, 2024

work page 2024

[8] [8]

Secure dual-functional radar-communication transmission: Hardware-efficient design,

N. Su, F. Liu, C. Masouros, T. Ratnarajah, and A. Petropulu, “Secure dual-functional radar-communication transmission: Hardware-efficient design,” in Proc. of ACSSC , 2021, pp. 629–633

work page 2021