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arxiv: 2604.20364 · v1 · submitted 2026-04-22 · 💻 cs.IT · math.IT

Trajectory Design for Fairness Enhancement in Movable Antennas-Aided Communications

Guojie Hu , Qingqing Wu , Lipeng Zhu , Kui Xu , Guoxin Li , Tong-Xing Zheng This is my paper

Pith reviewed 2026-05-09 23:17 UTC · model grok-4.3

classification 💻 cs.IT math.IT
keywords movable antennastrajectory designrate fairnessmax-min rateLagrangian dual methodmultiuser uplinkantenna movement speed
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The pith

Movable antennas achieve fairness by cycling through a finite set of deployment patterns with optimized durations.

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

This paper studies trajectory design for multiple movable antennas at a base station serving several single-antenna users in uplink. The goal is to maximize the minimum rate any user achieves over a finite time window. The authors first solve an idealized version where antennas move at infinite speed and show that the optimum reduces to selecting only a finite number of fixed geometries, each used for a computed fraction of time. They then build a practical heuristic for antennas that move at realistic limited speeds by following the ideal patterns. The work matters because the continuous-time movement problem becomes tractable once the solution is known to consist of discrete patterns and time shares.

Core claim

In the ideal case with infinitely fast MA movement, the relaxed problem can be optimally solved via the Lagrangian dual method, revealing that the BS should employ a finite set of MA deployment patterns, each allocated an optimal time duration. Building on this, the general case with limited MA movement speed is addressed by a heuristic trajectory design inspired by the optimal patterns identified in the ideal scenario.

What carries the argument

The finite set of MA deployment patterns with allocated time durations, found by the Lagrangian dual method applied to the infinite-speed relaxation of the max-min rate problem.

If this is right

  • The optimal fairness solution in the ideal case uses only a small number of discrete deployment patterns instead of continuous movement.
  • The Lagrangian dual method simultaneously identifies both the patterns and the exact time each pattern should be used.
  • The finite-pattern structure directly supplies the candidate positions for a heuristic when antenna speed is finite.
  • Simplified special cases of the system yield additional explicit insights into which patterns are selected.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • Practical MA systems could store a small library of precomputed geometries and simply switch among them to approach the fairness optimum.
  • The time-sharing structure identified here might be reused for other objectives such as sum-rate maximization or energy efficiency.
  • When movement speed is low the heuristic must balance lingering near each ideal pattern against the cost of traveling between patterns.

Load-bearing premise

The ideal-case analysis assumes that the movable antennas can change positions instantaneously without any transition time cost.

What would settle it

A numerical test that computes the minimum user rate achieved by the proposed limited-speed heuristic and checks whether it approaches the upper bound obtained from the infinite-speed ideal solution as the allowed movement speed increases.

Figures

Figures reproduced from arXiv: 2604.20364 by Guojie Hu, Guoxin Li, Kui Xu, Lipeng Zhu, Qingqing Wu, Tong-Xing Zheng.

Figure 1
Figure 1. Figure 1: Illustration of the considered system model. [PITH_FULL_IMAGE:figures/full_fig_p001_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: The achievable rate of different users w.r.t. [PITH_FULL_IMAGE:figures/full_fig_p007_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: The minimum average rate of the SSMT scheme w.r.t. [PITH_FULL_IMAGE:figures/full_fig_p009_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: The minimum average rate of three different schemes w [PITH_FULL_IMAGE:figures/full_fig_p010_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: The case without antenna movement speed constraint: [PITH_FULL_IMAGE:figures/full_fig_p011_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: The case without antenna movement speed constraint: [PITH_FULL_IMAGE:figures/full_fig_p011_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: The minimum average rate of the three different schem [PITH_FULL_IMAGE:figures/full_fig_p011_7.png] view at source ↗
Figure 9
Figure 9. Figure 9: The minimum average rate of the three different schem [PITH_FULL_IMAGE:figures/full_fig_p012_9.png] view at source ↗
read the original abstract

Through adaptive antenna repositioning, the movable antenna (MA) technology enables on-demand reconfiguration of wireless channels, thereby creating an additional spatial degree of freedom in improving communication performance. This paper investigates a multiuser uplink communication system aided by MAs, where a base station (BS) equipped with multiple MAs serves multiple single-antenna users. Specifically, given that an optimized array geometry cannot guarantee rate fairness, we focus on designing antenna trajectory at the BS to maximize the minimum achievable rate among all users over a finite time period. The resulting optimization problem is fundamentally challenging to solve due to the continuous-time nature. To address it, we first examine an ideal case with infinitely fast MA movement and demonstrate that the relaxed problem can be optimally solved via the Lagrangian dual method. The obtained trajectory solution reveals that the BS should employ a finite set of MA deployment patterns, each allocated an optimal time duration. Building on this, we then study the general case with limited MA movement speed and propose a heuristic trajectory design inspired by the optimal patterns identified in the ideal scenario. Several insights are also gained by examining the simplified special case. Finally, numerical results are provided to validate the effectiveness of the proposed designs compared to competitive benchmarks.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The paper investigates trajectory design for movable antennas (MAs) at a base station to maximize the minimum achievable rate (fairness) among multiple single-antenna users in an uplink system over a finite time horizon. For the ideal case of infinitely fast MA movement, the continuous-time problem is relaxed and claimed to be optimally solved via the Lagrangian dual method, yielding a finite number of MA deployment patterns each with an optimal time duration (via Carathéodory). For the practical limited-speed case, a heuristic trajectory is proposed that is inspired by the ideal-case patterns. Numerical results validate the designs against benchmarks.

Significance. If the central optimality claim holds, the finite-pattern structural result is a useful insight that reduces the infinite-dimensional trajectory problem to a finite time-sharing optimization, providing a principled foundation for practical MA trajectory heuristics. The work correctly identifies the continuous-time nature as the core difficulty and leverages standard dualization plus convex-hull arguments to obtain the finite-support property.

major comments (2)
  1. [§III] §III (Ideal-case analysis): The claim that the relaxed problem 'can be optimally solved via the Lagrangian dual method' is not supported, because each evaluation of the dual function g(λ) = max_p ∑_k λ_k r_k(p) requires globally maximizing a non-concave weighted sum-rate over the continuous position domain p. Since r_k(p) = log(1+SINR_k(p)) with distance-dependent path loss, the inner problem is non-convex; local search or gradient methods therefore do not guarantee the global optimum required for strong duality to recover the primal optimum. This is load-bearing for the optimality assertion and the subsequent finite-pattern claim.
  2. [§IV] §IV (Limited-speed heuristic): The proposed heuristic is motivated by the ideal-case patterns but provides no approximation guarantee, convergence analysis, or bound relative to the true optimum under finite MA speed. Because the ideal-case analysis already rests on the unresolved global-optimization issue above, the heuristic's performance claims require additional justification (e.g., via worst-case analysis or tighter comparisons).
minor comments (2)
  1. [§II] The system model should explicitly state all assumptions on the channel (path-loss exponent, fading distribution, noise power) and the precise definition of the instantaneous rate r_k(p) before the optimization is introduced.
  2. [§V] In the numerical results, label all curves clearly (including the proposed heuristic versus the ideal-case benchmark) and report the number of random channel realizations used for averaging.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the insightful and constructive comments, which help clarify the limitations of our analysis. We address the major comments point by point below, indicating the revisions we will incorporate.

read point-by-point responses

  1. Referee: [§III] §III (Ideal-case analysis): The claim that the relaxed problem 'can be optimally solved via the Lagrangian dual method' is not supported, because each evaluation of the dual function g(λ) = max_p ∑_k λ_k r_k(p) requires globally maximizing a non-concave weighted sum-rate over the continuous position domain p. Since r_k(p) = log(1+SINR_k(p)) with distance-dependent path loss, the inner problem is non-convex; local search or gradient methods therefore do not guarantee the global optimum required for strong duality to recover the primal optimum. This is load-bearing for the optimality assertion and the subsequent finite-pattern claim.

    Authors: We acknowledge that the inner weighted sum-rate maximization over continuous antenna positions is non-convex, and our numerical solution via discretization and local search does not guarantee global optimality of the dual function. The manuscript applies the Lagrangian dual to the time-relaxed problem and invokes Carathéodory's theorem on the convex hull of achievable rate vectors to establish the finite-support property. This structural result holds for any point in the convex hull regardless of whether the supporting hyperplane is found exactly. We will revise §III to explicitly note that the dual is solved numerically (with potential suboptimality in the inner maximization), clarify that the finite-pattern claim follows from the convex relaxation and Carathéodory theorem applied to observed rate vectors, and discuss the implications for strong duality. revision: yes

  2. Referee: [§IV] §IV (Limited-speed heuristic): The proposed heuristic is motivated by the ideal-case patterns but provides no approximation guarantee, convergence analysis, or bound relative to the true optimum under finite MA speed. Because the ideal-case analysis already rests on the unresolved global-optimization issue above, the heuristic's performance claims require additional justification (e.g., via worst-case analysis or tighter comparisons).

    Authors: We agree that the limited-speed heuristic carries no theoretical approximation guarantee or convergence proof, as obtaining such bounds for the continuous-time non-convex trajectory problem is intractable. The design discretizes the ideal-case patterns into time slots and connects them with feasible velocity-constrained paths. In the revision we will expand the numerical section with additional benchmarks (e.g., random and greedy trajectories), report more performance metrics, and add an explicit limitations paragraph stating that the heuristic is a practical, pattern-inspired approximation whose gains are validated empirically rather than theoretically. We will also note that the heuristic remains useful even when the ideal patterns are obtained numerically. revision: partial

Circularity Check

0 steps flagged

No circularity: derivation uses external duality and Carathéodory theorem on relaxed problem

full rationale

The claimed chain begins with the ideal-case relaxation (infinitely fast movement) and applies the Lagrangian dual method to the time-sharing problem max t s.t. average rates ≥ t. The finite-support structure (at most K patterns) follows from standard convex-hull arguments (Carathéodory) applied to the achievable rate region, which is an external theorem independent of the paper. The abstract and description state that the relaxed problem “can be optimally solved via the Lagrangian dual method” and that the solution “reveals” the finite-pattern form; neither step defines the output in terms of itself nor renames a fitted quantity as a prediction. No self-citation is load-bearing for the core claim, and the subsequent limited-speed heuristic is explicitly presented as inspired by, not derived from, the ideal solution. The noted non-convexity of the position subproblems affects whether global optimality is attained in practice but does not create a definitional loop or self-referential input.

Axiom & Free-Parameter Ledger

1 free parameters · 2 axioms · 0 invented entities

The central claim rests on standard convex-optimization assumptions and the modeling choice that channel gains depend only on instantaneous antenna positions; no new physical entities are introduced.

free parameters (1)
  • time durations allocated to each deployment pattern
    These durations are decision variables solved by the dual method in the ideal case and become part of the heuristic schedule.
axioms (2)
  • domain assumption The relaxed infinite-speed problem is convex and its dual yields the global optimum for the original fairness objective
    Invoked when the authors state that the relaxed problem can be optimally solved via the Lagrangian dual method.
  • domain assumption Channel coefficients are deterministic functions of antenna positions only
    Standard modeling assumption for movable-antenna systems used throughout the trajectory design.

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