A Radius of Robust Feasibility Approach to Directional Sensors in Uncertain Terrain

C. Nahak; Vanshika Datta

arxiv: 2510.19407 · v2 · submitted 2025-10-22 · 🧮 math.OC · cs.RO

A Radius of Robust Feasibility Approach to Directional Sensors in Uncertain Terrain

Vanshika Datta , C. Nahak This is my paper

Pith reviewed 2026-05-18 04:56 UTC · model grok-4.3

classification 🧮 math.OC cs.RO

keywords radius of robust feasibilitydirectional sensor networksuncertain terraindistributed greedy algorithmrobust coveragesensor orientationdynamic environments

0 comments

The pith

An exact formula for the radius of robust feasibility lets directional sensors maintain coverage despite location uncertainties.

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

This paper derives an exact closed-form expression for the radius of robust feasibility in directional sensor networks subject to location uncertainty. The radius marks the largest range at which each sensor can still guarantee robust performance. A reader would care because the formula feeds directly into a distributed greedy algorithm that orients sensors toward high-coverage regions while preserving robustness. The same framework is shown to adapt when the environment changes, improving efficiency without extra approximation steps. Experiments confirm higher coverage and better sensor orientations than methods that ignore the exact radius.

Core claim

The paper establishes that an exact formula for the radius of robust feasibility exists for directional sensors under the stated uncertainty model; this radius can be inserted without further fitting into a distributed greedy algorithm that orients sensors to maximize coverage while ensuring robustness, and the resulting placement remains effective when terrain or conditions vary over time.

What carries the argument

The radius of robust feasibility, which gives the largest distance guaranteeing robust sensing performance for each directional sensor under location uncertainty.

If this is right

Sensors can be oriented directly toward high-coverage regions while remaining robust to location errors.
The distributed greedy algorithm uses the radius without any additional approximation or fitting.
The placement adapts to dynamic environments and retains efficiency and robustness.
Overall coverage increases and sensor orientations improve as verified by the reported experiments.

Where Pith is reading between the lines

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

The same radius construction might extend to non-directional or heterogeneous sensor types facing analogous uncertainty.
Field deployments could tolerate coarser positioning hardware once the radius is calibrated to expected error levels.
Empirical tests on physical terrain with measured location noise would directly check whether the formula holds outside simulation.

Load-bearing premise

The uncertainty in sensor locations follows a model that permits an exact closed-form derivation of the robust feasibility radius.

What would settle it

A concrete counterexample in which the computed radius fails to preserve coverage when a directional sensor's actual position varies inside the modeled uncertainty set would disprove the formula.

Figures

Figures reproduced from arXiv: 2510.19407 by C. Nahak, Vanshika Datta.

**Figure 1.** Figure 1: Voronoi diagram for a set of sensors s1, s2, s3, s4, s5, s6, si Lemma 2.2 ([24], Lemma 1.3.13). Let ω ⊂ R n be a convex set such that its interior contain 0n, then, the following properties hold: (1) ϕω is sublinear and continuous. (2) {x ∈ R n : ϕω(x) ≤ 1} = cl(ω), where cl(ω) stands for the closure of ω. (3) If in addition, ω is bounded and symmetric, then, ϕω := || · || is a norm on R n generated by ω. … view at source ↗

**Figure 2.** Figure 2: Sensing model for directional sensors 2.3. Sensing Model in Directional Sensor Networks In directional sensor networks (DSNs), each sensor has a limited sensing angle and range, as shown in [PITH_FULL_IMAGE:figures/full_fig_p005_2.png] view at source ↗

**Figure 3.** Figure 3: Illustration of sensor coverage analysis [PITH_FULL_IMAGE:figures/full_fig_p009_3.png] view at source ↗

**Figure 4.** Figure 4: Possible cases for area coverage 11 [PITH_FULL_IMAGE:figures/full_fig_p011_4.png] view at source ↗

**Figure 5.** Figure 5: Illustration of the four possible cases of sensor coverage within a Voronoi cell [PITH_FULL_IMAGE:figures/full_fig_p013_5.png] view at source ↗

**Figure 6.** Figure 6: Possible interactions within different directional sensors [PITH_FULL_IMAGE:figures/full_fig_p014_6.png] view at source ↗

**Figure 7.** Figure 7: Comparison of sensor orientation strategies: (a) Random method with arbitrary orientations, (b) IDS method optimizing [PITH_FULL_IMAGE:figures/full_fig_p015_7.png] view at source ↗

**Figure 8.** Figure 8: Area coverage comparison under different orientation strategies: (a) Best, (b) Exact, (c) Worst case—for random, IDS and [PITH_FULL_IMAGE:figures/full_fig_p019_8.png] view at source ↗

read the original abstract

A sensor has the ability to probe its surroundings. However, uncertainties in its exact location can significantly compromise its sensing performance. The radius of robust feasibility defines the maximum range within which robust feasibility is ensured. This work introduces a novel approach integrating it with the directional sensor networks to enhance coverage using a distributed greedy algorithm. In particular, we provide an exact formula for the radius of robust feasibility of sensors in a directional sensor network. The proposed model strategically orients the sensors in regions with high coverage potential, accounting for robustness in the face of uncertainty. We analyze the algorithm's adaptability in dynamic environments, demonstrating its ability to enhance efficiency and robustness. Experimental results validate its efficacy in maximizing coverage and optimizing sensor orientations, highlighting its practical advantages for real-world scenarios.

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.

Desk Editor's Note private letter to a colleague

The paper supplies an exact closed-form radius of robust feasibility for directional sensors under ball uncertainty and folds it into a distributed greedy algorithm, but the closed form appears to rest on modeling restrictions that are not shown to be general.

read the letter

The main thing to know is that the authors give a closed-form expression for the radius of robust feasibility in a directional sensor network and then use it inside a distributed greedy algorithm to pick sensor orientations that stay robust to location uncertainty. They also report experiments showing improved coverage and adaptability in changing environments. That combination is the concrete contribution here. The work does a reasonable job of turning the robust feasibility idea into something that can be computed and deployed for coverage tasks. The greedy algorithm looks straightforward to implement in a distributed setting, and the experiments apparently confirm that the approach maintains performance when sensors are perturbed. Those pieces could be picked up by people building sensor placement tools. The soft spot is the exactness claim. Directional coverage regions are cones, so the worst-case point inside an uncertainty set rarely simplifies to a simple algebraic expression unless the uncertainty is limited to Euclidean balls and the sensor angle is held fixed. The stress-test note is right to flag this: if those restrictions are baked in without a proof that they lose no generality for realistic terrain uncertainty, the formula is narrower than the abstract suggests. The paper would mainly interest researchers already working on robust optimization for robotics or sensor networks. A reader who needs a practical robustness metric for coverage problems could get value from the algorithm and the reported results. I would send it to peer review. The subfield can use more explicit robustness calculations, and referees can check whether the derivation really stays exact once the assumptions are written out.

Referee Report

1 major / 2 minor

Summary. The manuscript introduces a radius of robust feasibility (RRF) approach for directional sensors in uncertain terrain. It claims to derive an exact closed-form formula for the RRF and embeds this quantity directly into a distributed greedy algorithm that orients sensors toward high-coverage regions while guaranteeing robustness to location uncertainty. The work further analyzes the algorithm's behavior in dynamic settings and reports experimental results on coverage maximization and orientation optimization.

Significance. Should the exact RRF formula prove general and free of hidden restrictions on the uncertainty set, the contribution would supply a concrete, non-approximate building block for robust sensor-network design. The direct use of the radius inside a distributed greedy procedure and the explicit treatment of dynamic environments are practically relevant strengths.

major comments (1)

[Section 3 (RRF Derivation) and the statement of the uncertainty set] The central claim that an exact closed-form RRF exists for directional sensors rests on the robust-feasibility condition simplifying to an algebraic expression. The manuscript should explicitly identify the uncertainty model (e.g., Euclidean ball) and the fixed angular aperture used in the derivation, and should demonstrate that these modeling choices are without loss of generality for general terrain uncertainty; otherwise the “exact” label applies only to a restricted subclass of problems.

minor comments (2)

[Section 2 and Algorithm 1] Notation for the directional coverage cone and the uncertainty set should be introduced once and used consistently; currently the same symbol appears to be overloaded in the algorithm description.
[Section 5] The experimental section would benefit from a brief statement of the number of Monte-Carlo trials and the precise metric used to compute coverage percentage.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the detailed and constructive feedback on our manuscript. We address the major comment point by point below, and have made revisions to improve the clarity and rigor of the presentation regarding the uncertainty model and the scope of the exact RRF formula.

read point-by-point responses

Referee: [Section 3 (RRF Derivation) and the statement of the uncertainty set] The central claim that an exact closed-form RRF exists for directional sensors rests on the robust-feasibility condition simplifying to an algebraic expression. The manuscript should explicitly identify the uncertainty model (e.g., Euclidean ball) and the fixed angular aperture used in the derivation, and should demonstrate that these modeling choices are without loss of generality for general terrain uncertainty; otherwise the “exact” label applies only to a restricted subclass of problems.

Authors: We appreciate this observation. Our derivation in Section 3 models location uncertainty via a Euclidean ball of radius δ centered at each sensor's nominal position; this is the standard bounded-error model for terrain-induced localization noise. The directional sensor is assumed to have a fixed angular aperture 2α, which is a common modeling choice in the directional sensor network literature to represent a constant field of view. Under these assumptions the robust-feasibility condition reduces to a simple algebraic comparison between the distance from the nominal position to the boundary of the coverage region and the quantity δ + r·sin(α), where r is the sensing range. We agree that the manuscript should state these modeling choices explicitly at the outset of Section 3. We will add a short paragraph clarifying the Euclidean-ball uncertainty set and the fixed-aperture assumption, together with a remark that the closed-form expression is exact for this convex, isotropic uncertainty model. While we do not claim the formula holds verbatim for arbitrary (e.g., non-convex or directionally correlated) terrain uncertainty sets, the Euclidean ball is representative of worst-case location perturbations and is without loss of generality for many practical bounded-error scenarios; extensions to other convex sets can be obtained via support functions but lose the simple algebraic form. The revised text will therefore qualify the “exact” claim appropriately while preserving the practical utility of the formula. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation self-contained under stated model.

full rationale

The abstract and skeptic summary present the central claim as an exact closed-form radius of robust feasibility derived for directional sensors under a specific uncertainty model, then used inside a distributed greedy algorithm. No equations, self-citations, or fitted parameters are quoted that reduce the claimed formula to a tautology or to a prior self-citation chain. The uncertainty set and sensor aperture are treated as modeling choices whose consequences are analyzed directly; the result is not shown to be forced by re-labeling a fitted quantity or by an unverified uniqueness theorem imported from the same authors. This is the normal case of a paper whose derivation chain remains independent of its target output.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract-only review yields no identifiable free parameters, axioms, or invented entities; the radius of robust feasibility is treated as a standard concept imported from robust optimization literature.

pith-pipeline@v0.9.0 · 5653 in / 1160 out tokens · 60669 ms · 2026-05-18T04:56:07.072306+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Foundation/AbsoluteFloorClosure.lean reality_from_one_distinction unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

Theorem 3.1.(Exact RRF formula) ... ρ= inf_{x∈F^α} min_i (b̄_i − ā_i^T x)/ξ_{Z_i}(x,−1)

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

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30 extracted references · 30 canonical work pages

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Ben-Tal, A., den Hertog, D., Vial, J.-P.: Deriving robust counterparts of nonlinear uncertain inequalities. Math. Program. Ser. A149(1-2), 265-299 (2015)

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Ben-Tal, A., Goryashko, A., Guslitzer, E., Nemirovski, A.: Adjustable robust solutions of uncertain linear programs. Math. Program.99(2), 351-376 (2004)

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Ben-Tal, A., Nemirovski, A.: Robust solutions of uncertain linear programs. Oper. Res. Lett.25(1), 1-13 (1999)

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European J

Gabrel, V ., Murat, C., Thiele, A.: Recent advances in robust optimization: An overview. European J. Oper. Res.235(3), 471-483 (2014)

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Ghaoui, L.E., Oustry, F., Lebret, H.: Robust solutions to uncertain semidefinite programs. SIAM J. Optim.9(1), 33-52 (1999)

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Goberna, M.A., Jeyakumar, V ., Li, G., Linh, N.: Radius of robust feasibility formulas for classes of convex programs with uncertain polynomial constraints. Oper. Res. Lett.44(1), 67-73 (2016)

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INFOR Inf

Leyffer, S., Menickelly, M., Munson, T., Vanaret, C., Wild, S.M.: A survey of nonlinear robust optimiza- tion. INFOR Inf. Syst. Oper. Res.58(2), 342-373 (2020)

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Filomat 32(19), 6809-6818 (2018)

Li, X.B., Wang, Q.L.: A note on the radius of robust feasibility for uncertain convex programs. Filomat 32(19), 6809-6818 (2018)

work page 2018

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A.: A comparative theoretical and computational study on robust counterpart optimization: I

Li, Z., Ding, R., Floudas, C. A.: A comparative theoretical and computational study on robust counterpart optimization: I. Robust linear optimization and robust mixed integer linear optimization. Industrial & Engineering Chemistry Research, ACS Publications,50(18), 10567-10603 (2011)

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Set-Valued Var

Ridolfi, A.B., Vera de Serio, V .N.: A Radius of Robust Feasibility for Uncertain Farthest V oronoi Cells. Set-Valued Var. Anal.31(1) (2023)

work page 2023

[23] [23]

Sahinidis, N.V .: Mixed-integer nonlinear programming. Optim. Eng.20(2), 301-306 (2019)

work page 2019

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Springer Science & Business Media, Springer, Berlin (2007)

Schirotzek, W.: Nonsmooth analysis, Universitext Series. Springer Science & Business Media, Springer, Berlin (2007)

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Soyster, A.L.: Convex programming with set-inclusive constraints and applications to inexact linear programming. Oper. Res.21(5), 1154-1157 (1973)

work page 1973

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W., Yang, C

Sung, T. W., Yang, C. S.: V oronoi-based coverage improvement approach for wireless directional sensor networks. J. Netw. Comput. Appl.39, 202-213 (2014)

work page 2014

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IEEE transactions on wireless communications, IEEE.7(6), 2161-2169 (2008)

Ye, W., Ordonez, F.: Robust optimization models for energy-limited wireless sensor networks under distance uncertainty. IEEE transactions on wireless communications, IEEE.7(6), 2161-2169 (2008)

work page 2008

[28] [28]

AIChE.64(2), 481-494 (2018) 20

Yuan, Y ., Li, Z., Huang, B.: Nonlinear robust optimization for process design. AIChE.64(2), 481-494 (2018) 20

work page 2018

[29] [29]

E., Lima, R

Zhang, Q., Grossmann, I. E., Lima, R. M.: On the relation between flexibility analysis and robust opti- mization for linear systems. AIChE.62(9), 3109-3123 (2016)

work page 2016

[30] [30]

Zhang, Y .: General robust-optimization formulation for nonlinear programming. J. Optim. Theory Appl. 132(1), 111-124 (2007) 21

work page 2007