lpviz: Interactive Linear Programming Visualization

Evan Grand; Michael Klamkin

arxiv: 2604.27518 · v1 · submitted 2026-04-30 · 💻 cs.HC · math.OC

lpviz: Interactive Linear Programming Visualization

Evan Grand , Michael Klamkin This is my paper

Pith reviewed 2026-05-07 09:17 UTC · model grok-4.3

classification 💻 cs.HC math.OC

keywords linear programmingvisualizationinteractive toolSimplex algorithmInterior-Point methodsalgorithm comparisoneducational software3D visualization

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The pith

lpviz lets users draw and edit linear programming feasible regions directly in a browser to compare how Simplex, Interior-Point, and other solvers progress.

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

The paper presents lpviz, a web tool that replaces numerical coefficient entry with direct graphical drawing of the feasible region and objective vector. Users can run and observe multiple algorithm classes including Simplex, Interior-Point, Primal-Dual Hybrid Gradient, and Central Path on the same problem. In 3D mode the tool lifts iterates vertically according to metadata such as complementarity gap or KKT residual, exposing behavior not visible in the primal plane alone. The authors report classroom and research use for developing intuition about solver strengths, weaknesses, and parameter effects. If the visual interface succeeds, it moves LP learning from static tables to live manipulation and side-by-side trajectory inspection.

Core claim

lpviz is a browser-based visualization tool for linear programming that allows direct drawing and editing of the feasible region and objective vector, and supports comparison of Simplex, Interior-Point, Primal-Dual Hybrid Gradient, and Central Path algorithms, with 3D visualization of iterates using metadata heights.

What carries the argument

Direct graphical editing of the feasible region and objective combined with 3D height-mapping of solver metadata such as complementarity gap or KKT residual.

Load-bearing premise

Direct graphical manipulation of the feasible region combined with 3D metadata visualization will meaningfully improve user intuition about LP algorithm behavior compared to standard numerical or textual interfaces.

What would settle it

A controlled user study in which participants using only solver text logs show equivalent or better understanding of algorithm differences than participants using lpviz.

Figures

Figures reproduced from arXiv: 2604.27518 by Evan Grand, Michael Klamkin.

**Figure 1.** Figure 1: Screenshots of lpviz in Trace mode, visualizing how convergence behavior changes as the objective rotates. Abstract—This paper presents lpviz, a browser-based visualization tool for linear programming. lpviz is deeply interactive, offering an intuitive interface where users can directly draw and edit the feasible region and objective vector, without requiring cumbersome manipulation of raw numerical coeffi… view at source ↗

**Figure 2.** Figure 2: System architecture diagram for lpviz. 2 SYSTEM DESCRIPTION This section describes the internal design of lpviz. A key feature of lpviz is that it runs fully on-device: all computations run entirely via static web assets in the browser. This design removes network latency while allowing lpviz to be accessible without installation. Many Content Delivery Networks (CDNs) also host static files for free, reduc… view at source ↗

**Figure 3.** Figure 3: Comparison of IPM and PDHG on an unbounded problem. view at source ↗

**Figure 4.** Figure 4: Phantom pivots caused by difference-of-positives reformulation. view at source ↗

**Figure 7.** Figure 7: In the 3D view, the Z axis is solver-specific: view at source ↗

**Figure 5.** Figure 5: Comparison of IPM with high and low corrector thresholds. view at source ↗

**Figure 6.** Figure 6: Comparison of Inequality and Equality modes in PDHG. view at source ↗

read the original abstract

This paper presents lpviz, a browser-based visualization tool for linear programming. lpviz is deeply interactive, offering an intuitive interface where users can directly draw and edit the feasible region and objective vector, without requiring cumbersome manipulation of raw numerical coefficients. lpviz lets users compare the behavior of several classes of linear programming algorithms, namely Simplex, Interior-Point, Primal-Dual Hybrid Gradient, and Central Path. In the 3D mode, lpviz places iterates at heights corresponding to important solver metadata such as complementarity gap or KKT residual, helping users gain further insight into algorithm behavior beyond the primal iterates alone. lpviz has been used in both research and classroom settings, to help develop intuition for the strengths and weaknesses of different solvers and the impact of solver settings on convergence behavior. lpviz is open-source, permissively licensed, and freely available on any device with a web browser at https://lpviz.net .

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

lpviz is a practical browser tool for drawing LP problems and comparing solvers with 3D metadata views, but the paper is mostly a feature list without evaluation.

read the letter

lpviz is a browser-based tool that lets users sketch the feasible region and objective vector directly, then run and compare Simplex, Interior-Point, Primal-Dual Hybrid Gradient, and Central Path solvers. The 3D mode lifts iterates by solver metadata such as complementarity gap or KKT residual so users can see more than just the path in the plane. The paper presents this as an open-source, no-install resource already tried in classrooms and research settings. That combination of direct graphical editing plus multi-solver comparison in one interface is the concrete new piece. The web delivery and permissive license also make it easy to try or adapt. The authors have shipped something that works and is publicly available, which counts as real output. The soft spots are straightforward. The paper gives almost no evidence that the 3D metadata or direct drawing actually improves intuition over standard plots or code. There are no user studies, no learning metrics, and no comparison against existing LP visualizers. Implementation details on how the solvers are hooked up or how the 3D rendering stays responsive are thin. The claims about helping users understand convergence rest on the tool existing rather than on measured outcomes. This is a tools paper aimed at optimization instructors, students learning algorithms, or HCI people who build educational demos. A reader who wants a ready-made example or starting code for similar visualizations can get value from it. It deserves peer review in a visualization or education track, but reviewers will reasonably ask for some usability data or at least clearer description of the backend before accepting. The tool itself is testable right now, so the conversation can start from actual use rather than promises.

Referee Report

2 major / 2 minor

Summary. The paper presents lpviz, a browser-based interactive visualization tool for linear programming. Users can directly draw and edit the feasible region and objective vector without manipulating raw coefficients. The tool supports comparison of algorithm classes including Simplex, Interior-Point, Primal-Dual Hybrid Gradient, and Central Path. In 3D mode, iterates are rendered at heights corresponding to solver metadata such as complementarity gap or KKT residual. lpviz is described as having been used in research and classroom settings to build intuition about solver strengths, weaknesses, and parameter effects; it is open-source and available at https://lpviz.net.

Significance. If the implemented features match the description, lpviz offers a potentially useful contribution to HCI and optimization education by enabling direct graphical interaction with LP problems and multi-algorithm comparison in an accessible web interface. The open-source release and claimed classroom/research usage are positive indicators of practical value, though the absence of any evaluation data limits assessment of whether the visualizations actually improve user intuition over standard interfaces.

major comments (2)

Abstract: the assertion that lpviz helps users 'gain further insight into algorithm behavior beyond the primal iterates alone' and has been used 'to help develop intuition' is presented without any supporting user studies, quantitative metrics, or qualitative feedback, which is load-bearing for the claimed educational impact.
Abstract and overall manuscript: no implementation details, architecture description, or validation of numerical accuracy are provided for how the listed algorithms (Simplex, Interior-Point, PDHG, Central Path) are computed and rendered, making it impossible to verify that the visualized behaviors are faithful to the underlying solvers.

minor comments (2)

The manuscript would benefit from a brief section or appendix describing the underlying LP solver libraries or custom implementations used to generate the iterates.
Figure captions and the 3D mode description could more explicitly link the vertical axis choices (e.g., complementarity gap) to specific algorithmic properties for readers unfamiliar with LP duality.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the constructive review and the recommendation for minor revision. We address each major comment below and will incorporate changes to strengthen the manuscript.

read point-by-point responses

Referee: Abstract: the assertion that lpviz helps users 'gain further insight into algorithm behavior beyond the primal iterates alone' and has been used 'to help develop intuition' is presented without any supporting user studies, quantitative metrics, or qualitative feedback, which is load-bearing for the claimed educational impact.

Authors: We agree that the manuscript presents no formal user studies, quantitative metrics, or collected qualitative feedback to substantiate the educational claims. The statements in the abstract reflect the authors' observations from deploying the tool in research and classroom settings. To prevent any overstatement and align the text with the available evidence, we will revise the abstract to qualify these assertions explicitly as based on informal usage observations rather than empirical evaluation. This change will be made in the next version. revision: yes
Referee: Abstract and overall manuscript: no implementation details, architecture description, or validation of numerical accuracy are provided for how the listed algorithms (Simplex, Interior-Point, PDHG, Central Path) are computed and rendered, making it impossible to verify that the visualized behaviors are faithful to the underlying solvers.

Authors: We acknowledge that the current manuscript does not include implementation details, architecture descriptions, or numerical validation for the visualized algorithms. While the paper's primary contribution is the interactive HCI interface, we will add a new subsection describing the software architecture, the specific libraries and custom implementations used for each solver class (including how iterates and metadata such as complementarity gaps are computed and rendered), and basic accuracy checks against reference solvers on standard test problems. This addition will enable verification of fidelity and will appear in the revised manuscript. revision: yes

Circularity Check

0 steps flagged

No significant circularity: descriptive tool paper with no derivations

full rationale

The paper describes an implemented browser-based visualization tool (lpviz) for linear programming, including interactive editing of feasible regions, comparison of named algorithms (Simplex, Interior-Point, etc.), and 3D rendering of solver metadata. No equations, predictions, fitted parameters, or derivation chains exist in the manuscript. Usage claims in research and classroom settings are presented without metrics or formal evaluation, but these are not load-bearing for any theoretical result. The contribution is the open-source tool itself, directly testable via the provided URL, making the paper self-contained with no opportunity for circular reasoning.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

The paper is a software tool description with no mathematical derivations, fitted parameters, axioms, or invented entities.

pith-pipeline@v0.9.0 · 5446 in / 1096 out tokens · 71371 ms · 2026-05-07T09:17:12.156049+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

13 extracted references · 13 canonical work pages

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work page doi:10.1287/moor.2.2.103 1977

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S. Cajot et al. Interactive optimization with parallel coordinates.Frontiers in ICT, 2019. doi: 10.3389/fict.2018.00032 2

work page doi:10.3389/fict.2018.00032 2019

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A. Chambolle and T. Pock. A first-order primal-dual algorithm for convex problems with applications to imaging.Journal of mathematical imaging and vision, 2011. doi: 10.1007/s10851-010-0251-1 4

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M. Espadoto, F. C. M. Rodrigues, N. S. T. Hirata, and A. C. Telea. Optmap: using dense maps for visualizing multidimensional optimization problems. InProceedings, 2021. doi: 10.5220/0010288501230132 2

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Liu and H

T. Liu and H. Lu. A new crossover algorithm for lp inspired by the spiral dynamic of pdhg.INFORMS Journal on Computing, 2026. doi: 10. 1287/ijoc.2024.0996 4

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work page doi:10.1007/978-3-031-11623-0_13 2022

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work page arXiv 2023

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C. A. Shaffer et al. Algorithm visualization.Transactions on Computing Education, 2010. doi: 10.1145/1821996.1821997 2

work page doi:10.1145/1821996.1821997 2010

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H. M. Wagner. A comparison of the original and revised simplex methods. Operations Research, 5(3), 1957. doi: 10.1287/opre.5.3.361 3

work page doi:10.1287/opre.5.3.361 1957

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M. J. Walter, D. J. Walker, and M. J. Craven. An explainable visuali- sation of the evolutionary search process. InGenetic and Evolutionary Computation, 2022. doi: 10.1145/3520304.3533984 2

work page doi:10.1145/3520304.3533984 2022