Least-elongated stellarator plasma boundary

Find the least elongated stellarator plasma boundary with aspect ratio at most 4, average triangularity at most −0.5 and edge iota per period at least 0.3.

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1	# Least-elongated stellarator plasma boundary
2
3	Find the outer shape of a stellarator plasma whose cross-sections are as round as possible while meeting fixed targets on compactness, triangularity and the twist of the magnetic field lines. It is the entry-level problem of the ConStellaration benchmark, posed to lower the entry barrier to stellarator design for optimization and machine-learning researchers.
4
5	## Task
6
7	A candidate is a stellarator-symmetric plasma boundary, written as a truncated Fourier series in cylindrical coordinates $(R,\phi,Z)$:
8
9	$$R(\theta,\phi)=\sum_{m,n}R_{mn}\cos(m\theta-N_{\mathrm{fp}}n\phi),\qquad Z(\theta,\phi)=\sum_{m,n}Z_{mn}\sin(m\theta-N_{\mathrm{fp}}n\phi),$$
10
11	where $\theta$ and $\phi$ are poloidal and toroidal angles and $N_{\mathrm{fp}}$, the number of field periods, is also chosen by the designer. The forward model computes the vacuum ideal-MHD equilibrium inside the boundary with VMEC++ and returns the metrics used below. The problem is
12
13	$$\min\ \epsilon_{\max}\quad\text{subject to}\quad A\le 4,\qquad \bar\delta\le-0.5,\qquad \lvert\tilde\iota\rvert\ge 0.3,$$
14
15	where $\epsilon_{\max}$ is the largest elongation among the boundary cross-sections, $A$ the aspect ratio (major over minor radius), $\bar\delta$ the triangularity averaged over the two symmetric cross-sections $\phi=0$ and $\phi=\pi/N_{\mathrm{fp}}$, and $\tilde\iota$ the edge rotational transform divided by $N_{\mathrm{fp}}$. A design is feasible when every normalized constraint violation $(c_i-c_i^)/\lvert c_i^\rvert$ is at most $0.01$. The score is
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17	$$s=\min\Big(1,\ \max\Big(0,\ 1-\frac{\epsilon_{\max}-1}{9}\Big)\Big)$$
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19	for a feasible design and $0$ otherwise; higher is better, and circular cross-sections would score 1.
20
21	## Background
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23	Stellarators confine a fusion plasma with external coils alone, which shape twisted magnetic surfaces, so they need no large plasma current and avoid current-driven disruptions; the price is a hard, high-dimensional shape-design problem. ConStellaration (Cadena et al., NeurIPS 2025) was released by Proxima Fusion with a dataset of about 158,000 quasi-isodynamic-like boundaries and their equilibria, and three optimization problems of increasing difficulty. This first one is purely geometric: no confinement or stability property is imposed, so it isolates the optimizer and the choice of boundary parameterization, and it shares its forward model with the two physics problems.
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25	## Data and evaluation
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27	- Instances: one fixed problem. The [dataset](https://huggingface.co/datasets/proxima-fusion/constellaration) and all public leaderboard submissions ([results dataset](https://huggingface.co/datasets/proxima-fusion/constellaration-bench-results)) can serve as starting points.
28	- Evaluation code: `constellaration==0.2.6`, `problems.GeometricalProblem().evaluate(boundary)`, which runs VMEC++ at its high-fidelity setting. The [leaderboard](https://huggingface.co/spaces/proxima-fusion/constellaration-bench) scores every submission with exactly this release; later releases give different numbers. A submission is one boundary as `SurfaceRZFourier` JSON.
29	- Hardware: none assumed; the score does not depend on run time.
30
31	## What is known
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33	- Paper baseline (Cadena et al., 2025): an augmented-Lagrangian method with the NGOpt optimizer, seeded from a rotating ellipse with three field periods and Fourier modes up to 4, reached $\epsilon_{\max}=1.27$ and a score of 0.969 after 18 hours on 96 vCPUs. SciPy's trust-constr and COBYQA found no feasible design.
34	- Weißberg et al. (2026) refined the top leaderboard designs with an augmented-Lagrangian loop around L-BFGS, using finite-difference gradients through VMEC++; one refined design scored 0.9763 in their study, and their submitted entry (0.97440 on the leaderboard) took first place in May 2026.
35	- Best on the leaderboard on 2026-10-20: 0.98048 (user canyon-ant, September 2026), which corresponds to $\epsilon_{\max}\approx1.176$.
36
37	## What would count as progress
38
39	- A boundary scoring above 0.98048 under `constellaration==0.2.6`, submitted as JSON; the check is a single forward-model evaluation that anyone can rerun.
40	- A method that reaches at least 0.98 from a rotating-ellipse start, without seeding from published leaderboard designs, with its code.
41	- A study of how the best achievable $\epsilon_{\max}$ depends on $N_{\mathrm{fp}}$, with the best boundary for each value.
42
43	## References
44
45	1. S. A. Cadena, A. Merlo, E. Laude, et al. ConStellaration: A dataset of QI-like stellarator plasma boundaries and optimization benchmarks. NeurIPS 2025 Datasets and Benchmarks Track. [arXiv:2506.19583](https://arxiv.org/abs/2506.19583)
46	2. J. Schilling. The numerics of VMEC++. 2025. [arXiv:2502.04374](https://arxiv.org/abs/2502.04374)
47	3. T. Weißberg, M. Heep, Z. Lähner, F. Bernard. Towards stellarator geometry optimisation for nuclear fusion. 2026. [arXiv:2608.28224](https://arxiv.org/abs/2608.28224)
48	4. Proxima Fusion. ConStellaration code, release v0.2.6. [GitHub](https://github.com/proximafusion/constellaration/releases/tag/v0.2.6)
49
50	## Source
51
52	Imported from [ConStellaration geometrical problem](https://github.com/proximafusion/constellaration/blob/v0.2.6/src/constellaration/problems.py#L113), curated by Proxima Fusion. Open according to the source as of 2026-10-20. The statement above is written in our own words; see the source for the original wording, history and discussion.