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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State the problem, from ConStellaration

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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
	16	+
	17	+$$s=\min\Big(1,\ \max\Big(0,\ 1-\frac{\epsilon_{\max}-1}{9}\Big)\Big)$$
	18	+
	19	+for a feasible design and $0$ otherwise; higher is better, and circular cross-sections would score 1.
	20	+
	21	+## Background
	22	+
	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.
	24	+
	25	+## Data and evaluation
	26	+
	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
	32	+
	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.