This repository contains verifier-backed numerical results produced by a cold-start optimization agent.
The system currently appears strongest on continuous, geometric, and lightly constrained black-box optimization problems, including circle packing, rectangle packing, spherical codes / Tammes-type problems, and energy minimization.
This is a public result archive, not the full optimization harness. The private harness is available for serious private benchmark evaluation, collaboration, or licensing discussions under a separate agreement.
- Largest current packing improvement: n=27 circle packing in a fixed-perimeter rectangle, improving the Berthold et al. Jan 2026 reference from 2.69015 to 2.691523369458056.
- Cleanest full-precision paper comparison: n=32 variable-radius circle packing in the unit square, improving on the Berthold et al. Jan 2026 full-precision reference by about 4.45e-8 and passing the public AlphaEvolve/DeepMind verifier.
- External mathematical table improvements: new best numerical spherical codes for S^5, N=86 and S^5, N=98, referenced from spherical-codes.org.
- Lennard-Jones morphology validation: matched canonical LJ global minima across three distinct structural families — double-funnel fcc LJ38, Marks-decahedral LJ75, and C2v Marks-icosahedral LJ104.
- Largest constrained-NLP improvement: AC OPF on the 13,659-bus European Pegase grid (199,281 vars, 191,097 indefinite QCQP constraints), improving the MINLPLib
p1primal from 386,108.81 to 386,106.54 in a structurally different generator-dispatch basin at constraint residual 1.79e-12. Accepted by MINLPLib asp2on 2026-05-25.
The hypothesis is simple:
If a problem has an executable evaluator, compact candidate solutions, and feasibility can be checked or repaired automatically, a cold-start optimizer may be able to find improved incumbents without domain-specific manual tuning.
This repository documents public benchmark results where that process produced new best-known numerical results, matched known reference optima, or failed to reach the best-known result.
The current system appears best suited for:
- geometric packing,
- spherical / Grassmannian code search,
- continuous nonconvex optimization,
- energy minimization,
- lightly constrained black-box search,
- large-scale constrained nonlinear and quadratically constrained programs (QCQP / NLP) when an incumbent warm-start and a solver chain (IPOPT / MUMPS / MA57) are available,
- problems where candidate solutions are compact and cheap to verify.
This is not a claim of a general-purpose optimizer.
So far, the system has been less successful on:
- routing / placement with many interacting constraints,
- problems dominated by feasibility engineering,
- objectives where the evaluator is hard to reproduce.
| Problem | This repo's result | Reference | Status | Details |
|---|---|---|---|---|
| n=26 circle packing in unit square (maximize sum of radii) | sum r = 2.6359830849175889 | 2.6359830822781625 (Aemon) | new SOTA at floating-point precision | details |
| n=32 circle packing in unit square (maximize sum of radii) | sum r = 2.9395727712007664 | 2.939572726664292 (Berthold et al., Jan 2026, arXiv:2601.05943; raw data at DominikKamp/Packing) | new SOTA at floating-point precision (+4.45e-8) | details |
| n=21 circle packing in a rectangle (perimeter 4) (maximize sum of radii) | sum r = 2.365832375910822 | 2.3658321334167627 (AlphaEvolve, DeepMind notebook B.13) | new SOTA at floating-point precision (+2.42e-7) | details |
| n=26 circle packing in a rectangle (perimeter 4) (maximize sum of radii) | sum r = 2.6393205704880214 | 2.63930 (Berthold et al., Jan 2026, arXiv:2601.05943) | new SOTA at 5-decimal precision (2.63932 vs 2.63930) | details |
| n=27 circle packing in a rectangle (perimeter 4) (maximize sum of radii) | sum r = 2.691523369458056 | 2.69015 (Berthold et al., Jan 2026, arXiv:2601.05943) | new SOTA (2.69152 vs 2.69015, +1.37e-3) | details |
| Spherical code / Tammes problem on S^5, N=86 (minimize max pairwise dot) | max dot = 0.548916479201208 | 0.548918184883 (Henry Cohn, spherical-codes.org, 2026, "needs more optimization") | new best numerical code at verifier precision | details |
| Spherical code / Tammes problem on S^5, N=98 (minimize max pairwise dot) | max dot = 0.571037778803683 | 0.571052839653 (Henry Cohn, spherical-codes.org, 2026, "needs more optimization") | new best numerical code at verifier precision | details |
| Problem | This repo's result | Reference | Status | Details |
|---|---|---|---|---|
| Lennard-Jones 38-atom cluster (minimum energy) | U = -173.92842659 | -173.928427 (Cambridge canonical, Gomez/Pillardy/Doye) | matches the canonical global minimum | details |
| Lennard-Jones 75-atom cluster (minimum energy) | U = -397.4923309829 | -397.492331 (Marks decahedral global, Doye/Wales/Locatelli) | matches the canonical global minimum | details |
| Lennard-Jones 104-atom cluster (minimum energy) | U = -582.0866420676 | -582.086642 (Doye2 C2v Marks-icosahedral global, Doye-Wales 1995) | matches the canonical global minimum | details |
MINLPLib is the COIN-OR / GAMS benchmark library
for mixed-integer nonlinear programs. The QCQP instance below is a
large-scale AC Optimal Power Flow model on the European Pegase grid, with a
linear cost objective subject to nonlinear (lifted bilinear) power-flow
constraints. MINLPLib's published acceptance gate for primal submissions is
infeas_max <= 1e-8 against the official .nl evaluator.
| Problem | This repo's result | Published primal | Status | Details |
|---|---|---|---|---|
| acopf_case13659pegase_qcqp (13,659-bus European AC OPF; 199,281 vars, 191,097 constraints; non-convex QCQP) | obj = 386106.5446322 (infeas 1.79e-12) | 386108.80970 (infeas 1e-10, MINLPLib p1) | new SOTA primal — accepted by MINLPLib as p2 on 2026-05-25 (different generator-dispatch basin than p1, Δ obj ≈ -2.265) | details |
| acopf_case1354pegase_qcqp (1354-bus European AC OPF; 19,236 vars, 21,580 constraints; non-convex QCQP) | obj = 74068.79660229431 (infeas 9.9997e-9) | 74069.35457 (infeas 8e-11, MINLPLib p1) | matches p1 SOTA — tolerance-budget farming inside the 1e-8 gate (polishes back to p1; not added by MINLPLib) |
details |
Each result folder contains the candidate data that is available for that problem, the relevant reference value, and any result-specific caveats.
For the n=26 circle-packing result, the repository includes a local strict checker:
python n26_circle_packing/verify.py n26_circle_packing/best_26_circles.jsonThe n=32 unit-square result and the three rectangle-packing results each ship
their own strict verifier; the verifier finds the adjacent solution.json
automatically:
python n32_circle_packing/verify.py
python n21_circle_packing_rectangle/verify.py
python n26_circle_packing_rectangle/verify.py
python n27_circle_packing_rectangle/verify.pyThe spherical-code entries (S^5, N=86 and N=98) include the coordinate files and the captured verification report from the experiment run. The codes themselves were verified by Henry Cohn: submissions to spherical-codes.org are rejected if they fail his correctness checks, so listing there is itself an external verification step. No separate Tammes verifier is currently included in this repository.
The public repository is a result and verification archive.
The optimization harness, operating rules, prompts, and full per-run traces are not included in this repository. They are kept separate to preserve the method for private benchmark evaluation, collaboration, or licensing.
For serious technical review, I can provide additional evidence under an appropriate agreement, including selected run logs, verifier outputs, and candidate-generation history.
I am interested in private executable benchmarks where:
- the problem is continuous, geometric, or lightly constrained,
- feasible solutions can be represented compactly,
- an incumbent or baseline solution is available,
- improvement can be independently verified,
- the evaluator is deterministic or reasonably stable,
- the evaluator can be run locally or in a controlled environment.
A useful private test has the following structure:
- You provide an executable evaluator and one or more incumbent solutions.
- I run the optimizer without access to your proprietary internal methods.
- I return candidate solutions, verifier logs, and a short technical report.
- If there is a verified improvement, we can discuss paid follow-up, licensing, or collaboration.
The optimization harness itself is private. Public result files, verifier scripts, candidate solutions, and technical reports can be shared where appropriate.
Contact: Matevz Kovacic
Email: matevz.celje@gmail.com
MIT - see LICENSE. Citation requested as a courtesy (see CITATION.cff), not
as a license condition.
Kovacic, M. (2026). Active Model: cold-start results on circle packing,
spherical codes, and Lennard-Jones cluster minimization.
https://github.com/matevz-kovacic/active-model