smopt
Stiefel Manifold Optimization: Penalty-Free First-Order Solvers
smopt minimizes a smooth function, optionally plus a nonsmooth
regularizer, over the matrices with orthonormal columns:
$$ \min_{X \in \mathbb{R}^{n\times p}} f(X) + r(X) \quad \text{subject to} \quad X^\top X = I_p. $$
The solvers are penalty-free first-order methods. Rather than retracting along geodesics, they work in the ambient space and dissolve the orthogonality constraint with a cheap feasibility restoring map, so an iteration costs little more than a gradient evaluation and a couple of small matrix products.
Everything numerical — the manifold geometry, the proximal operators, the Barzilai-Borwein step sizes and the solver loops themselves — is written in Fortran 77 and reached through f2py. Python supplies the objective through a callback and handles reporting. NumPy is the only runtime dependency; the extension links against nothing but the Fortran runtime.
Quick example
import numpy as np
from smopt import slpg_smooth, Stiefel
n, p = 1000, 10
M = Stiefel(n, p)
A = np.diag(np.arange(n, dtype=float))
def obj_fun(X):
"""Return the objective and its Euclidean gradient together."""
AX = A @ X
return float(np.sum(X * AX)), 2.0 * AX
X, out = slpg_smooth(obj_fun, M)
print(out["fval"], out["fea"])
Solvers
| Name | Comment | Call |
|---|---|---|
slpg_smooth |
penalty-free first-order method for smooth problems | slpg_smooth(obj_fun, M) |
slpg |
penalty-free first-order method for nonsmooth problems | slpg(obj_fun, M, prox=...) |
slpg_l21 |
penalty-free first-order method for $\ell_{2,1}$ regularized problems | slpg_l21(obj_fun, M, gamma=...) |
pencf |
constraint dissolving penalty method | pencf(xinit, obj_fun, M) |
Every solver returns (X, out), where out carries the fvals, kkts
and feas histories together with the final fval, kkt and fea.
Installation
pip install smopt
Requires Python 3.10+ and NumPy. See the full installation guide for uv, poetry, and source builds.
Building from source additionally needs a Fortran compiler; the wheels carry the Fortran runtime, so installing one does not.
Layout
src/
smblas.f dense kernels: matmul, Cholesky, Jacobi eigensolver, Gram-Schmidt
smman.f Stiefel geometry: C, JA, JC, the A map, the polar retraction
smprox.f proximal operators and the l_{2,1} multiplier
smslpg.f the SLPG solver drivers and the Arrow-Hurwicz inner iteration
smpencf.f the pencf driver
_smopt.pyf f2py signatures binding the above to Python
smopt/ the thin Python layer: argument marshalling and reporting
tests/
reference.py a NumPy transcription of the algorithm, used as a test oracle
Documentation
- Theory — the manifold, the constraint dissolving map, the algorithms
- Quickstart — runnable examples
- API Reference — class and function signatures and arguments
- References — literature citations
Acknowledgement
The algorithm ported here originates in the STOP toolbox by Nachuan
Xiao, Lei Wang, Bin Gao, Xin Liu and Ya-xiang Yuan
(https://stmopt.gitee.io/). smopt re-implements its numerics in
Fortran 77 behind the same solver interface.
License
GNU General Public License v3 (GPLv3) — see LICENSE.
Metadata
Release files for smopt 0.1.1
For a detailed explanation of source distributions (sdists) and built distributions (wheels), please see the package formats documentation.
Built distributions (wheels)
Total release size: 2.5 MB
Release files / smopt-0.1.1-cp315-cp315t-win_amd64.whl
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