\( \qquad\; \min_X \{f(X) + \mu\, \|X\|_* \mid X \in \Re^{m\times n}\} \\[5pt] \qquad \min_X \{f(X) + \mu\,{\rm trace}(X) \mid X \succeq 0, X\in {\cal S}^n\} \)

where \(\mu > 0\) is a regularization paramter, and \(f(X) = \frac{1}{2}\|A(X)-b\|^2\).Important note: this is a research software. It is not intended nor designed to be a general purpose software at the moment. The solver is expected to work well only for favorable problems such as nuclear norm regularized random matrix completion problems. Being a gradient method, it is quite sensitive to the various parameters used in the algorithm. The selection of the parameters typically depend on the class of problems being solved.

For more details, see:

- Copyright: This version of NNLS is distributed under the GNU General Public License 2.0. For commercial applications that may be incompatible with this license, please contact the authors to discuss alternatives.
- User guide: in preparation.
**Download NNLS-0.zip**

Please read. Welcome to NNLS-0! The software requires a few Mex files for execution. You can generate (only need to be done once) these Mex files as follows:- Firstly, unpack the software:

unzip NNLS-0.zip - Run Matlab in the directory NNLS-0
- In the Matlab command window, type:

>> Installmex

- After that, to see whether you have installed NNLS-0 correctly,
type:

>> startup

>> runRandomMatComp - By now, NNLS is ready for you to use.

- Firstly, unpack the software:

Some segmentation errors have been reported when running NNLS-0 on Windows 64-bit PCs. We have not observed such errors when running on MATLAB version 7.13.0.564 (R2011b). For example, when running the m-file runRandomMatComp, we get the following successful run:

>> runRandomMatComp Create matrix 1 with rank = 10, Problem: n = 5000, p = 600614, r = 10, p/dr = 6.01e+000, p/n^2 = 2.40e+000% mu = 1.37e-002, noise ratio, sigma = 0.00e+000, 0.00e+000 nr = 5000, nc = 5000, m = 600614 Lipschitz const = 1.00e+000 mu_target = 1.37e-002 matrix storage format = factors ------------------------------------------------------------------------------------------ it mu tau svp relDist relRes relObjdiff obj time ------------------------------------------------------------------------------------------+ 1 1.37e+002 1.00e+000 | 2| 0.00e+000 1.00e+000 1.00e+000| 3.0200e+006| 3.28e-001|+ [0.0e+000] 2 9.62e+001 1.00e+000 | 1 1| 9.86e-001 9.99e-001 5.56e-004| 3.0183e+006| 5.31e-001|+ [0.0e+000] 3 6.74e+001 1.00e+000 | 2 2| 6.83e-001 9.96e-001 3.71e-003| 3.0072e+006| 6.55e-001|+ [0.0e+000] 4 4.72e+001 1.00e+000 | 3 3| 4.70e-001 9.89e-001 1.07e-002| 2.9755e+006| 7.49e-001|+ [0.0e+000] 45 1.37e-002 4.40e-002 | 10 11| 2.61e-004 1.54e-004 1.97e-005| 6.8908e+002| 6.8911e+002| 1.56e+001| [1.1e+004] 46 1.37e-002 4.40e-002 | 10 11| 2.05e-004 1.48e-004 2.89e-005| 6.8906e+002| 6.8908e+002| 1.58e+001| [1.0e+004] 47 1.37e-002 4.40e-002 | 10 11| 1.45e-004 1.37e-004 4.40e-005| 6.8903e+002| 6.8904e+002| 1.60e+001| [8.4e+003] 48 1.37e-002 4.40e-002 | 10 11| 9.63e-005 1.35e-004 3.17e-005| 6.8901e+002| [a] relDist < 1.00e-004 Finished the main algorithm! Objective function = 6.89011e+002 Number of iterations = 48 Number of singular values = 10 max singular value = 5.23e+003 min singular value = 4.79e+003 CPU time = 1.61e+001 norm(X-Xold,'fro')/norm(X,'fro') = 9.63e-005 Problem: nr = 5000, nc = 5000, p = 600614, r = 10, p/dr = 6.01e+000, p/n^2 = 2.40e+000% mu = 1.37e-002, noise ratio, sigma = 0.00e+000, 0.00e+000 ----------------------------------------------------------------------------- problem type : NNLS randstate : 1 iterations : 48 # singular : 10 obj value : 6.89011e+002 cpu time : 1.62e+001 mean square error : 1.85e-004 ------------------------------------------------------------------------------