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