Geodesic
A conjugate subgradient algorithm with adaptive preconditioning for the least absolute shrinkage and selection operator minimization
Žurnal vyčislitelʹnoj matematiki i matematičeskoj fiziki, Tome 57 (2017) no. 4

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This paper describes a new efficient conjugate subgradient algorithm which minimizes a convex function containing a least squares fidelity term and an absolute value regularization term. This method is successfully applied to the inversion of ill-conditioned linear problems, in particular for computed tomography with the dictionary learning method. A comparison with other state-of-art methods shows a significant reduction of the number of iterations, which makes this algorithm appealing for practical use. 
@article{ZVMMF_2017_57_4_a12,
     author = {A. Mirone and P. Paleo},
     title = {A conjugate subgradient algorithm with adaptive preconditioning for the least absolute shrinkage and selection operator minimization},
     journal = {\v{Z}urnal vy\v{c}islitelʹnoj matematiki i matemati\v{c}eskoj fiziki},
     pages = {744},
     publisher = {mathdoc},
     volume = {57},
     number = {4},
     year = {2017},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/ZVMMF_2017_57_4_a12/}
}
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A. Mirone; P. Paleo. A conjugate subgradient algorithm with adaptive preconditioning for the least absolute shrinkage and selection operator minimization. Žurnal vyčislitelʹnoj matematiki i matematičeskoj fiziki, Tome 57 (2017) no. 4. http://geodesic.mathdoc.fr/item/ZVMMF_2017_57_4_a12/