Geodesic
Solving regularized linear least-squares problems by the alternating direction method with applications to image restoration
Electronic transactions on numerical analysis, Tome 40 (2013), pp. 356-372
We present and analyze ways to apply the Alternating Direction Method (ADM) to bound-constrained quadratic problems including $\ell_1$ and $\ell_2$ regularized linear least-squares problems. The resulting ADM schemes require the solution of two subproblems at each iteration: the first one is a linear system, the second one is a bound-constrained optimization problem with closed-form solution. Numerical results on image deblurring problems are provided and comparisons are made with a Newton-based method and a first-order method for bound-constrained optimization.
Classification : 65F22, 65K10, 65T50, 68U10, 90C25
Keywords: linear least-squares problems, $\ell_1$ and $\ell_2$ regularization, bound-constraints, alternating direction method, image deblurring 
@article{ETNA_2013__40__a7,
     author = {Zhang,  Jianjun and Morini,  Benedetta},
     title = {Solving regularized linear least-squares problems by the alternating direction method with applications to image restoration},
     journal = {Electronic transactions on numerical analysis},
     pages = {356--372},
     year = {2013},
     volume = {40},
     zbl = {1288.65090},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/ETNA_2013__40__a7/}
}
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Zhang,  Jianjun; Morini,  Benedetta. Solving regularized linear least-squares problems by the alternating direction method with applications to image restoration. Electronic transactions on numerical analysis, Tome 40 (2013), pp. 356-372. http://geodesic.mathdoc.fr/item/ETNA_2013__40__a7/