## Solving regularized linear least-squares problems by the alternating direction method with applications to image restoration

Jianjun Zhang and Benedetta Morini

### Abstract

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.

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### Key words

Linear least-squares problems, $\ell_1$ and $\ell_2$ regularization, bound-constraints, alternating direction method, image deblurring.

### AMS subject classifications

65F22, 65K10, 65T50, 68U10, 90C25

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