Alternating minimization algorithms with a shrinkage step, derived within the Split Bregman (SB) or Alternating Direction Method of Multipliers (ADMM) frameworks, have become very popular for ℓ1-regularized problems, including Total Variation and Basis Pursuit Denoising. It appears to be generally assumed that they deliver much better computational performance than older methods such as Iteratively Reweighted Least Squares (IRLS). We show, however, that IRLS type methods are computationally competitive with SB/ADMM methods for a variety of problems, and in some cases outperform them. © 2012 IEEE.