Solving Convex Optimization Problem Used for High Quality Denoising The highest voted answer to this question suggests that to denoise a signal while preserving sharp transitions one should 


  minimize the objective function:
  
  $$ |x-y|^2 + b|f(y)| $$
  
  where $x$ is the noisy signal, $y$ is the denoised signal, $b$ is the
  regularziation parameter, and $|f(y)|$ is some L1 norm penalty.
  Denoising is accomplished by finding the solution $y$ to this
  optimization problem, and $b$ depends on the noise level.


However, there is no indication of how one might accomplish that in practice as that is a problem in a very high dimensional space, especially if the signal is e.g. 10 million samples long. In practice how is this sort of problem solved computationally for large signals?
Boyd has A Matlab Solver for Large-Scale ℓ1-Regularized Least Squares Problems. The problem formulation in there is slightly different, but the method can be applied for the problem.

Classical majorization-minimization approach also works well. This corresponds to iteratively perform soft-thresholding (for TV, clipping).

The solutions can be seen from the links. However, there are many methods to minimize these functionals by the extensive use of optimisation literature.

PS: As mentioned in other comments, FISTA will work well. Another 'very fast' algorithm family is primal-dual algorithms. You can see the interesting paper of Chambolle for an example, however there are plethora of research papers on primal-dual methods for linear inverse problem formulations.