For almost 50 years l_2-Augmented Lagrangian algorithms have been around and they are still frequently used today. One way of looking at them is to consider them as a modification of the inexact quadratic penalty function (which requires that the penalty parameter becomes unbounded) by adding Lagrangian terms. The resulting advantage is that by way of updating the Lagrangian multipliers one can solve the original constrained problem whilst bounding the penalty parameter.
In this talk I will describe an l_1-Augmented Lagrangian which one could consider as a modification of the exact l_1-penalty function by adding Lagrangian terms. Since the penalty parameter for exact penalty functions remains bounded anyway and furthermore the l_1 exact penalty function is not differentiable, this does not sound like a good idea. I hope to convince you otherwise.
I will include, motivation, theory, context and some provisional numerical results.