Alternating Linearization for Structured Regularization Problems

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We adapt the alternating linearization method for proximal decomposition to structured regularization problems, in particular, to the generalized lasso problems. The method is related to two well-known operator splitting methods, the Douglas--Rachford and the Peaceman--Rachford method, but it has descent properties with respect to the objective function. This is achieved by employing a special update test, which decides whether it is beneficial to make a Peaceman--Rachford step, any of the two possible Douglas--Rachford steps, or none. The convergence mechanism of the method is related to that of bundle methods of nonsmooth optimization. We also discuss implementation for very large problems, with the use of specialized algorithms and sparse data structures. Finally, we present numerical results for several synthetic and real-world examples, including a three-dimensional fused lasso problem, which illustrate the scalability, efficacy, and accuracy of the method.