Relaxed regularization for linear inverse problems

We consider regularized least-squares problems of the form min x 1/2 kAx − bk22+ R(Lx). Recently, Zheng et al. [45] proposed an algorithm called Sparse Relaxed Regularized Regression (SR3) that employs a splitting strategy by introducing an auxiliary variable y and solves minx,y 1 2 kAx − bk22 + k/2 2 kLx − yk22 + R(x). By minimizing out the variable x, we obtain an equivalent optimization problem miny 1 2 kFy − gk22 + R(y). In our work, we view the SR3 method as a way to approximately solve the regularized problem. We analyze the conditioning of the relaxed problem in general and give an expression for the SVD of F as a function of κ. Furthermore, we relate the Pareto curve of the original problem to the relaxed problem and we quantify the error incurred by relaxation in terms of κ. Finally, we propose an efficient iterative method for solving the relaxed problem with inexact inner iterations. Numerical examples illustrate the approach.

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