Globally convergent B-semismooth Newton methods for l1-Tikhonov regularization
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Abstract
We are concerned with the globalization of a semismooth Newton method for l1-Tikhonov regularization. This regularization strategy for inverse problems with sparsity constraints leads to a nonsmooth minimization problem. Based on the generalized derivative concept of Newton differentiability, a locally superlinearly convergent, semismooth Newton method has been proposed in the literature. However, the convergence of local Newton methods is not guaranteed in general for an arbitrary initial guess. In order to globalize the algorithm, we consider a B(ouligand)-Newton method. We discuss the feasibility of the B-Newton method. The resulting algorithm is called a B-semismooth Newton method because it can also be interpreted as a semismooth Newton method. The algorithm converges locally superlinearly and the Newton equations are finite-dimensional. The B-Newton directions satisfy a descent property with respect to the square norm of the residual. We globalize the algorithm in a finite-dimensional setting by inexact line search. A drawback of this approach is that the sequence of iterates might begin to stagnate. Therefore, by a modification of the Newton equation, a globally convergent algorithm is proposed. We recommend a locally superlinearly convergent, hybrid algorithm that combines both methods. We present numerical results that demonstrate the efficiency of the methods.