A superlinearly convergent R‐regularized Newton scheme for variational models with concave sparsity‐promoting priors

A general class of variational models with concave priors is considered for obtaining certain sparse solutions, for which nonsmoothness and non‐Lipschitz continuity of the objective functions pose significant challenges from an analytical as well as numerical point of view. For computing a stationary point of the underlying variational problem, a Newton‐type scheme with provable convergence properties is proposed. The possible non‐positive definiteness of the generalized Hessian is handled by a tailored regularization technique, which is motivated by reweighting as well as the classical trust‐region method. Our numerical experiments demonstrate selected applications in image processing, support vector machines, and optimal control of partial differential equations.