A Smoothing Descent Method for Nonconvex TV $$^q$$ -Models

A novel class of variational models with nonconvex `q-normtype regularizations (0 < q < 1) is considered, which typically outperforms popular models with convex regularizations in restoring sparse images. Due to the fact that the objective function is nonconvex and nonLipschitz, such models are very challenging from an analytical as well as numerical point of view. In this work a smoothing descent method with provable convergence properties is proposed for computing stationary points of the underlying variational problem. Numerical experiments are reported to illustrate the effectiveness of the new method.