Classic inverse problems are formulated using smooth penalties and regularizations. However, nonsmooth and nonconvex penalties/regularizers have proved to be extremely useful in underdetermined and noisy settings. Problems with these features also arise naturally when modeling complex physical and chemical phenomena; including PDE-constrained optimization, phase retrieval, and structural resolution of bio-molecular models.

We propose a new technique for solving a broad range of nonsmooth, nonconvex problems. The technique is based on a relaxed reformulation, and can be implemented on a range of problems in a simple and scalable way. In particular, we typically need only solve least squares problems, as well as implement custom separable operators. We discuss the problem class, reformulation and algorithms, and give numerous examples of very promising numerical results in different applications.