Fast Methods for Nonsmooth Nonconvex Problems using Variable Projection
Abstract: Classic inverse problems are formulated by 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 biomolecular 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.