Relax-and-split method for nonsmooth nonconvex problems

We develop and analyze a new `relax-and-split' (RS) approach for compositions of separable nonconvex nonsmooth functions with linear maps. RS uses a relaxation technique together with partial minimization, and brings classic techniques including direct factorization, matrix decompositions, and fast iterative methods to bear on nonsmooth nonconvex problems. We also extend the approach to trimmed nonconvex-composite formulations; the resulting Trimmed RS (TRS) can fit models while detecting outliers in the data. We then test RS and TRS on a diverse set of applications: (1) phase retrieval, (2) stochastic shortest path problems, (3) semi-supervised classification, and (4) new clustering approaches. RS/TRS can be applied to models with very weak functional assumptions, are easy to implement, competitive with existing methods, and enable a new level of modeling formulations to be put forward to address emerging challenges in the mathematical sciences.