Proximal Newton-type methods for minimizing composite functions

We generalize Newton-type methods to minimize a sum of two convex functions: a smooth function and a nonsmooth function with a simple proximal mapping. We show these proximal Newton-type methods inherit the desirable convergence properties of Newton-type methods for minimizing smooth functions. Many popular methods tailored to problems arising in bioinformatics, machine learning, signal processing are cases of proximal Newton-type methods, and our results yield new convergence guarantees for some of these methods.