  We generalize Newton-type methods for minimizing smooth functions to handle a
sum of two convex functions: a smooth function and a nonsmooth function with a
simple proximal mapping. We show that the resulting proximal Newton-type
methods inherit the desirable convergence behavior of Newton-type methods for
minimizing smooth functions, even when search directions are computed
inexactly. Many popular methods tailored to problems arising in bioinformatics,
signal processing, and statistical learning are special cases of proximal
Newton-type methods, and our analysis yields new convergence results for some
of these methods.
