Convergence analysis of inexact proximal Newton-type methods

We study inexact proximal Newton-type methods to solve convex optimization problems in composite form: minimize x∈Rn f(x) := g(x) + h(x), where g is convex and continuously differentiable and h : R → R is a convex but not necessarily differentiable function whose proximal mapping can be evaluated efficiently. Proximal Newton-type methods require the solution of subproblems to obtain the search directions, and these subproblems are usually solved using first-order methods or coordinate descent methods, which converge to the solution linearly. In this paper, we analyze the convergence rate of inexact proximal Newton method, which solves the subproblems inexactly.