An inexact proximal Newton algorithm with partial Hessian information for nonsmooth separable optimization

We propose a regularized proximal Newton algorithm with partial Hessian information for minimizing the sum of a twice continuously differentiable function and a separable convex (possibly nonsmooth) function. We prove that every accumulation point of the iterates is a stationary point while allowing the inner minimization to be solved inexactly. Moreover, we prove that when the problem possesses the local Hölderian error bound property, the local convergence rate of the iterates is superlinear. The efficiency of the proposed algorithm is demonstrated by comparisons with several state-of-the-art algorithms on the ℓ1-regularized problems.