  This paper proposes new proximal Newton-type methods with a diagonal metric
for solving composite optimization problems whose objective function is the sum
of a twice continuously differentiable function and a proper closed
directionally differentiable function. Although proximal Newton-type methods
using diagonal metrics have been shown to be superior to the proximal gradient
method numerically, no theoretical results have been obtained to suggest this
superiority. Even though our proposed method is based on a simple idea, its
convergence rate suggests an advantage over the proximal gradient method in
certain situations. Numerical experiments show that our proposed algorithms are
effective, especially in the nonconvex case.
