Proximal Diagonal Newton Methods for Composite Optimization Problems

Abstract: 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.