Proximal Diagonal Newton Methods for Composite Optimization Problems

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

Inexact proximal DC Newton-type method for nonconvex composite functions

We consider a class of difference-of-convex (DC) optimization problems where the objective function is the sum of a smooth function and a possible nonsmooth DC function. The application of proximal DC algorithms to address this problem class is well-known. In this paper, we combine a proximal DC algorithm with an inexact proximal Newton-type method to propose an inexact proximal DC Newton-type method. We demonstrate global convergence properties of the proposed method. In addition, we give a memoryless quasi-Newton matrix for scaled proximal mappings and consider a two-dimensional system of semi-smooth equations that arise in calculating scaled proximal mappings. To efficiently obtain the scaled proximal mappings, we adopt a semi-smooth Newton method to inexactly solve the system. Finally, we present some numerical experiments to investigate the efficiency of the proposed method, showing that the proposed method outperforms existing methods

大規模な無制約最適化問題に対するメモリーレス準ニュートン法

東京理科大学201