  In this paper, we propose new methods to efficiently solve convex
optimization problems encountered in sparse estimation, which include a new
quasi-Newton method that avoids computing the Hessian matrix and improves
efficiency, and we prove its fast convergence. We also prove the local
convergence of the Newton method under weaker assumptions. Our proposed methods
offer a more efficient and effective approach, particularly for L1
regularization and group regularization problems, as they involve variable
selection with each update. Through numerical experiments, we demonstrate the
efficiency of our methods in solving problems encountered in sparse estimation.
Our contributions include theoretical guarantees and practical applications for
various problems.
