A modified local quadratic approximation algorithm for penalized optimization problems

In this paper, we propose an optimization algorithm called the modified local quadratic approximation algorithm for minimizing various ℓ1-penalized convex loss functions. The proposed algorithm iteratively solves ℓ1-penalized local quadratic approximations of the loss function, and then modifies the solution whenever it fails to decrease the original ℓ1-penalized loss function. As an extension, we construct an algorithm for minimizing various nonconvex penalized convex loss functions by combining the proposed algorithm and convex concave procedure, which can be applied to most nonconvex penalty functions such as the smoothly clipped absolute deviation and minimax concave penalty functions. Numerical studies show that the algorithm is stable and fast for solving high dimensional penalized optimization problems.