A row relaxation method for large l1 problems

Abstract

This paper presents a row relaxation method for solving the regularized l 1 problem minimize 1 2 ε‖ x‖ 2 2 +‖A x− b‖ 1. It is shown that the dual of this problem has the form minimize 1 2 ‖A T y‖ 2 2 −ε b T y subject to ‖ y‖ ∞⩽1, and if y solves the dual, then A T y ε solves the primal. The fact that the dual variables have simple bounds enables us to apply a wide range of methods. The method derived in this paper is a row relaxation method that resembles Kaczmarz's method. This feature makes it suitable for solving problems in which A is large, sparse, and unstructured. Another advantage is that the method is easily adapted to handle linear constraints. The paper introduces an iterative improvement technique that shifts the limit of the iterative process toward a solution of the unregularized problem minimize ‖A x− b‖ 1 .