Quadratic Convergence of Newton's Method for Convex Interpolation and Smoothing

Abstract. In this paper, we prove that Newton's method for convex best interpolation is locally quadratically convergent, giving an answer to a question of Irvine, Marin, and Smith [7] and strengthening a result of Andersson and Elfving [1] and our previous work [5]. A damped Newton-type method is presented which has global quadratic convergence. Analogous results are obtained for the convex smoothing problem. Numerical examples are presented.