Search

Recently, an idling mechanism has been introduced in the context of distributed first order methods for minimization of a sum of nodes’ local convex costs over a generic, connected network. With the idling mechanism, each node  $i$ , at each iteration  $k$ , is active—updates its solution estimate and exchanges messages with its network neighborhood—with probability  $p_k$ , and it stays idle with probability  $1-p_k$ , while the activations are independent both across nodes and across iterations. In this paper, we demonstrate that the idling mechanism can be successfully incorporated in distributed second-order methods also. Specifically, we apply the idling mechanism to the recently proposed distributed quasi-Newton (DQN) method. We show that, when $p_k$ grows to one across iterations geometrically, DQN with idling exhibits very similar theoretical convergence and convergence rates properties as the standard DQN method, thus achieving the same order of convergence rate (R-linear) as the standard DQN, but with significantly cheaper updates. [ABSTRACT FROM AUTHOR]

This paper proposes a new Newton-like method which defines new iterates using a linear system with the same coefficient matrix in each iterate, while the correction is performed on the right-hand-side vector of the Newton system. In this way a method is obtained which is less costly than the Newton method and faster than the fixed Newton method. Local convergence is proved for nonsingular systems. The influence of the relaxation parameter is analyzed and explicit formulae for the selection of an optimal parameter are presented. Relevant numerical examples are used to demonstrate the advantages of the proposed method.