A distributed Newton method for Network Utility Maximization

Abstract

Most existing work uses dual decomposition and first-order methods to solve Network Utility Maximization (NUM) problems in a distributed manner, which suffer from slow rate of convergence properties. This paper develops an alternative distributed Newton-type fast converging algorithm for solving NUM problems with self-concordant utility functions. By using novel matrix splitting techniques, both primal and dual updates for the Newton step can be computed using iterative schemes in a decentralized manner. We propose a stepsize rule and provide a distributed procedure to compute it in finitely many iterations. The key feature of our direction and stepsize computation schemes is that both are implemented using the same distributed information exchange mechanism employed by first order methods. We show that even when the Newton direction and the stepsize in our method are computed within some error (due to finite truncation of the iterative schemes), the resulting objective function value still converges superlinearly in terms of primal iterations to an explicitly characterized error neighborhood. Simulation results demonstrate significant convergence rate improvement of our algorithm relative to the existing first-order methods based on dual decomposition.

Distributed Newton Methods for Strictly Convex Consensus Optimization Problems in Multi-Agent Networks

“…By splitting the matrix AH −1 k A as the sum of C k +B k and B k −B k , the following theorem [21] can be obtained.…”

“…Note that the predetermined routes and full row rank coefficient matrix are necessary when running the distributed Newton method for the NUM problem according to Reference [21]. Unfortunately, this property is usually not met in the general multi-agents consensus optimization problems.…”

“…Wei et al [21] proposed a distributed Newton method for the NUM problem (1) by using a matrix splitting scheme to compute the dual vector ω k in Equation (5) in a distributed manner. Let C k be a diagonal matrix, with diagonal entries…”

“…Another second-order method-the distributed Newton method-was proposed by Wei et al [21] to solve the network utility maximization (NUM) problem in a distributed manner. NUM can be formulated as a convex optimization problem with equality constraints by introducing some slack variables and the coefficient matrix of the equality constraints having full row rank.…”

Network optimization with dynamic demands and link prices

“…The recent work by Zargham et al gives a more general method, Accelerated Dual Descent (ADD), that is based on the h th order Taylor approximation [5]. Wei et al propose a distributed consensus algorithm to compute the Newton direction in each iteration of the descent algorithm [6], [7]. A similar method based on gossiping and consensus techniques was proposed by Modiano et al [8].…”