Synchronous and Asynchronous Bertsekas Methods for Energy-Aware Networks and Solution of Regularized Linear Systems

Abstract

This paper develops and analyzes augmented Lagrangian-based methods for two classes of large-scale optimization problems relevant to modern computational systems: energy-aware network routing with bandwidth allocation and the solution of regularized linear systems. In the first problem, routing and bandwidth allocation are jointly optimized in communication networks under energy constraints, modeled as a mixed-integer nonlinear program. In the second, regularized linear systems are formulated to address ill-posed or ill-conditioned problems by introducing stabilization terms such as $\ell_2$ regularization. For both problems, synchronous and asynchronous distributed optimization schemes are designed using decomposition techniques grounded in augmented Lagrangian theory. Extensive numerical experiments across diverse datasets, including network flow instances and benchmark regularized linear systems, demonstrate that the asynchronous variants retain comparable solution quality while significantly improving computational performance, particularly under delay and scalability conditions. These findings reinforce the practical value of asynchronous augmented Lagrangian methods for distributed, high-dimensional, and delay-sensitive optimization problems.