Synchronous and asynchronous Bertsekas methods for energy-aware networks and solution of Regularized Linear Systems

Abstract

In this work, augmented Lagrangian-based techniques are developed and analyzed for two types of large-scale optimization problems that are pertinent to contemporary computing systems: the solution of regularized linear systems and energy-aware network routing with bandwidth allotment. In the first problem, a mixed-integer nonlinear algorithm is used to jointly optimize bandwidth allocation and routing in communication networks under energy restrictions. In the second, stability terms like Ψ2 regularization are introduced in regular-ized linear systems to solve ill-posed or ill-conditioned problems. For both problems, synchronous and asyn-chronous distributed optimization schemes are designed using problem characterized by discrete variables and inherent non-convexities. Such formulations pose significant computational challenges and demand algorithmic strategies that can scale while maintaining solution quality [3], [4].

In parallel, the solution of regularized linear systems has become increasingly important across domains such as signal processing, numerical optimization, and machine learning. Such problems arise when seeking stable solutions to ill-posed or ill-conditioned systems, where regularization terms (e.g., ℓ₂ penalties) are introduced to ensure robustness and 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 scala-bility conditions. These findings reinforce the practical value of asynchronous augmented Lagrangian methods for distributed, high-dimensional, and delay-sensitive optimization problems.