Distributed accelerated primal-dual neurodynamic approaches for resource allocation problem

Abstract

This paper investigates two distributed accelerated primal-dual neurodynamic approaches over undirected connected graphs for resource allocation problems (RAP) where the objective functions are generally convex. With the help of projection operators, a primal-dual framework, and Nesterov’s accelerated method, we first design a distributed accelerated primal-dual projection neurodynamic approach (DAPDP), and its convergence rate of the primal-dual gap is $O\left( {{1 \over {{t^2}}}} \right)$ by selecting appropriate parameters and initial values. Then, when the local closed convex sets are convex inequalities which have no closed-form solutions of their projection operators, we further propose a distributed accelerated penalty primal-dual neurodynamic approach (DAPPD) on the strength of the penalty method, primal-dual framework, and Nesterov’s accelerated method. Based on the above analysis, we prove that DAPPD also has a convergence rate $O\left( {{1 \over {{t^2}}}} \right)$ of the primal-dual gap. Compared with the distributed dynamical approaches based on the classical primal-dual framework, our proposed distributed accelerated neurodynamic approaches have faster convergence rates. Numerical simulations demonstrate that our proposed neurodynamic approaches are feasible and effective.