Distributed Inertial Continuous and Discrete Time Algorithms for Solving Resource Allocation Problem

You Zhao, Xiaofeng Liao, Xing He

Abstract:

In this article, we investigate several distributed inertial algorithms in continuous and discrete time for solving resource
allocation problem (RAP), where its objective function is convex
or strongly convex. First, the original RAP is equivalently transformed into a distributed unconstrained optimization problem by
introducing an auxiliary variable. Then, two distributed inertial
continuous time algorithms and two discrete time algorithms are
proposed and the rates of their convergence based on the gap
between the objective function and their optimal function are
determined. Our first distributed damped inertial continuous time
algorithm is designed for RAP with a convex function, it achieves
convergence rate at O( t
1
2 ) based on Lyapunov analysis method,
and then we design a rate-matching distributed damped inertial
discrete time algorithm by exploiting implicit and Nesterov’s discretization scheme. Our second distributed fixed inertial discrete
time algorithm is designed to deal with the RAP with a strongly
convex objective function. Noteworthy, the transformed distributed
problem is no longer strongly convex even though the original
objective function is strongly convex, but it satisfies the Polyak-
Łjasiewicz (PL) and quadratic growth (QG) conditions. Inspired
by the Heavy-Ball method, a distributed fixed inertial continuous
time algorithm is proposed, it has an explicit and accelerated
exponential convergence rate. Later, a rate-matching accelerated
distributed fixed inertial discrete time algorithm is also obtained by
applying explicit, semi-implicit Euler discretization and sufficient
decrease update schemes. Finally, the effectiveness of the proposed
distributed inertial algorithms is verified by simulation.