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

Abstract

In this paper, 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 <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$O( \frac{1}{ t^{2}})$</tex-math></inline-formula> 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-Ł jasiewicz ( <bold xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">PL</b> ) and quadratic growth ( <bold xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">QG</b> ) 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.