On the Convergence of Discrete-Time Linear Systems: A Linear Time-Varying Mann Iteration Converges IFF Its Operator Is Strictly Pseudocontractive

Abstract

We adopt an operator-theoretic perspective to study convergence of linear fixed-point iterations and discrete-time linear systems. We mainly focus on the so-called Krasnoselskij–Mann iteration, ${x}$ ( $k + 1$ ) = ( $1-\alpha _{k}$ ) ${x}$ ( ${k}$ ) + $\alpha _{k} A~{x}$ ( ${k}$ ), which is relevant for distributed computation in optimization and game theory, when $A$ is not available in a centralized way. We show that strict pseudocontractiveness of the linear operator $x \mapsto Ax$ is not only sufficient (as known) but also necessary for the convergence to a vector in the kernel of $I-A$ . We also characterize some relevant operator-theoretic properties of linear operators via eigenvalue location and linear matrix inequalities. We apply the convergence conditions to multi-agent linear systems with vanishing step sizes, in particular, to linear consensus dynamics and equilibrium seeking in monotone linear-quadratic games.