
Akian, M., Gaubert, S., & Hochart, A. (2020). A Game Theory Approach To The Existence And Uniqueness Of Nonlinear PerronFrobenius Eigenvectors. Discret. Contin. Dyn. Syst., 40(1), 207–231.
Abstract: We establish a generalized PerronFrobenius theorem, based on a combinatorial criterion which entails the existence of an eigenvector for any nonlinear orderpreserving and positively homogeneous map f acting on the open orthant R>0(n). This criterion involves dominions, i.e., sets of states that can be made invariant by one player in a twoperson game that only depends on the behavior of f “at infinity”. In this way, we characterize the situation in which for all alpha, beta > 0, the “slice space” Salpha(beta) :={x is an element of R>0(n) vertical bar alpha x <= f(x) <= beta x} is bounded in Hilbert's projective metric, or, equivalently, for all uniform perturbations g of f, all the orbits of g are bounded in Hilbert's projective metric. This solves a problem raised by Gaubert and Gunawardena (Trans. AMS, 2004). We also show that the uniqueness of an eigenvector is characterized by a dominion condition, involving a different game depending now on the local behavior of f near an eigenvector. We show that the dominion conditions can be verified by directed hypergraph methods. We finally illustrate these results by considering specific classes of nonlinear maps, including Shapley operators, generalized means and nonnegative tensors.



Bravo, M., Cominetti, R., & PavezSigne, M. (2019). Rates of convergence for inexact Krasnosel'skiiMann iterations in Banach spaces. Math. Program., 175(12), 241–262.
Abstract: We study the convergence of an inexact version of the classical Krasnosel'skiiMann iteration for computing fixed points of nonexpansive maps. Our main result establishes a new metric bound for the fixedpoint residuals, from which we derive their rate of convergence as well as the convergence of the iterates towards a fixed point. The results are applied to three variants of the basic iteration: infeasible iterations with approximate projections, the Ishikawa iteration, and diagonal Krasnosels'kiiMann schemes. The results are also extended to continuous time in order to study the asymptotics of nonautonomous evolution equations governed by nonexpansive operators.

