arXiv subjects

Roberto Cominetti

Publications and source records attributed to Roberto Cominetti.

26 records · Page 2Linked to original sources

arXiv2017

Rates of convergence for inexact Krasnosel'skii-Mann iterations in Banach spaces

We study the convergence of an inexact version of the classical Krasnosel'skii-Mann iteration for computing fixed points of nonexpansive maps. Our main result establishes a new metric bound for the fixed-point 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'kii-Mann schemes. The results are also extended to continuous time in order to study the asymptotics of nonautonomous evolution equations governed by nonexpansive operators.

math.OC↗

arXiv2017

Sharp convergence rates for averaged nonexpansive maps

We establish sharp estimates for the convergence rate of the Kranosel'ski\vı-Mann fixed point iteration in general normed spaces, and we use them to show that the asymptotic regularity bound recently proved in [11] (Israel Journal of Mathematics 199(2), 757-772, 2014) with constant $1/\sqrtπ$ is sharp and cannot be improved. To this end we consider the recursive bounds introduced in [3] (Proceedings of the 2nd International Conference on Fixed Point Theory and Applications, World Scientific Press, London, 27-66, 1992) which we reinterpret in terms of a nested family of optimal transport problems. We show that these bounds are tight by building a nonexpansive map $T:[0,1]^{\mathbb N}\to[0,1]^{\mathbb N}$ that attains them with equality, settling the main conjecture in [3]. The recursive bounds are in turn reinterpreted as absorption probabilities for an underlying Markov chain which is used to establish the tightness of the constant $1/\sqrtπ$.

math.OC↗

arXiv2016

On the Price of Anarchy of Highly Congested Nonatomic Network Games

We consider nonatomic network games with one source and one destination. We examine the asymptotic behavior of the price of anarchy as the inflow increases. In accordance with some empirical observations, we show that, under suitable conditions, the price of anarchy is asymptotic to one. We show with some counterexamples that this is not always the case. The counterexamples occur in very simple parallel graphs.

cs.GT↗

arXiv2014

Dynamic Equilibria in Fluid Queuing Networks

This paper continues the study of equilibria for flows over time in the fluid queueing model recently considered by Koch and Skutella [10]. We provide a constructive proof for the existence and uniqueness of equilibria in the case of a single origin-destination with piecewise constant inflow rates, through a detailed analysis of the static flows obtained as derivatives of a dynamic equilibrium. We also give a nonconstructive existence proof of equilibria when the inflow rates belong to $L^p$ including the extension to multiple origin-destinations.

math.OC↗

arXiv2014

Network Congestion Control with Markovian Multipath Routing

In this paper we consider an integrated model for TCP/IP protocols with multipath routing. The model combines a Network Utility Maximization for rate control based on end-to-end queuing delays, with a Markovian Traffic Equilibrium for routing based on total expected delays. We prove the existence of a unique equilibrium state which is characterized as the solution of an unconstrained strictly convex program. A distributed algorithm for solving this optimization problem is proposed, with a brief discussion of how it can be implemented by adapting the current Internet protocols.

cs.NI↗

arXiv2013

Additive consistency of risk measures and its application to risk-averse routing in networks

This paper investigates the use of risk measures and theories of choice for modeling risk-averse route choice and traffic network equilibrium with random travel times. We interpret the postulates of these theories in the context of routing, and we identify additive consistency as a plausible and relevant condition that allows to reduce risk-averse route choice to a standard shortest path problem. Within the classical theories of choice under risk, we show that the only preferences that satisfy this consistency property are the ones induced by the entropic risk measures.

math.OC↗

arXiv2013

A sharp uniform bound for the distribution of sums of Bernoulli trials

In this note we establish a uniform bound for the distribution of a sum $S_n=X_1+\cdots+X_n$ of independent non-homogeneous Bernoulli trials. Specifically, we prove that $σ_n \mathbb{P}(S_n\!=\!j)\leqη$ where $σ_n$ denotes the standard deviation of $S_n$ and $η$ is a universal constant. We compute the best possible constant $η\sim 0.4688$ and we show that the bound also holds for limits of sums and differences of Bernoullis, including the Poisson laws which constitute the worst case and attain the bound. We also investigate the optimal bounds for $n$ and $j$ fixed. An application to estimate the rate of convergence of Mann's fixed point iterations is presented.

math.PR↗

arXiv2012

On the rate of convergence of Krasnoselski-Mann iterations and their connection with sums of Bernoullis

In this paper we establish an estimate for the rate of convergence of the Krasnosel'ski\vı-Mann iteration for computing fixed points of non-expansive maps. Our main result settles the Baillon-Bruck conjecture [3] on the asymptotic regularity of this iteration. The proof proceeds by establishing a connection between these iterates and a stochastic process involving sums of non-homogeneous Bernoulli trials. We also exploit a new Hoeffding-type inequality to majorize the expected value of a convex function of these sums using Poisson distributions.

math.OC↗