A note on strong convergence to common fixed points of nonexpansive mappings in Hilbert spaces

The aim of this paper is to investigate the links between ${\cal T}_C$-class algorithms, CQ Algorithm and shrinking projection methods. We show that strong convergence of these algorithms are related to coherent ${\cal T}_C$-class sequences of mapping. Some examples dealing with nonexpansive finite set of mappings and nonexpansive semigroups are given. They extend some existing theorems

Extensions and applications of ACF mappings

Using a definition of ASF sequences derived from the definition of asymptotic contractions of the final type of ACF, we give some new fixed points theorem for cyclic mappings and alternating mapping which extend results from T.Suzuki and X.Zhang

A novel code generation methodology for block diagram modeler and simulators Scicos and VSS

Block operations during simulation in Scicos and VSS environments can naturally be described as Nsp functions. But the direct use of Nsp functions for simulation leads to poor performance since the Nsp language is interpreted, not compiled. The methodology presented in this paper is used to develop a tool for generating efficient compilable code, such as C and ADA, for Scicos and VSS models from these block Nsp functions. Operator overloading and partial evaluation are the key elements of this novel approach. This methodology may be used in other simulation environments such as Matlab/Simulink

Iterative schemes for computing fixed points of nonexpansive mappings in Banach spaces

Let $X$ be a real Banach space with a normalized duality mapping uniformly norm-to-weak$^\star$ continuous on bounded sets or a reflexive Banach space which admits a weakly continuous duality mapping $J_{\Phi}$ with gauge $\phi$. Let $f$ be an {\em $\alpha$-contraction} and $\{T_n\}$ a sequence of nonexpansive mapping, we study the strong convergence of explicit iterative schemes x_{n+1} = \alpha_n f(x_n) + (1-\alpha_n) T_n x_n with a general theorem and then recover and improve some specific cases studied in the literatur

T{\cal T}-class algorithms for pseudocontractions and κ\kappa-strict pseudocontractions in Hilbert spaces

In this paper we study iterative algorithms for finding a common element of the set of fixed points of $\kappa$-strict pseudocontractions or finding a solution of a variational inequality problem for a monotone, Lipschitz continuous mapping. The last problem being related to finding fixed points of pseudocontractions. These algorithms were already studied in [G.L. Acedo, H.-K. Xu] and [N. Nadezhkina, W. Takahashi] but our aim here is to provide the links between these know algorithms and the general framework of ${\cal T}$-class algorithms studied in [H.H. Bauschke, P.L. Combettes]

Time Blocks Decomposition of Multistage Stochastic Optimization Problems

Multistage stochastic optimization problems are, by essence, complex because their solutions are indexed both by stages (time) and by uncertainties (scenarios). Their large scale nature makes decomposition methods appealing.The most common approaches are time decomposition --- and state-based resolution methods, like stochastic dynamic programming, in stochastic optimal control --- and scenario decomposition --- like progressive hedging in stochastic programming. We present a method to decompose multistage stochastic optimization problems by time blocks, which covers both stochastic programming and stochastic dynamic programming. Once established a dynamic programming equation with value functions defined on the history space (a history is a sequence of uncertainties and controls), we provide conditions to reduce the history using a compressed "state" variable. This reduction is done by time blocks, that is, at stages that are not necessarily all the original unit stages, and we prove areduced dynamic programming equation. Then, we apply the reduction method by time blocks to \emph{two time-scales} stochastic optimization problems and to a novel class of so-called \emph{decision-hazard-decision} problems, arising in many practical situations, like in stock management. The \emph{time blocks decomposition} scheme is as follows: we use dynamic programming at slow time scale where the slow time scale noises are supposed to be stagewise independent, and we produce slow time scale Bellman functions; then, we use stochastic programming at short time scale, within two consecutive slow time steps, with the final short time scale cost given by the slow time scale Bellman functions, and without assuming stagewise independence for the short time scale noises

Iterative schemes for computing fixed points of nonexpansive mappings in Banach spaces

Let $X$ be a real Banach space with a normalized duality mapping uniformly norm-to-weak$^\star$ continuous on bounded sets or a reflexive Banach space which admits a weakly continuous duality mapping $J_{\Phi}$ with gauge $\phi$. Let $f$ be an {\em $\alpha$-contraction} and $\{T_n\}$ a sequence of nonexpansive mapping, we study the strong convergence of explicit iterative schemes \begin{equation} x_{n+1} = \alpha_n f(x_n) + (1-\alpha_n) T_n x_n \end{equation} with a general theorem and then recover and improve some specific cases studied in the literatur