Andrzej Cegielski

Iterative methods for fixed point problems in Hilbert spaces

1 Introduction.- 2 Algorithmic Operators.- 3 Convergence of Iterative Methods.- 4 Algorithmic Projection Operators.- 5 Projection methods.

A method of projection onto an acute cone with level control in convex minimization

We study a projection method for convex minimization problems. In each step we construct a certain polyhedral function that approximates the objective function from below. The affine pieces of the approximating function are linearizations of the objective function obtained in previous steps. The linearization are determined by a linearly independent system of subgradients which generates an obtuse cone. The current approximation of the solution is projected onto a sublevel set of the polyhedral function with level which estimates the optimal value and is updated in each iteration. The projection can be easily obtained by solving a system of linear equations and is, actually, the projection onto a translated acute cone dual to the constructed obtuse cone. The construction ensures long steps between succesive approximations of the solution. Such long steps are important for the good behavior of the method.

Relaxed Alternating Projection Methods

In this paper we deal with the von Neumann alternating projection method

Methods for Variational Inequality Problem Over the Intersection of Fixed Point Sets of Quasi-Nonexpansive Operators

x_{k+1}=P_{A}P_{B}x_{k}

Obtuse cones and Gram matrices with nonnegative inverse

and with its generalization of the form

Projection Methods: An Annotated Bibliography of Books and Reviews

x_{k+1}=P_{A}(x_{k}+\lambda _{k}(P_{A}P_{B}x_{k}-x_{k}))

General Method for Solving the Split Common Fixed Point Problem

, where

Opial-Type Theorems and the Common Fixed Point Problem

A,B

Residual Selection in A Projection Method for Convex Minimization Problems

are closed and convex subsets of a Hilbert space

Application of Quasi-Nonexpansive Operators to an Iterative Method for Variational Inequality

\mathcal{H}