Victoria Martín-Márquez

Solving the split feasibility problem without prior knowledge of matrix norms

The split feasibility problem (SFP) consists in finding a point in a given closed convex subset of a Hilbert space such that its image under a bounded linear operator belongs to a given closed convex subset of another Hilbert space. Iterative methods can be employed to solve the SFP. The most popular iterative method is Byrne’s CQ algorithm. However, to employ Byrne’s CQ algorithm, one needs to know a priori the norm (or at least an estimate of the norm) of the bounded linear operator (matrix in the finite-dimensional framework). It is the purpose of this paper to introduce a way of selecting the stepsizes such that the implementation of the CQ algorithm does not need any prior information about the operator norm. We also practise this way of selecting stepsizes for variants of the CQ algorithm, including a relaxed CQ algorithm where the two closed convex sets are both level sets of convex functions, and a Halpern-type algorithm. Both weak and strong convergence are investigated. Numerical experiments are included to illustrate the applications in signal processing of the CQ algorithm with stepsizes selected in an adaptive way.

Alternative iterative methods for nonexpansive mappings, rates of convergence and applications

Alternative iterative methods for a nonexpansive mapping in a Banach space are proposed and proved to be convergent to a common solution to a fixed point problem and a variational inequality. We give rates of asymptotic regularity for such iterations using proof-theoretic techniques. Some applications of the convergence results are presented.

Existence and Approximation of Fixed Points of Right Bregman Nonexpansive Operators

We study the existence and approximation of fixed points of right Bregman nonexpansive operators in reflexive Banach space. We present, in particular, necessary and sufficient conditions for the existence of fixed points and an implicit scheme for approximating them.

Approximation Methods for Nonexpansive Type Mappings in Hadamard Manifolds

Nonexpansrive type mappings defined on Hadamard manifolds and iterative methods for approximating fixed points of these mappings are surveyed. The close relationship with monotone vector fields is pointed out and some numerical examples are included.