Genaro López

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.

Stable and Total Fenchel Duality for Convex Optimization Problems in Locally Convex Spaces

We consider the optimization problem

Convergence analysis of inexact proximal point algorithms on Hadamard manifolds

(P_{A})

Proximal Point Algorithms on Hadamard Manifolds: Linear Convergence and Finite Termination

\inf_{x\in X}\{f(x)+g(Ax)\}

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

where

The Proximal Point Method for Locally Lipschitz Functions in Multiobjective Optimization with Application to the Compromise Problem

f

Approximation Methods for Nonexpansive Type Mappings in Hadamard Manifolds

and

Monotone vector fields and the proximal point algorithm on Hadamard manifolds

g

Equilibrium problems in Hadamard manifolds

are proper convex functions defined on locally convex Hausdorff topological vector spaces