The Asymptotic Behavior of the Composition of Firmly Nonexpansive Mappings

In this paper we provide a unified treatment of some convex minimization problems, which allows for a better understanding and, in some cases, improvement of results in this direction proved recently in spaces of curvature bounded above. For this purpose, we analyze the asymptotic behavior of compositions of finitely many firmly nonexpansive mappings in the setting of $p$-uniformly convex geodesic spaces focusing on asymptotic regularity and convergence results

Iterative oblique projection onto convex sets and the split feasibility problem,”

Abstract Let C and Q be nonempty closed convex sets in R N and R M , respectively, and A an M by N real matrix. The split feasibility problem (SFP) is to find x ∈ C with Ax ∈ Q, if such x exist. An iterative method for solving the SFP, called the CQ algorithm, has the following iterative step: with L the largest eigenvalue of the matrix A T A and P C and P Q denote the orthogonal projections onto C and Q, respectively; that is, P C x minimizes c − x , over all c ∈ C. The CQ algorithm converges to a solution of the SFP, or, more generally, to a minimizer of P Q Ac − Ac over c in C, whenever such exist. The CQ algorithm involves only the orthogonal projections onto C and Q, which we shall assume are easily calculated, and involves no matrix inverses. If A is normalized so that each row has length one, then L does not exceed the maximum number of nonzero entries in any column of A, which provides a helpful estimate of L for sparse matrices. Particular cases of the CQ algorithm are the Landweber and projected Landweber methods for obtaining exact or approximate solutions of the linear equations Ax = b; the algebraic reconstruction technique of Gordon, Bender and Herman is a particular case of a block-iterative version of the CQ algorithm. One application of the CQ algorithm that is the subject of ongoing work is dynamic emission tomographic image reconstruction, in which the vector x is the concatenation of several images corresponding to successive discrete times. The matrix A and the set Q can then be selected to impose constraints on the behaviour over time of the intensities at fixed voxels, as well as to require consistency (or near consistency) with measured data

Legendre Functions and the Method of Random Bregman Projections

The convex feasibility problem, that is, finding a point in the intersection of finitely many closed convex sets in Euclidean space, arises in various areas of mathematics and physical sciences. It can be solved by the classical method of cyclic orthogonal projections, where, by projecting cyclically onto the sets, a sequence is generated that converges to a point in the intersection. In 1967, Bregman extended this method to non-orthogonal projections based on a new notion of distance, nowadays called "Bregman distance". The Bregman distance is induced by a convex function. If this function is a so-called "zone consistent Bregman function", then Bregman's method works; however, deciding on this can be difficult. In this paper, Bregman's method is studied within the powerful framework of Convex Analysis. New insights are obtained and the rich class of "Bregman/Legendre functions" is introduced. Bregman's method still works, if the underlying function is Bregman/Legendre or more generally if it is Legendre but some constraint qualification holds additionally. The key advantage is the broad applicability and verifiability of these concepts. The results presented here are complementary to recent work by Censor and Reich on the method of random Bregman projections (where the sets are projected onto infinitely often -not necessarily cyclically). Special attention is given to examples, some of which connect to Pythagorean means and to Convex Analysis on the Hermitian or symmetric matrices