Convergence of Bregman Projection Methods for Solving Consistent Convex Feasibility Problems in Reflexive Banach Spaces

The problem that we consider is whether or under what conditions sequences generated in reflexive Banach spaces by cyclic Bregman projections on finitely many closed convex subsets Q _i with nonempty intersection converge to common points of the given sets.     

Iterative Methods of Solving Stochastic Convex Feasibility Problems and Applications

The stochastic convex feasibility problem (SCFP) is the problem of finding almost common points of measurable families of closed convex subsets in reflexive and separable Banach spaces. In this paper we prove convergence criteria for two iterative algorithms devised to solve SCFPs. To do that, we first analyze the concepts of Bregman projection and Bregman function with emphasis on the properties of their local moduli of convexity. The areas of applicability of the algorithms we present include optimization problems, linear operator equations, inverse problems, etc., which can be represented as SCFPs and solved as such. Examples showing how these algorithms can be implemented are also given.     

On the convexity coefficient of Musielak–Orlicz function spaces equipped with the Orlicz norm

In Hudzik and Landes, the convexity coefficient of Musielak–Orlicz function spaces over a non-atomic measure space equipped with the Luxemburg norm is computed whenever the Musielak–Orlicz functions are strictly convex see [6]. In this paper, we extend this result to the case of Musielak–Orlicz spaces equipped with the Orlicz norm. Also, a characterization of uniformly convex Musielak–Orlicz function spaces as well as k-uniformly convex Musielak–Orlicz spaces equipped with the Orlicz norm is given.     

Iterative Averaging of Entropic Projections for Solving Stochastic Convex Feasibility Problems

The problem considered in this paper is that of finding a point which iscommon to almost all the members of a measurable family of closed convexsubsets of R₊₊ ⁿ , provided that such a point exists.The main results show that this problem can be solved by an iterative methodessentially based on averaging at each step the Bregman projections withrespect to f(x)=_i=1 ⁿx_i· ln x_i ofthe current iterate onto the given sets.