Iterative algorithms for the multiple-sets split feasibility problem in Hilbert spaces

In this paper, for the multiple-sets split feasibility problem, that is to find a point closest to a family of closed convex subsets in one space such that its image under a linear bounded mapping will be closest to another family of closed convex subsets in the image space, we study several iterative methods for finding a solution, which solves a certain variational inequality. We show that particular cases of our algorithms are some improvements for existing ones in literature. We also give two numerical examples for illustrating our algorithms.     

Perturbed projections and subgradient projections for the multiple-sets split feasibility problem

We study the multiple-sets split feasibility problem that requires to find a point closest to a family of closed convex sets in one space such that its image under a linear transformation will be closest to another family of closed convex sets in the image space. By casting the problem into an equivalent problem in a suitable product space we are able to present a simultaneous subgradients projections algorithm that generates convergent sequences of iterates in the feasible case. We further derive and analyze a perturbed projection method for the multiple-sets split feasibility problem and, additionally, furnish alternative proofs to two known results.     

Optimization for Inconsistent Split Feasibility Problems

The split feasibility problem deals with finding a point in a closed convex subset of the domain space of a linear operator such that the image of the point under the linear operator is in a prescribed closed convex subset of the image space. The split feasibility problem and its variants and generalizations have been widely investigated as a means for resolving practical inverse problems in various disciplines. Many iterative algorithms have been proposed for solving the problem. This article discusses a split feasibility problem which does not have a solution, referred to as an inconsistent split feasibility problem. When the closed convex set of the domain space is the absolute set and the closed convex set of the image space is the subsidiary set, it would be reasonable to formulate a compromise solution of the inconsistent split feasibility problem by using a point in the absolute set such that its image of the linear operator is closest to the subsidiary set in terms of the norm. We show that the problem of finding the compromise solution can be expressed as a convex minimization problem over the fixed point set of a nonexpansive mapping and propose an iterative algorithm, with three-term conjugate gradient directions, for solving the minimization problem.     

At infinity of finite-dimensional CAT(0) spaces

We show that any filtering family of closed convex subsets of a finite-dimensional CAT(0) space X has a non-empty intersection in the visual bordification ${ \overline{X} = X \cup \partial X}We show that any filtering family of closed convex subsets of a finite-dimensional CAT(0) space X has a non-empty intersection in the visual bordification [`(X)] = X è?X{ \overline{X} = X \cup \partial X} . Using this fact, several results known for proper CAT(0) spaces may be extended to finite-dimensional spaces, including the existence of canonical fixed points at infinity for parabolic isometries, algebraic and geometric restrictions on amenable group actions, and geometric superrigidity for non-elementary actions of irreducible uniform lattices in products of locally compact groups.     

On lightly and countably compact spaces in ZF

Abstract

Given a topological space X = (X, T ), we show in the Zermelo-Fraenkel set theory ZF that:

1. Every locally finite family of open sets of X is finite iff every pairwise disjoint, locally finite family of open sets is finite.

2. Every locally finite family of subsets of X is finite iff every pairwise disjoint, locally finite family of subsets of X is finite iff every locally finite family of closed subsets of X is finite.

3. The statement “every locally finite family of closed sets of X is finite” implies the proposition “every locally finite family of open sets of X is finite”. The converse holds true in case X is T₄ and the countable axiom of choice holds true.

   We also show:

4. It is relatively consistent with ZF the existence of a non countably compact T₁ space such that every pairwise disjoint locally finite family of closed subsets is finite but some locally finite family of subsets is infinite.

5. It is relatively consistent with ZF the existence of a countably compact T₄ space including an infinite pairwise disjoint locally finite family of open (resp. closed) sets.

     

Topological convexities, selections and fixed points

A convexity on a set X is a family of subsets of X which contains the whole space and the empty set as well as the singletons and which is closed under arbitrary intersections and updirected unions. A uniform convex space is a uniform topological space endowed with a convexity for which the convex hull operator is uniformly continuous. Uniform convex spaces with homotopically trivial polytopes (convex hulls of finite sets) are absolute extensors for the class of metric spaces; if they are completely metrizable then a continuous selection theorem à la Michael holds. Upper semicontinuous maps have approximate selections and fixed points, under the usual assumptions.     

D.C. Representability of closed sets in reflexive Banach spaces and applications to optimization problems

A closed subsetM of a Hausdorff locally convex space is called d.c. representable if there are an extended-real valued lsc convex functionf and a continuous convex functionh such that $$M = \{ x \in X:f(x) - h(x) \leqslant 0\} .$$ Using the existence of a locally uniformly convex norm, we prove that any closed subset in a reflexive Banach space is d.c. representable. For d.c. representable subsets, we define an index of nonconvexity, which can be regarded as an indicator for the degree of nonconvexity. In fact, we show that a convex closed subset is weakly closed when it has a finite index of nonconvexity, and optimization problems on closed subsets with a low index of nonconvexity are less difficult from the viewpoint of computation.     

On Hypertopologies

In this article a unified approach is presented to hypertopologies on collections of nonempty closed subsets of a Hausdorff uniform space generated by a saturated and separating family of pseudo-metrics. One identifies here a suitable topology on the family of proper, convex and lower semicontinuous functions defined on a Hausdorff locally convex space for which the Young Fenchel transform is bicontinuous. This improves a well known result due to Mosco, Joly and Beer.     

Motzkin predecomposable sets

We introduce and study the family of sets in a finite dimensional Euclidean space which can be written as the Minkowski sum of a compact and convex set and a convex cone (not necessarily closed). We establish several properties of the class of such sets, called Motzkin predecomposable, some of which hold also for the class of Motzkin decomposable sets (i.e., those for which the convex cone in the decomposition is requested to be closed), while others are specific of the new family.     

The space clcv(? n ) with the Hausdorff-Bebutov metric and statistically invariant sets of control systems

We obtain conditions that allow one to evaluate the relative frequency of occurrence of the reachable set of a control system in a given set. If the relative frequency of occurrence in this set is 1, then the set is said to be statistically invariant. It is assumed that the images of the right-hand side of the differential inclusion corresponding to the given control system are convex, closed, but not necessarily compact. We also study the basic properties of the space clcv(? ⁿ ) of nonempty closed convex subsets of ? ⁿ with the Hausdorff-Bebutov metric.     

Bregman distances and Chebyshev sets

A closed set of a Euclidean space is said to be Chebyshev if every point in the space has one and only one closest point in the set. Although the situation is not settled in infinite-dimensional Hilbert spaces, in 1932 Bunt showed that in Euclidean spaces a closed set is Chebyshev if and only if the set is convex. In this paper, from the more general perspective of Bregman distances, we show that if every point in the space has a unique nearest point in a closed set, then the set is convex. We provide two approaches: one is by nonsmooth analysis; the other by maximal monotone operator theory. Subdifferentiability properties of Bregman nearest distance functions are also given.     

Notions of boundaries for function spaces

Sibony and the author independently defined a higher order generalization of the usual Shilov boundary of a function algebra which yielded extensions of results about analytic structure from one dimension to several dimensions. Tonev later obtained an alternative characterization of this generalized Shilov boundary by looking at closed subsets of the spectrum whose image under the spectral mapping contains the topological boundary of the joint spectrum. In this note we define two related notions of what it means to be a higher order/higher dimensional boundary for a space of functions without requiring that the boundary be a closed set. We look at the relationships between these two boundaries, and in the process we obtain an alternative proof of Tonev's result. We look at some examples, and we show how the same concepts apply to convex sets and linear functions.

     

On the convergence of von Neumann''s alternating projection algorithm for two sets

We give several unifying results, interpretations, and examples regarding the convergence of the von Neumann alternating projection algorithm for two arbitrary closed convex nonempty subsets of a Hilbert space. Our research is formulated within the framework of Fejér monotonicity, convex and set-valued analysis. We also discuss the case of finitely many sets.     

A relaxed self-adaptive CQ algorithm for the multiple-sets split feasibility problem

The multiple-sets split feasibility problem (MSFP) is to find a point belongs to the intersection of a family of closed convex sets in one space, such that its image under a linear transformation belongs to the intersection of another family of closed convex sets in the image space. Many iterative methods can be employed to solve the MSFP. Jinling Zhao et al. proposed a modification for the CQ algorithm and a relaxation scheme for this modification to solve the MSFP. The strong convergence of these algorithms are guaranteed in finite-dimensional Hilbert spaces. Recently López et al. proposed a relaxed CQ algorithm for solving split feasibility problem, this algorithm can be implemented easily since it computes projections onto half-spaces and has no need to know a priori the norm of the bounded linear operator. However, this algorithm has only weak convergence in the setting of infinite-dimensional Hilbert spaces. In this paper, we introduce a new relaxed self-adaptive CQ algorithm for solving the MSFP where closed convex sets are level sets of some convex functions such that the strong convergence is guaranteed in the framework of infinite-dimensional Hilbert spaces. Our result extends and improves the corresponding results.     

Fixed point theorems for multivalued nonexpansive mappings satisfying inwardness conditions

Let X be a Banach space whose characteristic of noncompact convexity is less than 1 and satisfies the nonstrict Opial condition. Let C be a bounded closed convex subset of X, KC(X) the family of all compact convex subsets of X and T a nonexpansive mapping from C into KC(X) with bounded range. We prove that T has a fixed point. The nonstrict Opial condition can be removed if, in addition, T is an 1-χ-contractive mapping.     

On the convergence of von Neumann's alternating projection algorithm for two sets

We give several unifying results, interpretations, and examples regarding the convergence of the von Neumann alternating projection algorithm for two arbitrary closed convex nonempty subsets of a Hilbert space. Our research is formulated within the framework of Fejér monotonicity, convex and set-valued analysis. We also discuss the case of finitely many sets.     

Best proximity points,cyclic Kannan maps and geodesic metric spaces

We introduce the concept of cyclic Kannan orbital C-nonexpansive mappings and obtain the existence of a best proximity point on a pair of bounded, closed and convex subsets of a strictly convex metric space by using the geometric notion of seminormal structure. We also study the structure of minimal sets for cyclic Kannan C-nonexpansive mappings and show that results similar to the celebrated Goebel– Karlovitz lemma for nonexpansive self-mappings can be obtained for cyclic Kannan C-nonexpansive mappings.     

Dentability and extreme points in Banach spaces

It is shown that a Banach space E has the Radon-Nikodym property (equivalently, every bounded subset of E is dentable) if and only if every bounded closed convex subset of E is the closed convex hull of its strongly exposed points. Using recent work of Namioka, some analogous results are obtained concerning weak¹ strongly exposed points of weak¹ compact convex subsets of certain dual Banach spaces.     

An Eberlein–Šmulian type result for the weak* topology

We present a result on relative weak* compactness in the dual of a Banach space X that allows a short proof of both the Eberlein–Šmulian theorem and Šmulian’s characterisation of weak compactness of closed convex subsets of X.     

The characteristic of noncompact convexity and random fixed point theorem for set-valued operators

Let (Ω, Σ) be a measurable space, X a Banach space whose characteristic of noncompact convexity is less than 1, C a bounded closed convex subset of X, KC(C) the family of all compact convex subsets of C. We prove that a set-valued nonexpansive mapping T: C → KC(C) has a fixed point. Furthermore, if X is separable then we also prove that a set-valued nonexpansive operator T: Ω × C → KC(C) has a random fixed point.