Finding best approximation pairs for two intersections of closed convex sets

Computational Optimization and Applications - The problem of finding a best approximation pair of two sets, which in turn generalizes the well known convex feasibility problem, has a long history...     

Finding best approximation pairs relative to two closed convex sets in Hilbert spaces

We consider the problem of finding a best approximation pair, i.e., two points which achieve the minimum distance between two closed convex sets in a Hilbert space. When the sets intersect, the method under consideration, termed AAR for averaged alternating reflections, is a special instance of an algorithm due to Lions and Mercier for finding a zero of the sum of two maximal monotone operators. We investigate systematically the asymptotic behavior of AAR in the general case when the sets do not necessarily intersect and show that the method produces best approximation pairs provided they exist. Finitely many sets are handled in a product space, in which case the AAR method is shown to coincide with a special case of Spingarn's method of partial inverses.     

Convergence on closed convex sets in a locally convex space

The purpose of this paper is to compare several kinds of convergences on the space C(X) of nonempty closed convex subsets of a locally convex space X. First we verify that the AW-convergence on C(X) is weaker than the metric Attouch-Wets convergence on C(X) of a metrizable locally convex space X. Moreover, we show that X is normable if and only if the two convergences on C(X × R) are equivalent. Secondly we define two convergences on C(X) analogous to the corresponding ones in a normed linear space, and investigate some basic properties of these convergences and compare them.     

Motzkin decomposition of closed convex sets

Theodore Motzkin proved, in 1936, that any polyhedral convex set can be expressed as the (Minkowski) sum of a polytope and a polyhedral convex cone. This paper provides five characterizations of the larger class of closed convex sets in finite dimensional Euclidean spaces which are the sum of a compact convex set with a closed convex cone. These characterizations involve different types of representations of closed convex sets as the support functions, dual cones and linear systems whose relationships are also analyzed in the paper. The obtaining of information about a given closed convex set F and the parametric linear optimization problem with feasible set F from each of its different representations, including the Motzkin decomposition, is also discussed.     

On convergence of closed convex sets

In this paper we introduce a convergence concept for closed convex subsets of a finite-dimensional normed vector space. This convergence is called C-convergence. It is defined by appropriate notions of upper and lower limits. We compare this convergence with the well-known Painlevé-Kuratowski convergence and with scalar convergence. In fact, we show that a sequence (An)_n∈NC-converges to A if and only if the corresponding support functions converge pointwise, except at relative boundary points of the domain of the support function of A, to the support function of A.     

harmonic approximation on closed sets

In this paper the harmonic approximation ( ) on relatively closed subsets of a domain in the complex plane is characterized under the same conditions given by S. Gardiner for the uniform case. Thus, the result of P. Paramonov on harmonic polynomial approximation for compact subsets is extended to closed sets. Moreover, the problem of uniform approximation with continuous extension to the boundary for harmonic functions and similar questions in harmonic approximation are also studied.

     

Minimum recession-compatible subsets of closed convex sets

A subset B of a closed convex set A is recession-compatible with respect to A if A can be expressed as the Minkowski sum of B and the recession cone of A. We show that if A contains no line, then there exists a recession-compatible subset of A that is minimal with respect to set inclusion. The proof only uses basic facts of convex analysis and does not depend on Zorn’s Lemma. An application of this result to the error bound theory in optimization is presented.     

Proper holomorphic discs avoiding closed convex sets

Denote by the open unit disc in . Let C be a closed convex subset of . We prove that for each there is a proper holomorphic map such that and if and only if either C is a complex line or C does not contain any complex line. Received: 17 July 2001; in final form: 22 November 2001 / Published online: 5 September 2002     

On the illumination of unbounded closed convex sets

In this note we prove that the illumination of an almost bounded closed convex set by minimum number of affine subspaces of given dimension can be reduced to the illumination of a bounded closed convex set of lower dimension. The work was supported by Hung. Nat. Found. for Sci. Research No. 326-0213     

Automorphism groups of lattices of closed convex sets

An isomorphism theorem for certain lattices of closed convex sets is proved.     

Multivalued differential equations on closed convex sets in Banach spaces

New derivation results for integrands and multifunctions via the Lipschitzean approximations are obtained. Applications to multivalued differential equations on closed convex sets are presented.     

On inclusion and summands of bounded closed convex sets

Summary This article introduces coset extensions and group coextensions of S-sets.     

Generalized matching theorems for closed coverings of convex sets

In this paper we use fixed point and coincidence theorems due to Park [8] to give matching theorems concerning closed coverings of nonempty convex sets in a real topological vector space. Our new results extend previously given ones due to Ky Fan [2], [3], Shih [10], Shih and Tan [11], and Park [7].