Banach空间中的平均非扩张映象:不动点的存在定理

<正> 设X是Banach空间,E是X中的集合,T是映集合E到自身的映象.若T满足条件(称为平均非扩张条件)其中x,y∈E,a,b,c≥0且a+2b+2c≤1,则称T是平均非扩张映象. 文[1]概括了近年来研究关于平均非扩张映象不动点的一些主要结果.本文进     

Separable Determination of the Fixed Point Property of Convex Sets in Banach Spaces

In this paper, we first show that for every mapping $f$ from a metric space $Ω$ to itself which is continuous off a countable subset of $Ω,$ there exists a nonempty closed separable subspace $S ⊂ Ω$ so that $f|_S$ is again a self mapping on $S.$ Therefore, both the fixed point property and the weak fixed point property of a nonempty closed convex set in a Banach space are separably determined. We then prove that every separable subspace of $c_0(\Gamma)$ (for any set $\Gamma$) is again lying in $c_0.$ Making use of these results, we finally presents a simple proof of the famous result: Every non-expansive self-mapping defined on a nonempty weakly compact convex set of $c_0(\Gamma)$ has a fixed point.     

On the range sets of variational inequalities

This paper investigates the closedness and convexity of the range sets of the variational inequality (VI) problem defined by an affine mappingM and a nonempty closed convex setK. It is proved that the range set is closed ifK is the union of a polyhedron and a compact convex set. Counterexamples are given such that the range set is not closed even ifK is a simple geometrical figure such as a circular cone or a circular cylinder in a three-dimensional space. Several sufficient conditions for closedness and convexity of the range set are presented. Characterization for the convex hull of the range set is established in the case whereK is a cone, while characterization for the closure of the convex hull of the range set is established in general. Finally, some applications to stability of VI problems are derived.This work was supported by the Australian Research Council.We are grateful to Professors M. Seetharama Gowda, Olvi Mangasarian, Jong-Shi Pang, and Steve Robinson for references. We are thankful to Professor Jim Burke for discussions on Theorem 2.1 and Counterexample 3.5.     

Strong Restricted-Orientation Convexity

Strong restricted-orientation convexity is a generalization of standard convexity. We explore the properties of strongly convex sets in multidimensional Euclidean space and identify major properties of standard convex sets that also hold for strong convexity. We characterize strongly convex flats and halfspaces, and establish the strong convexity of the affine hull of a strongly convex set. We then show that, for every point in the boundary of a strongly convex set, there is a supporting strongly convex hyperplane through it. Finally, we show that a closed set with nonempty interior is strongly convex if and only if it is the intersection of strongly convex halfspaces; we state a condition under which this result extends to sets with empty interior.     

ON THE THEOREM OF ARROW-BARANKIN-BLACKWELL FOR WEAKLY COMPACT CONVEX SET

This papeer studies the known density theorem of Arrow-Barankin-Blackwell. The following main result is obtained: If X is a Hausdorff locally convex Topological space and C belong to X is a closed convex cone with bounded base, then for every nonempty weakly compact convex subset A, the set of positive proper efficient points of A is dense in the set of efficient points of A.     

Intersections by hyperplanes

LetA be a finite nonempty family of nonempty disjoint closed and bounded sets in a Banach spaceE which is either separable and the conjugate of some Banach spaceX (i.e.E=X*) or, reflexive and locally uniformly convex. IfC denotes the weak*-closed convex hull of ∪ {A:A ∈A} then the set of points inE ∼C through which there is no hyperplane intersecting exactly one member ofA is discrete (or empty). This research was supported by the National Research Council of Canada, Grant A-3999.     

ON THE THEOREM OF ARROW-BARANKIN-BLACKWELL FOR WEAKIY COMPACT CONVEX SET

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Convex Hulls of Integral Points

The convex hull of all integral points contained in a compact polyhedron C is obviously a compact polyhedron. If C is not compact, then the convex hull K of its integral points need not be a closed set. However, under some natural assumptions, K is a closed set and a generalized polyhedron. Bibliography: 11 titles.     

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.     

广义共同逼近问题的适定性

设C是实Banach空间X中有界闭凸子集且0是C的内点，G是X中非空闭的有界相对弱紧子集.记K(X)为X的非空紧凸子集全体并赋Hausdorff距离，KG(X)为集合｛A∈K(X)；A∩G=｝的闭包.称广义共同逼近问题minC(A,G)是适定的是指它有唯一解(x0,z0)，且它的每个极小化序列均强收敛到(x0,z0).在C是严格凸和Kadec的假定下，证明了｛A∈K(X)；minC(A，G)是适定的｝含有KG(X)中稠Gδ子集，这本质地推广和延拓了包括De Blasi,Myjak and Papini［1］、Li［2］和De Blasi and Myjak［3］等人在内的近期相应结果.     

On various semiconvex hulls in the calculus of variations

We show that for any compact set , if and only if , , and being the closed lamination convex hull, quasiconvex hull and convex hull of respectively. When , we show that if and only if , where is the polyconvex hull of . We give some estimates of these relations by using quasiconvexifications of distance functions. Received April 9, 1996 / In final revised form November 8, 1996 / Accepted December 17, 1996     

关于真有效点集的连通性

在局部有界的Ｈａｕｓｄｏｒｆｆ局部凸空间中讨论了集合的真有效点集的连通性问题。证明了当序锥具有基底时,任何非空紧凸集的真有效点集是连通的。     

Super weak compactness and uniform Eberlein compacta

We prove that a topological space is uniform Eberlein compact iff it is homeomorphic to a super weakly compact subset C of a Banach space such that the closed convex hull coC of C is super weakly compact. We also show that a Banach space X is super weakly compactly generated iff the dual unit ball B_X* of X^* in its weak star topology is affinely homeomorphic to a super weakly compactly convex subset of a Banach space.     

Maximal Points of Convex Sets in Locally Convex Topological Vector Spaces: Generalization of the Arrow–Barankin–Blackwell Theorem

In 1953, Arrow, Barankin, and Blackwell proved that, if C is a nonempty compact convex set in Rⁿ with its standard ordering, then the set of points in C maximizing strictly positive linear functionals is dense in the set of maximal points of C. In this paper, we present a generalization of this result. We show that that, if C is a compact convex set in a locally convex topological space X and if K is an ordering cone on X such that the quasi-interiors of K and the dual cone K* are nonempty, then the set of points in C maximizing strictly positive linear functionals is dense in the set of maximal points of C. For example, our work shows that, under the appropriate conditions, the density results hold in the spaces Rⁿ, L^p(, ), 1p, l^p, 1p, and C (), a compact Hausdorff space, when they are partially ordered with their natural ordering cones.     

积分凸性及其应用

该文在Banach空间中通过向量值函数的Bochner积分引进集合与泛函的积分凸性以及集合的积分端点等概念. 文章主要证明有限维凸集、开凸集和闭凸集均是积分凸集，下半连续凸泛函与开凸集上的上半连续凸泛函均是积分凸的, 非空紧集具有积分端点, 对紧凸集来说其积分端点集与端点集一致, 最后给出积分凸性在最优化理论方面的两个应用.     

Some Fundamental Properties of Successive Convex Relaxation Methods on LCP and Related Problems

General successive convex relaxation methods (SRCMs) can be used to compute the convex hull of any compact set, in an Euclidean space, described by a system of quadratic inequalities and a compact convex set. Linear complementarity problems (LCPs) make an interesting and rich class of structured nonconvex optimization problems. In this paper, we study a few of the specialized lift-and-project methods and some of the possible ways of applying the general SCRMs to LCPs and related problems.     

Nondifferentiable reverse convex programs and facetial convexity cuts via a disjunctive characterization

Disjunctive Programs can often be transcribed as reverse convex constrained problems with nondifferentiable constraints and unbounded feasible regions. We consider this general class of nonconvex programs, called Reverse Convex Programs (RCP), and show that under quite general conditions, the closure of the convex hull of the feasible region is polyhedral. This development is then pursued from a more constructive standpoint, in that, for certain special reverse convex sets, we specify a finite linear disjunction whose closed convex hull coincides with that of the special reverse convex set. When interpreted in the context of convexity/intersection cuts, this provides the capability of generating any (negative edge extension) facet cut. Although this characterization is more clarifying than computationally oriented, our development shows that if certain bounds are available, then convexity/intersection cuts can be strengthened relatively inexpensively.     

Generating Sum-of-Ratios Test Problems in Global Optimization

A method is presented for the construction of test problems involving the minimization over convex sets of sums of ratios of affine functions. Given a nonempty, compact convex set, the method determines a function that is the sum of linear fractional functions and attains a global minimum over the set at a point that can be found by convex programming and univariate search. Generally, the function will have also local minima over the set that are not global minima.     

Approximation of fixed points and variational solutions for pseudo-contractive mappings in banach spaces

Let K be a nonempty, closed and convex subset of a real reflexive Banach space E which has a uniformly Gâteaux differentiable norm. Assume that every nonempty closed convex and bounded subset of K has the fixed point property for nonexpansive mappings. Strong convergence theorems for approximation of a fixed point of Lipschitz pseudo-contractive mappings which is also a unique solution to variational inequality problem involving ?-strongly pseudo-contractive mappings are proved. The results presented in this article can be applied to the study of fixed points of nonexpansive mappings, variational inequality problems, convex optimization problems, and split feasibility problems. Our result extends many recent important results.     

Sufficient conditions for existence and uniqueness of a Chebyshev center of a nonempty bounded set in a geodesic space

We obtain sufficient conditions for existence and uniqueness of a Chebyshev center of a nonempty bounded set in a geodesic space. We also establish conditions under which the unique Chebyshev center of a nonempty bounded set in a geodesic space belongs to the closure of the convex hull of this set.