凸分析若干结果探讨

在实线性空间中引进了一类新广义凸集.讨论了它的一些性质,与之相联的Minkowski泛函与分离定理等.我们所得结果是经典凸分析中若干相应结果的推广,改进或等价表达.     

广义凸函数的特征性质

提出广义凸集、广义凸函数、中间点广义凸函数、端点广义凸函数四个定义,通过定义条件P1,研究条件P1所蕴含的等式关系,进而得到一个基础性定理一稠密性定理和一个相对条件较弱的推论,最后将结果应用于若干不同类型的广义凸函数类,尤其是s-凸函数、几何凸函数、rp-凸函数,得到它们所共有的一个特征性质,即满足稠密性定理．     

The isodiametric problem for lattice-point-free convex sets

The isodiametric problem in the Euclidean plane is solved for lattice-point-free convex sets: we characterize the planar convex sets containing no points of the rectangular lattice in their interior, which have maximum area, for each given value of the diameter. Then we use this result for giving also an answer to the isodiametric problem in the case of an arbitrary lattice. The first author is supported by Dirección General de Investigación (MEC) MTM2004-04934-C04-02 and by Fundación Séneca (C.A.R.M.) 00625/PI/04.     

E-Convex Sets, E-Convex Functions, and E-Convex Programming

A class of sets and a class of functions called E-convex sets and E-convex functions are introduced by relaxing the definitions of convex sets and convex functions. This kind of generalized convexity is based on the effect of an operator E on the sets and domain of definition of the functions. The optimality results for E-convex programming problems are established.     

Differences of Convex Compact Sets in the Space of Directed Sets. Part II: Visualization of Directed Sets

This paper is a continuation of the author's first paper (Set-Valued Anal. 9 (2001), pp. 217–245), where the normed and partially ordered vector space of directed sets is constructed and the cone of all nonempty convex compact sets in R ⁿ is embedded. A visualization of directed sets and of differences of convex compact sets is presented and its geometrical components and properties are studied. The three components of the visualization are compared with other known differences of convex compact sets.     

On Some Generalized Polyhedral Convex Constructions

Generalized polyhedral convex sets, generalized polyhedral convex functions on locally convex Hausdorff topological vector spaces, and the related constructions such as sum of sets, sum of functions, directional derivative, infimal convolution, normal cone, conjugate function, subdifferential are studied thoroughly in this paper. Among other things, we show how a generalized polyhedral convex set can be characterized through the finiteness of the number of its faces. In addition, it is proved that the infimal convolution of a generalized polyhedral convex function and a polyhedral convex function is a polyhedral convex function. The obtained results can be applied to scalar optimization problems described by generalized polyhedral convex sets and generalized polyhedral convex functions.     

Duality for almost convex optimization problems via the perturbation approach

We deal with duality for almost convex finite dimensional optimization problems by means of the classical perturbation approach. To this aim some standard results from the convex analysis are extended to the case of almost convex sets and functions. The duality for some classes of primal-dual problems is derived as a special case of the general approach. The sufficient regularity conditions we need for guaranteeing strong duality are proved to be similar to the ones in the convex case. The research of the first and third authors was partially supported by DFG (German Research Foundation), project WA 922/1. The research of the second author was supported by the grant PN II, ID 523/2007.     

A sequential iteration algorithm with non-monotoneous behaviour in the method of projections onto convex sets

The method of projections onto convex sets to find a point in the intersection of a finite number of closed convex sets in a Euclidean space, may lead to slow convergence of the constructed sequence when that sequence enters some narrow “corridor” between two or more convex sets. A way to leave such corridor consists in taking a big step at different moments during the iteration, because in that way the monotoneous behaviour that is responsible for the slow convergence may be interrupted. In this paper we present a technique that may introduce interruption of the monotony for a sequential algorithm, but that at the same time guarantees convergence of the constructed sequence to a point in the intersection of the sets. We compare experimentally the behaviour concerning the speed of convergence of the new algorithm with that of an existing monotoneous algorithm.     

Reduction of quasidifferentials and minimal representations

Some criterias for the non-minimality of pairs of compact convex sets of a real locally convex topological vector space are proved, based on a reduction technique via cutting planes and excision of compact convex subsets. Following an example of J. Grzybowski, we construct a class of equivalent minimal pairs of compact convex sets which are not connected by translations.Corresponding author.     

Barycentric coordinates for convex sets

In this paper we provide an extension of barycentric coordinates from simplices to arbitrary convex sets. Barycentric coordinates over convex 2D polygons have found numerous applications in various fields as they allow smooth interpolation of data located on vertices. However, no explicit formulation valid for arbitrary convex polytopes has been proposed to extend this interpolation in higher dimensions. Moreover, there has been no attempt to extend these functions into the continuous domain, where barycentric coordinates are related to Green’s functions and construct functions that satisfy a boundary value problem. First, we review the properties and construction of barycentric coordinates in the discrete domain for convex polytopes. Next, we show how these concepts extend into the continuous domain to yield barycentric coordinates for continuous functions. We then provide a proof that our functions satisfy all the desirable properties of barycentric coordinates in arbitrary dimensions. Finally, we provide an example of constructing such barycentric functions over regions bounded by parametric curves and show how they can be used to perform freeform deformations.      

Description of Continuous Isometry Covariant Valuations on Convex Sets

We present a characterization of continuous isometry covariant valuations on convex sets. The main result generalizes previous results of Hadwiger and Hadwiger and Schneider.     

一类局部凸空间及其映照特征

称局部凸空间（E,(?)0）为WCM空间若对于任何弱于(?)0的局部凸拓扑(?),（E,(?)）与（E,(?)0）具相同的弱紧圆凸集.本文研究了WCM空间的存在性及其与其他类型局部凸空间之间的关系,还给出了WCM空间的一种映照特征.     

Convex functions with continuous epigraph or continuous level sets

Convex functions with continuous epigraph in the sense of Gale and Klée have been studied recently by Auslender and Coutat in a finite-dimensional setting. Here, we provide characterizations of such functionals in terms of the Legendre-Fenchel transformation in general locally convex spaces. Also, we show that the concept of continuous convex sets is of interest in these spaces. We end with a characterization of convex functions on Euclidean spaces with continuous level sets.     

A Krasnosel’skii-type result for planar sets starshaped via orthogonally convex paths

A Krasnosel’skii-type theorem for compact sets that are starshaped via staircase paths may be extended to compact sets that are starshaped via orthogonally convex paths: Let S be a nonempty compact planar set having connected complement. If every two points of S are visible via orthogonally convex paths from a common point of S, then S is starshaped via orthogonally convex paths. Moreover, the associated kernel Ker S has the expected property that every two of its points are joined in Ker S by an orthogonally convex path. If S is an arbitrary nonempty planar set that is starshaped via orthogonally convex paths, then for each component C of Ker S, every two of points of C are joined in C by an orthogonally convex path. Communicated by Imre Bárány     

Generalized Monotone Operators on Dense Sets

In the present work we show that the local generalized monotonicity of a lower semicontinuous set-valued operator on some certain type of dense sets ensures the global generalized monotonicity of that operator. We achieve this goal gradually by showing at first that the lower semicontinuous set-valued functions of one real variable, which are locally generalized monotone on a dense subsets of their domain are globally generalized monotone. Then, these results are extended to the case of set-valued operators on arbitrary Banach spaces. We close this work with a section on the global generalized convexity of a real valued function, which is obtained out of its local counterpart on some dense sets.     

Characterizations of strictly convex sets by the uniqueness of support points

Abstract

This short paper characterizes strictly convex sets by the uniqueness of support points (such points are called unique support points or exposed points) under appropriate assumptions. A class of so-called regular sets, for which every extreme point is a unique support point, is introduced. Closed strictly convex sets and their intersections with some other sets are shown to belong to this class. The obtained characterizations are then applied to set-valued maps and to the separation of a convex set and a strictly convex set. Under suitable assumptions, so-called set-valued maps with path property are characterized by strictly convex images of the considered set-valued map.     

Efficient solutions in generalized linear vector optimization

This paper establishes several new facts on generalized polyhedral convex sets and shows how they can be used in vector optimization. Among other things, a scalarization formula for the efficient solution sets of generalized linear vector optimization problems is obtained. We also prove that the efficient solution set of a generalized linear vector optimization problem in a locally convex Hausdorff topological vector space is the union of finitely many generalized polyhedral convex sets and it is connected by line segments.     

Extrapolation algorithm for affine-convex feasibility problems

The convex feasibility problem under consideration is to find a common point of a countable family of closed affine subspaces and convex sets in a Hilbert space. To solve such problems, we propose a general parallel block-iterative algorithmic framework in which the affine subspaces are exploited to introduce extrapolated over-relaxations. This framework encompasses a wide range of projection, subgradient projection, proximal, and fixed point methods encountered in various branches of applied mathematics. The asymptotic behavior of the method is investigated and numerical experiments are provided to illustrate the benefits of the extrapolations.     

Methods of Chebyshev points of convex sets and their applications

Chebyshev points of bounded convex sets, search algorithms for them, and various applications to convex programming are considered for simple approximations of reachable sets, optimal control, global optimization of additive functions on convex polyhedra, and integer programming. The problem of searching for Chebyshev points in multicriteria models of development and operation of electric power systems is considered.     

闭凸集上的一类非线性半变分不等式解的存在性

考虑一类定义在闭凸集上的非线性半变分不等式问题,通过运用闭凸集上的临界点理论、Clarke次微分性质以及非光滑紧性条件等,得到了这类半变分不等式解的存在性．