近似凸集的某些性质

主要研究了两类近似凸集的关系和性质.首先,举例说明两类近似凸集没有相互包含关系.其次,在近似凸集(nearly convex)条件下,证明了在一定条件下函数上图是近似凸集与凸集的等价关系.同时,考虑了近似凸函数与函数上图是近似凸集的等价刻画、近似凸函数与函数水平集是近似凸集的必要性,并用例子说明近似凸函数与函数水平集是...     

On iterative convexity of diffeomorphism

Abstract

A function f is said to be iteratively convex if f possesses convex iterative roots of all orders. We give several constructions of iteratively convex diffeomorphisms and explain the phenomenon that the non-existence of convex iterative roots is a typical property of convex functions. We show also that a slight perturbation of iteratively convex functions causes loss of iterative convexity. However, every convex function can be approximate by iteratively convex functions.     

凸对策核仁的非空性及合成凸对策核仁的单调性

本文给出了核仁与核及最小核心之间的关系 ,且证明了凸对策核仁的存在性和唯一性 ,证明了凸对策的合成对策仍是凸对策 .最后 ,我们讨论了合成凸对策的核仁不满足单调性 .     

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.     

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．     

Convex Duality and Calculus: Reduction to Cones

An attempt is made to justify results from Convex Analysis by means of one method. Duality results, such as the Fenchel-Moreau theorem for convex functions, and formulas of convex calculus, such as the Moreau-Rockafellar formula for the subgradient of the sum of sublinear functions, are considered. All duality operators, all duality theorems, all standard binary operations, and all formulas of convex calculus are enumerated. The method consists of three automatic steps: first translation from the given setting to that of convex cones, then application of the standard operations and facts (the calculi) for convex cones, finally translation back to the original setting. The advantage is that the calculi are much simpler for convex cones than for other types of convex objects, such as convex sets, convex functions and sublinear functions. Exclusion of improper convex objects is not necessary, and recession directions are allowed as points of convex objects. The method can also be applied beyond the enumeration of the calculi.     

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.     

Optimality conditions in convex optimization revisited

The phrase convex optimization refers to the minimization of a convex function over a convex set. However the feasible convex set need not be always described by convex inequalities. In this article we consider a convex feasible set which is described by inequality constraints that are locally Lipschitz and not necessarily convex or differentiable. We show that if the Slater constraint qualification and a simple non-degeneracy condition is satisfied then the Karush–Kuhn–Tucker type optimality condition is both necessary and sufficient.     

关于K—极凸Banach空间

引进Ｋ－极凸Ｂａｎａｃｈ空间，证明了ＸＫ－极凸当且仅当Ｘ自反、Ｋ－严格凸且有（Ｈ）性质，得到了Ｋ－极凸空间的一些性质，并讨论了Ｋ－极凸与Ｋ－Ｋ－强光滑、Ｋ－一致凸及完全Ｋ－凸的关系。     

Some properties of convex hulls of integer points contained in general convex sets

In this paper, we study properties of general closed convex sets that determine the closedness and polyhedrality of the convex hull of integer points contained in it. We first present necessary and sufficient conditions for the convex hull of integer points contained in a general convex set to be closed. This leads to useful results for special classes of convex sets such as pointed cones, strictly convex sets, and sets containing integer points in their interior. We then present a sufficient condition for the convex hull of integer points in general convex sets to be a polyhedron. This result generalizes the well-known result due to Meyer (Math Program 7:223–225, 1974). Under a simple technical assumption, we show that these sufficient conditions are also necessary for the convex hull of integer points contained in general convex sets to be a polyhedron.     

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.     

Locally convex quotient lattice cones

Walter Roth has investigated certain equivalence relations on locally convex cones in [W. Roth, Locally convex quotient cones, J. Convex Anal. 18, No. 4, 903–913 (2011)] which give rise to the definition of a locally convex quotient cone. In this paper, we investigate some special equivalence relations on a locally convex lattice cone by which the locally convex quotient cone becomes a lattice. In the case of a locally convex solid Riesz space, this reduces to the known concept of locally convex solid quotient Riesz space. We prove that the strict inductive limit of locally convex lattice cones is a locally convex lattice cone. We also study the concept of locally convex complete quotient lattice cones.     

Convex envelopes generated from finitely many compact convex sets

We consider the problem of constructing the convex envelope of a lower semi-continuous function defined over a compact convex set. We formulate the envelope representation problem as a convex optimization problem for functions whose generating sets consist of finitely many compact convex sets. In particular, we consider nonnegative functions that are products of convex and component-wise concave functions and derive closed-form expressions for the convex envelopes of a wide class of such functions. Several examples demonstrate that these envelopes reduce significantly the relaxation gaps of widely used factorable relaxation techniques.     

Some generalizations of locally and weakly locally uniformly convex space

In this paper, we prove that strongly convex space and almost locally uniformly rotund space, very convex space and weakly almost locally uniformly rotund space are respectively equivalent. We also investigate a few properties of k-strongly convex space and k-very convex space, and discuss the applications of strongly convex space and very convex space in approximation theory.     

非常极凸空间

本文使用非常极凸的定义，证明了非常极凸和非常光滑是互为对偶空间且严格介于弱k凸和非常凸之间的空间，最后得到了非常极凸的一些特征．     

The affine representation theorem for abstract convex geometries

A convex geometry is a combinatorial abstract model introduced by Edelman and Jamison which captures a combinatorial essence of “convexity” shared by some objects including finite point sets, partially ordered sets, trees, rooted graphs. In this paper, we introduce a generalized convex shelling, and show that every convex geometry can be represented as a generalized convex shelling. This is “the representation theorem for convex geometries” analogous to “the representation theorem for oriented matroids” by Folkman and Lawrence. An important feature is that our representation theorem is affine-geometric while that for oriented matroids is topological. Thus our representation theorem indicates the intrinsic simplicity of convex geometries, and opens a new research direction in the theory of convex geometries.     

广义凸模糊映射

给出广义凸模糊映射、广义弱凸模糊映射等概念和若干特例。其次,构造集合Axf,y、Af,证明当f为下半连续广义弱凸模糊映射时Afx,y为闭弱凸集,进而得到广义凸模糊映射的充分条件。最后,给出广义凸模糊映射的性质,并指出半严格广义凸模糊映射成为严格广义凸模糊映射的条件。     

An alternative perspective on copositive and convex relaxations of nonconvex quadratic programs

We study convex relaxations of nonconvex quadratic programs. We identify a family of so-called feasibility preserving convex relaxations, which includes the well-known copositive and doubly nonnegative relaxations, with the property that the convex relaxation is feasible if and only if the nonconvex quadratic program is feasible. We observe that each convex relaxation in this family implicitly induces a convex underestimator of the objective function on the feasible region of the quadratic program. This alternative perspective on convex relaxations enables us to establish several useful properties of the corresponding convex underestimators. In particular, if the recession cone of the feasible region of the quadratic program does not contain any directions of negative curvature, we show that the convex underestimator arising from the copositive relaxation is precisely the convex envelope of the objective function of the quadratic program, strengthening Burer’s well-known result on the exactness of the copositive relaxation in the case of nonconvex quadratic programs. We also present an algorithmic recipe for constructing instances of quadratic programs with a finite optimal value but an unbounded relaxation for a rather large family of convex relaxations including the doubly nonnegative relaxation.

     

Banach空间的p— Asplund 伴随空间

我们称一个定义在Banach空间E上的连续凸函数f具有Frechet可微性质（FDP）,如果E上的每个实值凸函数g≤f均在E一个稠密的Gδ－子集上Frechet可微。本文主要证明了：对任何Banach空间E,均存在一个局部凸相容拓扑p使得1）（E,p)是Hausdorff局部凸空间;2） E上的每个范数连续具有FDP的凸函数均是p－连续的;3）每个p-连续的凸函数均具有FDP ;4）p等价某个范数拓扑当且仅不E是Asplund空间。     

Convexity theories IV. Klein-Hilbert parts in convex modules

The Klein-Hilbert part relation, which was introduced by Gleason in function algebras and investigated for convex subsets of real vector spaces by Bear and Bauer in [3], [5], [2], is defined for convex modules. It turns out that all results that were proved for convex sets can also be proved for convex modules, which constitute the algebraic theory generated by convex sets and which have a close connection to physics and mathematical economics.