Piecewise Constant Roughly Convex Functions

This paper investigates some kinds of roughly convex functions, namely functions having one of the following properties: -convexity (in the sense of Klötzler and Hartwig), -convexity and midpoint -convexity (in the sense of Hu, Klee, and Larman), -convexity and midpoint -convexity (in the sense of Phu). Some weaker but equivalent conditions for these kinds of roughly convex functions are stated. In particular, piecewise constant functions satisfying f(x) = f([x]) are considered, where [x] denotes the integer part of the real number x. These functions appear in numerical calculation, when an original function g is replaced by f(x):=g([x]) because of discretization. In the present paper, we answer the question of when and in what sense such a function f is roughly convex.     

Strictly and Roughly Convexlike Functions

A function is said to be strictly and roughly convexlike with respect to the roughness degree r > 0 (for short, strictly r-convexlike) provided that, for all x ₀, x ₁ D satisfying ||x ₀ – x ₁|| > r, there exists a ]0, 1[ such that

Convex Defining Functions for Convex Domains

We give three proofs of the fact that a smoothly bounded, convex domain in ℝ ⁿ has defining functions whose Hessians are non-negative definite in a neighborhood of the boundary of the domain.     

关于半连续函数与凸函数的注记

在半连续前提下,给出凸函数和严格凸函数的不等式刻划．指出非空凸集上的半连续函数满足中间点凸性时,成为凸函数,满足中间点严格凸性时,成为严格凸函数．最后定义F—G广义凸函数和条件p1,p2等概念,列举若干满足条件p1,p2的数量函数和向量函数,并指出,对于F—G广义凸函数,在条件p1,p2及一定连续性条件下,可以得到类似结果．     

Lipschitz-continuity of Spherically Convex Functions

Acta Mathematica Sinica, English Series - In this article, some basic and important properties of spherically convex functions, such as the Lipschitz-continuity, are investigated. It is shown that,...     

度量凸函数的延拓

每个度量空间都能等距嵌入到实Banach空间,所以度量凸函数可视为Banach空间子集上的函数．本文举出反例说明不是所有度量凸函数都能延拓为凸函数,并给出度量凸函数能延拓为凸函数的充分条件．     

Banach空间上凸函数项级数

本文研究了Ｂａｎａｃｈ空间上凸函数项级数，给出了Ｍｏｒｅａｕ Ｒｏｃｋａｆｅｌａｒ定理的推广，做为它的应用，获得了Ｋｕｈｎ Ｔｕｃｋｅｒ定理的一个部分推广．     

Well-positioned Closed Convex Sets and Well-positioned Closed Convex Functions

We characterize the class of those closed convex sets which have a barrier cone with a nonempty interior. As a consequence, we describe the set of those proper extended-real-valued functionals for which the domain of their Fenchel conjugate has a nonempty interior. As an application, we study the stability of the solution set of a semi-coercive variational inequality.     

Duality Unary Convex Functions Programming

Most nonliner programming problems consist of functions which are sums of unary convex functions of linear fuctions.In this paper,we derive the duality forms of the unary convex optimization,and these technuques are applied to the geometric programming and minimum discriminaiton information problems.     

Convex Functions with Unbounded Gradient

We show that domains, that allow for convex functions with unbounded gradient at their boundary, are convex.     

p-Harmonic Maps and Convex Functions

p-harmonic maps (p ' 2) between Riemannian manifolds are investigated. Some theorems of Liouville type are given for such maps when target manifolds have convex functions.     

凸函数与半连续函数的关系

通过研究凸函数与半连续函数的关系,给出了凸函数的一个与上半连续性相结合的等价定义.     

Two Inequalities for Convex Functions

Let a ₀ < a ₁ < ··· < a _n be positive integers with sums $ {\sum\nolimits_{i = 0}^n {\varepsilon _{i} a_{i} {\left( {\varepsilon _{i} = 0,1} \right)}} } Let a ₀ < a ₁ < ··· < a _n be positive integers with sums distinct. P. Erd?s conjectured that The best known result along this line is that of Chen: Let f be any given convex decreasing function on [A, B] with α ₀, α ₁, ... , α _n , β ₀, β ₁, ... , β _n being real numbers in [A, B] with α ₀ ≤ α ₁ ≤ ··· ≤ α _n , Then In this paper, we obtain two generalizations of the above result; each is of special interest in itself. We prove: Theorem 1 Let f and g be two given non-negative convex decreasing functions on [A, B], and α ₀, α ₁, ... , α _n , β ₀, β ₁, ... , β _n , α' ₀, α' ₁, ... , α' _n , β' ₀ , β' ₁ , ... , β' _n be real numbers in [A, B] with α ₀ ≤ α ₁ ≤ ··· ≤ α _n , α' ₀ ≤ α' ₁ ≤ ··· ≤ α' _n , Then Theorem 2 Let f be any given convex decreasing function on [A, B] with k ₀, k ₁, ... , k _n being nonnegative real numbers and α ₀, α ₁, ... , α _n , β ₀, β ₁, ... , β _n being real numbers in [A, B] with α ₀ ≤ α ₁ ≤ ··· ≤ α _n , Then      

关于凸函数的新不等式

考虑由Fejér不等式的右边部分生成的差值.通过建立积分恒等式,在导函数满足M-Lipschitz条件和导函数有界这两种情况下,给出这个差值的界的估计.     

亚纯p叶凸象函数的子类

本文引进单位圆盘内亚纯ｐ叶凸函数的新子类Ｃ（ｎ,ｐ,Ａ,Ｂ）和Ｋ（ｎ,ｐ,Ａ,Ｂ）,分别研究其包含关系与类中函数的积分变换等性质．     

Duality of Unary Convex Functions Programming

Most nonliner programming problems consist of functions which are sums of unary,convexfunctions of linear fuctions.In this paper.we derive the duality forms of the unary oonvex optimization,and these technuqucs are applied to the geometric programming and minimum discrimination informationproblems.     

The Strong Continuity of Convex Functions

A convex function defined on an open convex set of a finite-dimensional space is known to be continuous at every point of this set. In fact, a convex function has a strengthened continuity property. The notion of strong continuity is introduced in this study to show that a convex function has this property. The proof is based on only the definition of convexity and Jensen’s inequality. The definition of strong continuity involves a constant (the constant of strong continuity). An unimprovable value of this constant is given in the case of convex functions. The constant of strong continuity depends, in particular, on the form of a norm introduced in the space of arguments of a convex function. The polyhedral norm is of particular interest. It is straightforward to calculate the constant of strong continuity when it is used. This requires a finite number of values of the convex function.     

Directed Projection Functions of Convex Bodies

For 1 ≤ i < j < d, a j-dimensional subspace L of and a convex body K in , we consider the projection K|L of K onto L. The directed projection function v _i,j (K;L,u) is defined to be the i-dimensional size of the part of K|L which is illuminated in direction u ∈ L. This involves the i-th surface area measure of K|L and is motivated by Groemer’s [17] notion of semi-girth of bodies in . It is well-known that centrally symmetric bodies are determined (up to translation) by their projection functions, we extend this by showing that an arbitrary body is determined by any one of its directed projection functions. We also obtain a corresponding stability result. Groemer [17] addressed the case i = 1, j = 2, d = 3. For j > 1, we then consider the average of v _1,j (K;L,u) over all spaces L containing u and investigate whether the resulting function determines K. We will find pairs (d,j) for which this is the case and some pairs for which it is false. The latter situation will be seen to be related to some classical results from number theory. We will also consider more general averages for the case of centrally symmetric bodies. The research of the first author was supported in part by NSF Grant DMS-9971202 and that of the second author by a grant from the Volkswagen Foundation.     

Random Gradient-Free Minimization of Convex Functions

In this paper, we prove new complexity bounds for methods of convex optimization based only on computation of the function value. The search directions of our schemes are normally distributed random Gaussian vectors. It appears that such methods usually need at most n times more iterations than the standard gradient methods, where n is the dimension of the space of variables. This conclusion is true for both nonsmooth and smooth problems. For the latter class, we present also an accelerated scheme with the expected rate of convergence \(O\Big ({n^2 \over k^2}\Big )\), where k is the iteration counter. For stochastic optimization, we propose a zero-order scheme and justify its expected rate of convergence \(O\Big ({n \over k^{1/2}}\Big )\). We give also some bounds for the rate of convergence of the random gradient-free methods to stationary points of nonconvex functions, for both smooth and nonsmooth cases. Our theoretical results are supported by preliminary computational experiments.