Strong Duality for Generalized Convex Optimization Problems

In this paper, strong duality for nearly-convex optimization problems is established. Three kinds of conjugate dual problems are associated to the primal optimization problem: the Lagrange dual, Fenchel dual, and Fenchel-Lagrange dual problems. The main result shows that, under suitable conditions, the optimal objective values of these four problems coincide. The first author was supported in part by Gottlieb Daimler and Karl Benz Stiftung 02-48/99. This research has been performed while the second author visited Chemnitz University of Technology under DAAD (Deutscher Akademischer Austauschdienst) Grant A/02/12866. Communicated by T. Rapcsák     

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.     

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.     

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．     

A general Farkas lemma

An abstract version of the classical Farkas lemma for locally convex spaces is given. The abstract Farkas lemma is shown to imply Farkas-type results which have been obtained by Shimizu-Aiyoshi-Katayama, Schecter, Eisenberg, Zalinescu, and Smiley.     

Cyclic Hypomonotonicity,Cyclic Submonotonicity,and Integration

Rockafellar has shown that the subdifferentials of convex functions are always cyclically monotone operators. Moreover, maximal cyclically monotone operators are necessarily operators of this type, since one can construct explicitly a convex function, which turns out to be unique up to a constant, whose subdifferential gives back the operator. This result is a cornerstone in convex analysis and relates tightly convexity and monotonicity. In this paper, we establish analogous robust results that relate weak convexity notions to corresponding notions of weak monotonicity, provided one deals with locally Lipschitz functions and locally bounded operators. In particular, the subdifferentials of locally Lipschitz functions that are directionally hypomonotone [respectively, directionally submonotone] enjoy also an additional cyclic strengthening of this notion and in fact are maximal under this new property. Moreover, every maximal cyclically hypomonotone [respectively, maximal cyclically submonotone] operator is always the Clarke subdifferential of some directionally weakly convex [respectively, directionally approximately convex] locally Lipschitz function, unique up to a constant, which in finite dimentions is a lower C² function [respectively, a lower C¹ function].     

Toland-singer formula cannot distinguish a global minimizer from a choice of stationary points *

Necessary and sufficient optimality conditions are developed for specific classes of constrained and unconstrained global optimization problems. In particular conditions are developed applicable to the unconstrained minimization of the difference and quotient of convex functions for specific special cases. The conditions are developed using the concept of an excess function. Duality results are also presented.     

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.     

Three kinds of generalized convexity

This paper gives some properties of quasiconvex, strictly quasiconvex, and strongly quasiconvex functions. Relationships between them are discussed.This research was supported in part by the National Natural Science Foundation of China. The author would like to thank Professor M. Avriel for valuable comments about this paper.     

Approximately unital operator systems

The main theme of this paper is to consider a notion of 'approximately unital operator systems' including both C*-algebras and unital operator systems.The goals are to prove a version of the Choi-Effros theorem for these systems,to introduce a functorial process for forming an approximately unital operator systems from a given matrix ordered vector space with a proper approximate order unit,to study second duals of these objects and to prove that a C*-algebra can be characterized as an approximately unital ...     

Vector Variational Inequalities for Nondifferentiable Convex Vector Optimization Problems

In this paper, we consider a nondifferentiable convex vector optimization problem (VP), and formulate several kinds of vector variational inequalities with subdifferentials. Here we examine relations among solution sets of such vector variational inequalities and (VP). Mathematics Subject classification (2000). 90C25, 90C29, 65K10 This work was supported by the Brain Korea 21Project in 2003. The authors wish to express their appreciation to the anonymous referee for giving valuable comments.     

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.     

Some Remarks on the Class of Continuous (Semi-) Strictly Quasiconvex Functions

We introduce the notion of variational (semi-) strict quasimonotonicity for a multivalued operator T  : X⇉X ^* relative to a nonempty subset A of X which is not necessarily included in the domain of T. We use this notion to characterize the subdifferentials of continuous (semi-) strictly quasiconvex functions. The proposed definition is a relaxation of the standard definition of (semi-) strict quasimonotonicity, the latter being appropriate only for operators with nonempty values. Thus, the derived results are extensions to the continuous case of the corresponding results for locally Lipschitz functions.     

Differences of Convex Compact Sets in the Space of Directed Sets. Part I: The Space of Directed Sets

A normed and partially ordered vector space of so-called directed sets is constructed, in which the convex cone of all nonempty convex compact sets in R ⁿ is embedded by a positively linear, order preserving and isometric embedding (with respect to a new metric stronger than the Hausdorff metric and equivalent to the Demyanov one). This space is a Banach and a Riesz space for all dimensions and a Banach lattice for n=1. The directed sets in R ⁿ are parametrized by normal directions and defined recursively with respect to the dimension n by the help of a support function and directed supporting faces of lower dimension prescribing the boundary. The operations (addition, subtraction, scalar multiplication) are defined by acting separately on the support function and recursively on the directed supporting faces. Generalized intervals introduced by Kaucher form the basis of this recursive approach. Visualizations of directed sets will be presented in the second part of the paper.