On the E-Epigraph of an E-Convex Function

In Ref 1, Yang shows that some of the results obtained in Ref. 2 on E-convex programming are incorrect, but does not prove that the results which make the connection between an E-convex function and its E-epigraph are incorrect. In this note, we show that the results obtained in Ref. 2 concerning the characterization of an E-convex function f in terms of its E-epigraph are incorrect. Afterward, some characterizations of E-convex functions using a different notion of epigraph are given.     

On Geodesic <Emphasis Type="Italic">E</Emphasis>-Convex Sets,Geodesic <Emphasis Type="Italic">E</Emphasis>-Convex Functions and <Emphasis Type="Italic">E</Emphasis>-Epigraphs

In this paper, we introduce a new class of sets and a new class of functions called geodesic E-convex sets and geodesic E-convex functions on a Riemannian manifold. The concept of E-quasiconvex functions on R ⁿ is extended to geodesic E-quasiconvex functions on Riemannian manifold and some of its properties are investigated. Afterwards, we generalize the notion of epigraph called E-epigraph and discuss a characterization of geodesic E-convex functions in terms of its E-epigraph. Some properties of geodesic E-convex sets are also studied.     

E-凸函数的若干特征

讨论了一类广义的凸集和凸函数:E-凸集和E-凸函数的若干性质,并给出E-凸函数的一个判别准则．     

Nonlinear programming with E-preinvex and local E-preinvex functions

In this paper, we extend the class of E-convex sets, E-convex and E-quasiconvex functions introduced by [Youness, E.A., 1999. E-convex sets, E-convex functions and E-convex programming. Journal of Optimization Theory and Applications 102, 439–450], respectively by [Syau, Yu-Ru, Lee, E. Stanley, 2005. Some properties of E-convex functions. Applied Mathematics Letters 18, 1074–1080] to E-invex set, E-preinvex, E-prequasiinvex and corresponding local concepts. Some properties of these classes are studied. As an application of our results, we consider the nonlinear programming problem for which, we establish that, under mild conditions, a local minimum is a global minimum.     

Two Characterizations of Super-Reflexive Banach Spaces by the Behaviour of Differences of Convex Functions

A function defined on a Banach space X is called Δ-convex if it can be represented as a difference of two continuous convex functions. In this work we study the relationship between some geometrical properties of a Banach space X and the behaviour of the class of all Δ-convex functions defined on it. More precisely, we provide two new characterizations of super-reflexivity in terms Δ-convex functions.     

A note on three-parameter families and generalized convex functions

The concept of generalized convex functions introduced by Beckenbach [E.F. Beckenbach, Generalized convex functions, Bull. Amer. Math. Soc. 43 (1937) 363–371] is extended to the two-dimensional case. Using three-parameter families, we define generalized convex (midconvex, M-convex) functions and show some continuity properties of them.     

Proximity theorems of discrete convex functions

A proximity theorem is a statement that, given an optimization problem and its relaxation, an optimal solution to the original problem exists in a certain neighborhood of a solution to the relaxation. Proximity theorems have been used successfully, for example, in designing efficient algorithms for discrete resource allocation problems. After reviewing the recent results for L-convex and M-convex functions, this paper establishes proximity theorems for larger classes of discrete convex functions, L₂-convex functions and M₂-convex functions, that are relevant to the polymatroid intersection problem and the submodular flow problem.Mathematics Subject Classification (2000): 90C27, 05B35     

Epigraph of operator functions

It is known that a real function f is convex if and only if the set E(f) = {(x, y) ∈ ? × ?; f (x) ≤ y}, the epigraph of f is a convex set in ?². We state an extension of this result for operator convex functions and C^?-convex sets as well as operator log-convex functions and C^?-log-convex sets. Moreover, the C^?-convex hull of a Hermitian matrix has been represented in terms of its eigenvalues.     

Strict Topologies as Topological Algebras

Let X be a completely regular Hausdorff space, C_b(X) the space of all scalar-valued bounded continuous functions on X with strict topologies. We prove that these are locally convex topological algebras with jointly continuous multiplication. Also we find the necessary and sufficient conditions for these algebras to be locally m-convex.