On uniformly convex functions

In this paper we study uniformly convex functions and uniformly convex functions at a point, giving some properties and characterizations of them. Further, we give some examples and applications of these types of functions.     

Explicit convex and concave envelopes through polyhedral subdivisions

In this paper, we derive explicit characterizations of convex and concave envelopes of several nonlinear functions over various subsets of a hyper-rectangle. These envelopes are obtained by identifying polyhedral subdivisions of the hyper-rectangle over which the envelopes can be constructed easily. In particular, we use these techniques to derive, in closed-form, the concave envelopes of concave-extendable supermodular functions and the convex envelopes of disjunctive convex functions.     

Geometric conditions for optimization in a linear space

In this paper we give new properties of convex functions whose domains are subsets of a linear space; we use them in order to get geometric characterizations of a minimant of a convex function.     

Set containment characterization with strict and weak quasiconvex inequalities

Dual characterizations of the containment of a convex set, defined by infinite quasiconvex constraints, in an evenly convex set, and in a reverse convex set, defined by infinite quasiconvex constraints, are provided. Notions of quasiconjugate for quasiconvex functions, λ-quasiconjugate and λ-semiconjugate, play important roles to derive the characterizations of the set containments.     

Set containment characterization for quasiconvex programming

Dual characterizations of containment of a convex set, defined by quasiconvex constraints, in a convex set, and in a reverse convex set, defined by a quasiconvex constraint, are provided. Notions of quasiconjugate for quasiconvex functions, H-quasiconjugate and R-quasiconjugate, play important roles to derive characterizations of the set containments.     

一致星象和凸象超几何函数的一些性质

介绍了(高斯)超几何函数属于一致星象函数和一致凸函数的某些子类的一些性质.还考虑了与超几何函数相关的算子.     

Conditioning and Regularization of Nonsymmetric Operators

In this paper, we study conditioning problems for convex and nonconvex functions defined on normed linear spaces. We extend the notion of upper Lipschitzness for multivalued functions introduced by Robinson, and show that this concept ensures local conditioning in the nonconvex case via an abstract subdifferential; in the convex case, we obtain complete characterizations of global conditioning in terms of an extension of the upper-Lipschitz property.     

SOC-monotone and SOC-convex functions vs. matrix-monotone and matrix-convex functions

The SOC-monotone function (respectively, SOC-convex function) is a scalar valued function that induces a map to preserve the monotone order (respectively, the convex order), when imposed on the spectral factorization of vectors associated with second-order cones (SOCs) in general Hilbert spaces. In this paper, we provide the sufficient and necessary characterizations for the two classes of functions, and particularly establish that the set of continuous SOC-monotone (respectively, SOC-convex) functions coincides with that of continuous matrix monotone (respectively, matrix convex) functions of order 2.     

Convex functions, subdifferentiability and renormings

This paper through discussing subdifferentiability and convexity of convex functions shows that a Banach space admits an equivalent uniformly [locally uniformly, strictly] convex norm if and only if there exists a continuous uniformly [locally uniformly, strictly] convex function on some nonempty open convex subset of the space and presents some characterizations of super-reflexive Banach spaces. Supported by NSFC     

Motzkin decomposition of closed convex sets

Theodore Motzkin proved, in 1936, that any polyhedral convex set can be expressed as the (Minkowski) sum of a polytope and a polyhedral convex cone. This paper provides five characterizations of the larger class of closed convex sets in finite dimensional Euclidean spaces which are the sum of a compact convex set with a closed convex cone. These characterizations involve different types of representations of closed convex sets as the support functions, dual cones and linear systems whose relationships are also analyzed in the paper. The obtaining of information about a given closed convex set F and the parametric linear optimization problem with feasible set F from each of its different representations, including the Motzkin decomposition, is also discussed.     

Dual Representations for Convex Risk Measures via Conjugate Duality

The aim of this paper is to give dual representations for different convex risk measures by employing their conjugate functions. To establish the formulas for the conjugates, we use on the one hand some classical results from convex analysis and on the other hand some tools from the conjugate duality theory. Some characterizations of so-called deviation measures recently given in the literature turn out to be direct consequences of our results.     

Farkas-Type Results for Vector-Valued Functions with Applications

The main purpose of this paper consists of providing characterizations of the inclusion of the solution set of a given conic system posed in a real locally convex topological space into a variety of subsets of the same space defined by means of vector-valued functions. These Farkas-type results are used to derive characterizations of the weak solutions of vector optimization problems (including multiobjective and scalar ones), vector variational inequalities, and vector equilibrium problems.     

Rank functions of strict cg-matroids

A matroid-like structure defined on a convex geometry, called a cg-matroid, is defined by Fujishige et al. [Matroids on convex geometries (cg-matroids), Discrete Math. 307 (2007) 1936-1950]. A cg-matroid whose rank function is naturally defined is called a strict cg-matroid. In this paper, we give characterizations of strict cg-matroids by their rank functions.     

半严格不变凸函数

文章在Banach空间中定义了一种新的广义凸函数—半严格不变凸函数.对于满足局部Lipschitz条件的半严格不变凸函数,得到了它的广义Clarke次微分性质.文中还讨论了半严格不变凸函数与不变凸函数及半严格预不变凸函数之间的关系,得到了半严格不变凸函数的一些性质.     

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.     

Second-Order Global Optimality Conditions for Optimization Problems

Second-order optimality conditions are studied for the constrained optimization problem where the objective function and the constraints are compositions of convex functions and twice strictly differentiable functions. A second-order sufficient condition of a global minimizer is obtained by introducing a generalized representation condition. Second-order minimizer characterizations for a convex program and a linear fractional program are derived using the generalized representation condition     

Second-Order Global Optimality Conditions for Optimization Problems

Second-order optimality conditions are studied for the constrained optimization problem where the objective function and the constraints are compositions of convex functions and twice strictly differentiable functions. A second-order sufficient condition of a global minimizer is obtained by introducing a generalized representation condition. Second-order minimizer characterizations for a convex program and a linear fractional program are derived using the generalized representation condition     

Concepts and techniques of optimization on the sphere

In this paper some concepts and techniques of Mathematical Programming are extended in an intrinsic way from the Euclidean space to the sphere. In particular, the notion of convex functions, variational problem and monotone vector fields are extended to the sphere and several characterizations of these notions are shown. As an application of the convexity concept, necessary and sufficient optimality conditions for constrained convex optimization problems on the sphere are derived.     

半严格不变凸函数

文章在Banach空间中定义了一种新的广义凸函数—半严格不变凸函数.对于满足局部Lipschitz条件的半严格不变凸函数,得到了它的广义Clarke次微分性质.文中还讨论了半严格不变凸函数与不变凸函数及半严格预不变凸函数之间的关系,得到了半严格不变凸函数的一些性质.     

Characterizations of optimality in convex programming: the nondifferentiable case

Primal, dual and saddle-point characterizations of optimality are given for convex programming in the general case (nondifferentiable functions and no constraint qualification).