变双障碍问题解的抽象稳定性

本文考虑了主部为非线性变双障碍问题解的抽象稳定性 (连续依赖性 ) .由于采用了弱收敛原理和文 [2 ]中取检验函数的技巧 ,我们的证明无需像 [1 ]那样应用Minty引理 .     

广义Minty 向量似变分不等式解的性质

在拓扑向量空间中讨论下Dini方向导数形式的广义Minty向量似变分不等式问题. 可微形式的Minty变分不等式、Minty似变分不等式和Minty向量变分不等式是其特殊形式. 该文分别讨论了Minty向量似变分不等式的解与径向递减函数, 与向量优化问题的最优解或有效解之间的关系问题, 以及Minty向量似变分不等式的解集的仿射性质. 这些定理推广了文献中Minty变分不等式的一些重要的已知结果.     

On vector variational-like inequalities and vector optimization problems with (G,α)-invexity

The aim of this paper is to study the relationship among Minty vector variationallike inequality problem, Stampacchia vector variational-like inequality problem and vector optimization problem involving(G, α)-invex functions. Furthermore, we establish equivalence among the solutions of weak formulations of Minty vector variational-like inequality problem,Stampacchia vector variational-like inequality problem and weak efficient solution of vector optimization problem under the assumption of(G, α)-invex functions. Examples are provided to elucidate our results.     

Farkas引理的几个等价形式及其推广

本文考虑了Farkas引理,Gordan引理及其拓展形式之间的关系,从理论上证明了其等价性并说明了Farkas引理在各种等价形式中的重要地位,并指出了Gordan引理实际是叮看作是Farkas引理的弱形式,然后研究了Farkas引理及其它形式在锥线性不等式组中的推广.     

加权拟变分不等式

本文介绍了Stampacchia型和Minty型加权拟变分不等式以及加权拟变分不等式组,并讨论了它们之间的关系.利用这种关系研究了加权拟变分不等式和加权拟变分不等式组的解的存在性.     

KLee和Minty例的性质

本文证明了:用单纯形法迭代求解Kell和Minty例（式（1）的线性规划问题）时，如果用最大增量法确定入基变量，则从其任一顶点出发，迭代次数不超过n次．     

Bihali引理的证明

根据Bihali引理可以确定微分方程解存在的更大一些的区间.该引理显然是十分重要的,然而目前找不到这个引理的证明.本文给出了一种证明.并举实例说明了这个引理的重要性.     

Rn中单位球上满足Poisson方程的边界Schwarz引理

Schwarz引理在全纯映射和调和映射理论中扮演着重要的角色.本文建立了Rn中单位球上满足Poisson方程解的边界Schwarz引理.作为应用,给出了单位球上调和自映射的边界Schwarz引理,将平面上的多重调映射结果和边界Schwarz引理推广到了高维调和映射.     

大范围反函数定理

在非线性方程组数值解研究中,比较重要的问题之一是确定某一类函数是否是全空间到全空间的同胚映射。在这方面,比较经典的一个结果是Minty定理([1]p·22,[2]p·167)。我们在这里证明了一个较为广泛的大范围反函数定理(定理1),Minty定理也就是一致单调算子定理可以作为它的一个推论(定理2)。在证明过程中,主要运用了拓扑度论的一个重要定理——开域不变定理:在开集上定义的一对一连续映射把开集映为开集。     

两个引理及其推广的再拓广

文[1]介绍了有关不等式的两个引理及其推广命题1-4．文[2]将两个引理及其推广命题作出了进一步推广．本文将推广的两个引理及其推广的命题再进一步作出拓广．两个引理的再拓广如下：     

Nonsmooth Vector Optimization Problems and Minty Vector Variational Inequalities

The vector optimization problem may have a nonsmooth objective function. Therefore, we introduce the Minty vector variational inequality (Minty VVI) and the Stampacchia vector variational inequality (Stampacchia VVI) defined by means of upper Dini derivative. By using the Minty VVI, we provide a necessary and sufficient condition for a vector minimal point (v.m.p.) of a vector optimization problem for pseudoconvex functions involving Dini derivatives. We establish the relationship between the Minty VVI and the Stampacchia VVI under upper sign continuity. Some relationships among v.m.p., weak v.m.p., solutions of the Stampacchia VVI and solutions of the Minty VVI are discussed. We present also an existence result for the solutions of the weak Minty VVI and the weak Stampacchia VVI.     

Some remarks on the Minty vector variational principle

In scalar optimization it is well known that a solution of a Minty variational inequality of differential type is a solution of the related optimization problem. This relation is known as “Minty variational principle.” In the vector case, the links between Minty variational inequalities and vector optimization problems were investigated in [F. Giannessi, On Minty variational principle, in: New Trends in Mathematical Programming, Kluwer Academic, Dordrecht, 1997, pp. 93-99] and subsequently in [X.M. Yang, X.Q. Yang, K.L. Teo, Some remarks on the Minty vector variational inequality, J. Optim. Theory Appl. 121 (2004) 193-201]. In these papers, in the particular case of a differentiable objective function f taking values in Rm and a Pareto ordering cone, it has been shown that the vector Minty variational principle holds for pseudoconvex functions. In this paper we extend such results to the case of an arbitrary ordering cone and a nondifferentiable objective function, distinguishing two different kinds of solutions of a vector optimization problem, namely ideal (or absolute) efficient points and weakly efficient points. Further, we point out that in the vector case, the Minty variational principle cannot be extended to quasiconvex functions.     

A non-smooth version of Minty variational principle

The aim of this article is to study the relationship between generalized Minty vector variational inequalities and non-smooth vector optimization problems. Under pseudoconvexity or pseudomonotonicity, we establish the relationship between an efficient solution of a non-smooth vector optimization problem and a generalized Minty vector variational inequality. This offers a non-smooth version of existing Minty variational principle.     

Differential and sensitivity properties of gap functions for Minty vector variational inequalities

The purpose of this paper is to investigate differential properties of a class of set-valued maps and gap functions involving Minty vector variational inequalities. Relationships between their contingent derivatives are discussed. An explicit expression of the contingent derivative for the class of set-valued maps is established. Optimality conditions of solutions for Minty vector variational inequalities are obtained.     

Existence of Solutions and Star-shapedness in Minty Variational Inequalities

Minty Variational Inequalities (for short, Minty VI) have proved to characterize a kind of equilibrium more qualified than Stampacchia Variational Inequalities (for short, Stampacchia VI). This conclusion leads to argue that, when a Minty VI admits a solution and the operator F admits a primitive f (that is F= f′), then f has some regularity property, e.g. convexity or generalized convexity. In this paper we put in terms of the lower Dini directional derivative a problem, referred to as Minty VI(f′_,K), which can be considered a nonlinear extension of the Minty VI with F=f′ (K denotes a subset of ℝⁿ). We investigate, in the case that K is star-shaped, the existence of a solution of Minty VI(f’_,K) and increasing along rays starting at x^* property of (for short, F ɛIAR (K,x^*)). We prove that Minty VI(f’_,K) with a radially lower semicontinuous function fhas a solution x^* ɛker K if and only if FɛIAR(K, x^*). Furthermore we investigate, with regard to optimization problems, some properties of increasing along rays functions, which can be considered as extensions of analogous properties holding for convex functions. In particular we show that functions belonging to the class IAR(K,x^*) enjoy some well-posedness properties.     

A redundant Klee–Minty construction with all the redundant constraints touching the feasible region

We have previously shown, by redundant Klee–Minty constructions, that the central path may arbitrarily closely visit every vertex of the Klee–Minty cube. In those constructions, the redundant constraints are far away from the feasible region. In this paper, we provide a construction in which all redundant constraints touch the feasible region.     

Solvability of the Minty Variational Inequality

We consider the existence of solutions to the Minty variational inequality, as it plays a key role in a projection-type algorithm for solving the variational inequality. It is shown that, if the underlying mapping has a separable structure with each component of the mapping being quasimonotone, then the Minty variational inequality has a solution. An example shows that the underlying mapping itself is not necessarily quasimonotone, although each of its components is.     

Minty lemma for inverted vector variational inequalities

In this paper, we introduce Stampacchia-type inverted vector variational inequalities and Minty-type inverted vector variational inequalities and discuss Minty lemma for the inequalities showing the existence of solutions to them in Banach spaces. Next, we consider the equivalence of our Minty lemma with Brouwer’s fixed point theorem as an application.