Exact Formulae for Coderivatives of Normal Cone Mappings to Perturbed Polyhedral Convex Sets

In this paper, without using any regularity assumptions, we derive a new exact formula for computing the Fréchet coderivative and an exact formula for the Mordukhovich coderivative of normal cone mappings to perturbed polyhedral convex sets. Our development establishes generalizations and complements of the existing results on the topic. An example to illustrate formulae is given.     

Affine Variational Inequalities on Normed Spaces

This paper studies infinite-dimensional affine variational inequalities on normed spaces. It is shown that infinite-dimensional quadratic programming problems and infinite-dimensional linear fractional vector optimization problems can be studied by using affine variational inequalities. We present two basic facts about infinite-dimensional affine variational inequalities: the Lagrange multiplier rule and the solution set decomposition.     

Random Nonlinear Variational Inequalities for Mappings of Monotone Type in Banach Spaces

Abstract

Some new concepts of β-semimonotone, and weakly semiaccretive random mappings are introduced and then some nonlinear variational inequalities for such random mappings are solved.     

Nonlinear Elliptic Variational Inequalities

We consider the problem of finding a solution to a class of nonlinear elliptic variational inequalities. These inequalities may be defined on bounded or unbounded domains Ω, and the nonlinearity can depend on gradient terms. Appropriate definitions of sub-and supersolutions relative to the constraint sets are given. By using a mixture of maximal monotone operator theory and compactness arguments we prove the existence of a H²(Ω) solution lying between a given subsolution φ₁ and a given supersolution φ₂≧φ₁, when Ω is bounded, and a H¹(Ω) solution when Ω is unbounded.     

The Compression Property for Affine Variational Inequalities

In 1965, Gale and Nikaidô showed that for any n × n P-matrix A, the only nonnegative vector that A sends into a nonpositive vector is the origin. They applied that result to derive various results including univalence properties of certain nonlinear functions. In this article, we show that an extension of their result holds with the nonnegative orthant replaced by any nonempty polyhedral convex cone. In place of the P-matrix condition, we require a determinantal condition that we call the compression property. When the polyhedral convex cone is the nonnegative orthant, the compression property reduces to the property of being a P-matrix and we recover the Gale-Nikaidô result. We apply the extended theorem to derive tools useful in the analysis of affine variational inequalities over polyhedral convex cones.     

A Class of Iterative Algorithms and Solvability of Nonlinear Variational Inequalities Involving Multivalued Mappings

The solvability of the following class of nonlinear variational inequality (NVI) problems based on a class of iterative procedures, which possess an equivalence to a class of projection formulas, is presented.Determine an element x ^* K and u ^* T(x ^*) such that u ^*, x – x ^* 0 for all x K where T: K P(H) is a multivalued mapping from a real Hilbert space H into P(H), the power set of H, and K is a nonempty closed convex subset of H. The iterative procedure adopted here is represented by a nonlinear variational inequality: for arbitrarily chosen initial points x ⁰, y ⁰ K, u ⁰ T(y ⁰) and v ⁰ T(x ⁰), we have u ^k + x ^k+1 – y ^k , x – x ^k+1 0, x K, for u ^k T(y ^k ) and for k 0where v ^k + y ^k – x ^k , x – y ^k 0, x K and for v ^k T(x ^k ).     

多值单调映象的Browder-Hartman-Stampacchia型变分不等式

在Ｂａｎａｃｈ空间中,证明了多值单调映象的Ｂｒｏｗｄｅｒ－Ｈａｒｔｍａｎ－Ｓｔａｍｐａｃｃｈｉａ型变分不等式解的存在性定理,统一并推广了「４,５,６,９」中的主要结果。     

Multivalued Tikhonov Trajectories of General Affine Variational Inequalities

The Tikhonov trajectory of a general, not necessarily monotone, affine variational inequality is analyzed via the basic properties like single-valuedness, finite-valuedness, continuity, and convergence. We study the multivalued trajectory, which is obtained, by the Tikhonov regularization method, on the whole interval of positive parameters.     

Generalized Variational Inequalities with Pseudomonotone Operators Under Perturbations

Variational inequalities and generalized variational inequalities with perturbed operators and constraints are considered and convergence of solutions to such problems is proved under an assumption of pseudomonotonicity. The paper extends previous results given by the authors proved in the setting of monotone operators.     

A Result on Vector Variational Inequalities with Polyhedral Constraint Sets

In this note, by using some well-known results on properly efficient solutions of vector optimization problems, we show that the Pareto solution set of a vector variational inequality with a polyhedral constraint set can be expressed as the union of the solution sets of a family of (scalar) variational inequalities.     

非扩张映射和广义变分不等式的粘滞逼近法

应用已提出的非扩张映射的粘滞逼近方法,给定初值x_0∈C,考虑一般迭代过程{x_n},g(x_(n+1))=α_nf(x_n)+(1-α_n)SP_C(g(x_n)-λ_nAx_n),n≥0,其中{α_n}■(0,1),S:C→C是非扩张映射,C是实Hilbert空间H的非空闭凸子集.在{α_n}满足合适的条件下可证明,{x_n}强收敛到非扩张映射的不动点集和广义变分不等式解的公共元,且满足某变分不等式.     

Generalized Variational Inequalities and Associated Nonlinear Equations

Here we consider the solvability based on iterative algorithms of the generalized variational inequalities and associated nonlinear equations.     

Continuity of the Solution Map in Parametric Affine Variational Inequalities

A systematic study of the upper semicontinuity and the lower semicontinuity of the solution map in parametric affine variational inequalities is given in this paper. Several examples are constructed to analyze the results. This work was supported by Korea Research Foundation Grant (KRF 2001-015-DP0049), the APEC Postdoctoral Fellowships Program, and the KOSEF Brain Pool Program.     

法向锥与集值映射的单调极大性

研究局部凸空间凸集的法向锥及单调映射与下半连续凸函数的次微分之和的单调极大性．并讨论集值映射循环单调的极大性．     

参数集值变分不等式的解映射的有界性和连续性

本文借助于非空闭凸集序列的外极限集和极大单调集值映射的外极限研究了参数集值变分不等式的集值解映射的有界性和外半连续性．     

一类非线性拟变分不等式

本文利用伪单调算子理论研究如下变分不等式问题：求ｘ∈Ｍ，使得〈Ａｘ，ｙ－ｘ〉＋〈Ｇｘ，ｙ－ｘ〉≥〈ｆ，ｙ－ｘ〉，?ｙ∈Ｍ．并将所得结果应用于拟线性椭圆型边值问题的求解．     

渐近非扩张映射变分不等式的不动点解

设X是实的Banach空间且有一致Gateaux可微范数和一致正规结构，C是X的非空闭凸子集，T，f分别是C上的渐近非扩张映射与压缩映射，X0∈C，xn＋1=anT^mXn＋（1-an）f（xn），n=0，1，2，…，当an∈（O，1）满足适当条件时，则{Xn）强收敛到某变分不等式的不动点解．     

Semicontinuity Results on Parametric Vector Variational Inequalities with Polyhedral Constraint Sets

New results on the lower semicontinuity and upper semicontinuity properties of the Pareto solutions to a parametric vector variational inequality with a polyhedral constraint set are obtained by scalarization approaches.     

关于一类Fuzzy映射的广义变分不等式

本文在拓扩向量空间中研究一类Fuzzy映射的广义变分不等式问题,讨论了这类变分不等式解集的性质及满射性。本文结果改进、推广了作者在[1,2]中的相应结果。     

Isotonicity of the Metric Projection by Lorentz Cone and Variational Inequalities

In this paper, we first discuss the geometric properties of the Lorentz cone and the extended Lorentz cone. The self-duality and orthogonality of the Lorentz cone are obtained in Hilbert spaces. These properties are fundamental for the isotonicity of the metric projection with respect to the order, induced by the Lorentz cone. According to the Lorentz cone, the quasi-sublattice and the extended Lorentz cone are defined. We also obtain the representation of the metric projection onto cones in Hilbert quasi-lattices. As an application, solutions of the classic variational inequality problem and the complementarity problem are found by the Picard iteration corresponding to the composition of the isotone metric projection onto the defining closed and convex set and the difference in the identity mapping and the defining mapping. Our results generalize and improve various recent results obtained by many others.