Gap Functions and Error Bounds for Set-Valued Vector Quasi Variational Inequality Problems

One of the classical approaches in the analysis of a variational inequality problem is to transform it into an equivalent optimization problem via the notion of gap function. The gap functions are useful tools in deriving the error bounds which provide an estimated distance between a specific point and the exact solution of variational inequality problem. In this paper, we follow a similar approach for set-valued vector quasi variational inequality problems and define the gap functions based on scalarization scheme as well as the one with no scalar parameter. The error bounds results are obtained under fixed point symmetric and locally α-Holder assumptions on the set-valued map describing the domain of solution space of a set-valued vector quasi variational inequality problem.

Uniformly Bounded Set-Valued Composition Operators in the Spaces of Functions of Bounded Variation in the Sense of Riesz

We show that the lateral regularizations of the generator of any uniformly bounded set-valued composition Nemytskij operator acting in the spaces of functions of bounded variation in the sense of Riesz, with nonempty bounded closed and convex values, are an affine function.

Monotone Set-Valued Function Defined by Set-Valued Choquet Integral

Structural characteristics and absolute continuities of monotone set-valued function defined by set- valued Choquet integral are discussed. Similar to the single-valued monotone set function, several important structural characteristics of set-valued function are defined and have been proven the same as those in the original set functions, such as null-additivity, weakly null-additivity, order continuity, strong order continuity and property（S）. A counterexample shows that order continuity and strong order continuity of the original set functions are no longer kept in a monotone set-valued function when Choquet integrably bounded assumption is abandoned. Four kinds of absolute continuities are defined for set-valued function, and all been proven valid with respect to the original set functions.

KUHN-TUCKER CONDITION AND WOLFE DUALITY OF PREINVEX SET-VALUED OPTIMIZATION

The optimality Kuhn-Tucker condition and the wolfe duality for the preinvex set-valued optimization are investigated. Firstly, the concepts of alpha-order G-invex set and the alpha-order S-preinvex set-valued function were introduced, from which the properties of the corresponding contingent cone and the alpha-order contingent derivative were studied. Finally, the optimality Kuhn-Tucker condition and the Wolfe duality theorem for the alpha-order S-preinvex set-valued optimization were presented with the help of the alpha-order contingent derivative.

Optimality for Henig Proper Efficiency in Vector Optimization Involving Dini Set-Valued Directional Derivatives

This note studies the optimality conditions of vector optimization problems involving generalized convexity in locally convex spaces. Based upon the concept of Dini set-valued directional derivatives, the necessary and sufficient optimality conditions are established for Henig proper and strong minimal solutions respectively in generalized preinvex vector optimization problems.

A survey of ergodicity on the set-valued mappings

In this paper, we give a survey on the PhD thesis of the first author. There theexistence and ergodicity on invariant measures of set-valued mappings are discused.

Henig Efficiency in Vector Optimization with Nearly Cone-subconvexlike Set-valued Functions

In this paper, we study Henig efficiency in vector optimization with nearly cone-subconvexlike set-valued function. The existence of Henig efficient point is proved and characterization of Henig efficiency is established using the method of Lagrangian multiplier. As an interesting application of the results in this paper, we establish a Lagrange multiplier theorem for super efficiency in vector optimization with nearly conesubconvexlike set-valued function.

Approximate Weak Minimal Solutions of Set-Valued Optimization Problems

This paper deals with approximate weak minimal solutions of set-valued optimization problems under vector and set optimality criteria.The relationships between various concepts of approximate weak minimal solutions are investigated.Some topological properties and existence theorems of these solutions are given.It is shown that for set-valued optimization problems with upper(outer)cone-semicontinuous objective values or closed objective maps the approximate weak minimal and strictly approximate lower weak minimal solution sets are closed.By using the polar cone and two scalarization processes,some necessary and sufficient optimality conditions in the sense of vector and set criteria are provided.

The Approximation Theorem of Convolution Operator in △~p Set-valued Function Space

The paper is a contribution to the problem of approximating random set with values in a separable Banach space. This class of set-valued function is widely used in many areas.We investigate the properties of p-bounded integrable random set. Based on this we endow it with △p metric which can be viewed as a integral type hausdorff metric and present some approximation theorem of a class of convolution operators with respect to △p metric. Moreover we also can establish analogous theorem for other integral type operator in △p space.

弱伪连续向量值函数及其在多目标博弈中的应用

基于对弱伪连续实值函数的研究,提出向量值函数关于锥的弱伪连续定义,并建立了向量值函数R_＋~k弱伪连续与实值函数弱伪连续的对应关系.作为弱伪连续向量值函数的应用,给出了弱伪连续下的向量值Ky Fan点的存在性定理,并通过该定理证明了伪连续向量值支付下的多目标博弈弱Pareto-Nash平衡点的存在性定理.

Arterial spin labeling in neuroimaging

Arterial spin labeling(ASL) is a magnetic resonance imaging technique for measuring tissue perfusion using a freely diffusible intrinsic tracer.As compared with other perfusion techniques,ASL offers several advantages and is now available for routine clinical practice in many institutions.Its noninvasive nature and ability to quantitatively measure tissue perfusion make ASL ideal for research and clinical studies.Recent technical advances have increased its sensitivity and also extended its potential applications.This review focuses on some basic knowledge of ASL perfusion,emerging techniques and clinical applications in neuroimaging.

一类非连续多目标博弈的Hadamard良定性

通过引入n人非合作多目标博弈的若有效反应映射,证明了多目标博弈的弱Preto-Nash平衡点集在支付函数为伪连续条件下是非空紧集,并且研究了在支付函数和策略集均扰动情况下,伪连续多目标博弈的弱Pre-to-Nash平衡点的Hadamard良定性.

Continuous solutions for fractional integral inclusion in locally convex topological space

The existence of continuous solutions for fractional integral inclusion via its singlevalued problem and fixed point theorem for set-valued function in locally convex topological spaces is discussed. The proof of the single-valued problem will be based on the Leray- Schauder fixed point theorem. Moreover, the controllability of this solution is studied.

Solving Smooth Generalized Equations Using Modified Gauss-Type Proximal Point Method

Consider X and Y are two real or complex Banach spaces. We introduce and study a modified Gauss-type proximal point algorithm (in short modified G-PPA) for solving the generalized equations of the form 0 ∈h(x) + H(x), where h : X → Y is a smooth function on Ω ⊆X and H : X ⇉2Y is a set valued mapping with closed graph. When H is metrically regular and under some sufficient conditions, we analyze both semi-local and local convergence of the modified G-PPA. Moreover, the convergence results of the modified G-PPA are justified by presenting a numerical example.

ON SET-VALUED MARTINGALES, SUBMARTINGALES AND SUPERMARTINGALES

Based on Refs. [1—8], we discuss the following problem in this note.Let (Ω, A, P)be a complete probability space and X be a separable Banach space with the dual X~*.

On Ekeland's Variational Principle for Set-Valued Mappings

In this paper, we derive a general vector Ekeland variational principle for set-valued mappings, which has a dosed relation to εk^0 -efficient points of set-valued optimization problems. The main result presented in this paper is a generalization of the corresponding result in [3].

Existence of Competitive Equilibria without Standard Boundary Behavior

We study the existence of competitive equilibria when the excess demand function fails to satisfy the standard boundary behavior. We introduce alternative boundary conditions and we examine their role in proving the existence of strictly positive solutions to a system of non-linear equations (competitive equilibium prices). In addition, we slightly generalize a well-known theorem on the existence of maximal elements, and we unveil the link between the hypothesis of our theorem and one of the boundary conditions introduced in this work.

集值伪连续函数及其在非合作与合作博弈中的应用

首先,我们引入一类集值伪连续函数.在此基础上,我们给出集值伪连续函数的性质特征刻画.然后,作为应用,我们在具有集值伪连续支付函数的博弈问题中,分别证明了非合作博弈均衡与合作博弈均衡的存在性.最后给出一个具体算例,验证了其可行性.

Optimal Controls of Nonlinear Evolution Inclusions withPseudomonotone Operator

In this paper, we study the optimal control problem of nonlinear differentialinclusions with principle operator being pseudomonotone. First, we give some propertiesof solutions of certain evolution equations. Further, we prove the existence of admissibletrajectories for evolution inclusions. Then, we extend the Fillipov's selection theoremand discuss a general Lagrange type optimal control problem. Finally, we present anexample that demonstrates the appplicability of our results.

Partial asymptotic null-controllability with mixed state-input constraints

This paper is devoted to the problem of partial asymptotic null-controllability of control sys-tems governed by ordinary differential equations,subjected to possibly mixed state-input con-straints.Using Lyapunov functions within the framework of viability theory,feedback controls are designed in such a way a part of system’s state can be driven to the origin asymptotically,taking into account the mixed constraints.By using Michael selection theorem,the existence of such controls is proved,in the case of convex constraints,and their expressions are given as continu-ous selections of an appropriate constructed multifunction.Finally,two examples are processed numerically in order to illustrate the theoretical results.