Lower Semicontinuity for Parametric Weak Vector Variational Inequalities in Reflexive Banach Spaces

In this paper, we study the solution stability of parametric weak Vector Variational Inequalities with set-valued and single-valued mappings, respectively. We obtain the lower semicontinuity of the solution mapping for the parametric set-valued weak Vector Variational Inequality with strictly C-pseudomapping in reflexive Banach spaces. Moreover, under some requirements that the mapping satisfies the degree conditions, we establish the lower semicontinuity of the solution mapping for a parametric single-valued weak Vector Variational Inequality in reflexive Banach spaces, by using the degree-theoretic approach. The results presented in this paper improve and extend some known results due to Kien and Yao (Set-Valued Anal. 16:399–412, 2008) and Wong (J. Glob. Optim. 46:435–446, 2010).     

Well-posedness and stability of solutions for set optimization problems

In this paper, some characterizations for the generalized l-B-well-posedness and the generalized u-B-well-posedness of set optimization problems are given. Moreover, the Hausdorff upper semi-continuity of l-minimal solution mapping and u-minimal solution mapping are established by assuming that the set optimization problem is l-H-well-posed and u-H-well-posed, respectively. Finally, the upper semi-continuity and the lower semi-continuity of solution mappings to parametric set optimization problems are investigated under some suitable conditions.     

The stability of the solution sets for set optimization problems via improvement sets

ABSTRACT

The aim of this paper is to investigate the stability of the solution sets for set optimization problems via improvement sets. Firstly, we consider the relations among the solution sets for optimization problem with set optimization criterion. Then, the closeness and the convexity of solution sets are discussed. Furthermore, the upper semi-continuity, Hausdorff upper semi-continuity and lower semi-continuity of solution mappings to parametric set optimization problems via improvement sets are established under some suitable conditions. These results extend and develop some recent works in this field.     

Optimization and Variational Inequalities with Pseudoconvex Functions

In the present paper we consider a pseudoconvex (in an extended sense) function f using higher order Dini directional derivatives. A Variational Inequality, which is a refinement of the Stampacchia Variational Inequality, is defined. We prove that the solution set of this problem coincides with the set of global minimizers of f if and only if f is pseudoconvex. We introduce a notion of pseudomonotone Dini directional derivatives (in an extended sense). It is applied to prove that the solution sets of the Stampacchia Variational Inequality and Minty Variational Inequality coincide if and only if the function is pseudoconvex. At last, we obtain several characterizations of the solution set of a program with a pseudoconvex objective function.     

集优化问题的广义适定性与解的稳定性

本文研究了集优化问题的适定性与解的稳定性. 首次利用嵌入技术引入了集优化问题的广义适定性概念, 得到了此类适定性的一些判定准则和特征, 并给出其充分条件. 此外, 借助一类广义Gerstewitz 函数, 建立了此类适定性与一类标量优化问题广义适定性之间的等价关系. 最后, 在适当条件下研究了含参集优化问题弱有效解映射的上半连续性和下半连续性.     

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．     

Hadamard's variational formula for a mixed problem and an application to a problem related to a signorini-like variational inequality

We give a variational formula for the solution of a Mixed Boundary Value Problem when the geometrical domain is perturbed (Hadamard's Variational Formula) ; we show that this formula contains singularities ; the result is used to construct a new approach to the optimal control problem for a problem related to a Signorini-like Elliptic Variational Inequality.     

Some classes of projected dynamical systems in Banach spaces and variational inequalities

We introduce some Projected Dynamical Systems based on metric and generalized Projection Operator in a strictly convex and smooth Banach Space. Then we prove that critical points of these systems coincide with the solution of a Variational Inequality.     

Algorithms for the Split Variational Inequality Problem

We propose a prototypical Split Inverse Problem (SIP) and a new variational problem, called the Split Variational Inequality Problem (SVIP), which is a SIP. It entails finding a solution of one inverse problem (e.g., a Variational Inequality Problem (VIP)), the image of which under a given bounded linear transformation is a solution of another inverse problem such as a VIP. We construct iterative algorithms that solve such problems, under reasonable conditions, in Hilbert space and then discuss special cases, some of which are new even in Euclidean space.     

Parametric vector optimization

In this paper we consider a sequence of vector optimization problems. We aim to generalize a vector condition that relates the parametric function and the limit function. In particular, we recover our condition given in the scalar case. Our stability approach is such that the limit of the sequence of solutions that correspond to vector optimization problems to be a solution of a limit vector optimization problem. Therefore, one can view our statement as an existence result. This general framework has been used in several previous works. In our main theorem, we use the notion of strong lower cone-semi-continuity. An example is given to illustrate why only cone-lower semi-continuity for the limit function is not sufficient for our result.     

Split Monotone Variational Inclusions

Based on the very recent work by Censor-Gibali-Reich (), we propose an extension of their new variational problem (Split Variational Inequality Problem) to monotone variational inclusions. Relying on the Krasnosel’skii-Mann Theorem for averaged operators, we analyze an algorithm for solving new split monotone inclusions under weaker conditions. Our weak convergence results improve and develop previously discussed Split Variational Inequality Problems, feasibility problems and related problems and algorithms.     

Random attractor and upper semi-continuity for Zakharov lattice system with multiplicative white noise

In this paper, we study the long-term asymptotic behaviour of solutions to the stochastic Zakharov lattice equation with multiplicative white noise. We first transfer the stochastic lattice equation into a random lattice equation and prove the existence and uniqueness of solutions which generate a random dynamical system. Then we consider the existence of a tempered random bounded absorbing set and a random attractor for the system. Finally we establish the upper semi-continuity of random attractor to the global attractor of the limiting system as the coefficients of the white noise terms tend to zero.     

Convergence des EDSRs et homogéneisation des inégalités variationnelles semilinéaires dans un convexe

We study the limit of the solution of a Semi-linear Variational Inequality (SVI for short) involving a second order differential operator of parabolic type with periodic coefficients and highly oscillating term. Our basic tool is the approach given by Pardoux [16]. In particular, we use the weak convergence of an associated reflected Backward Stochastic Differential Equation (BSDE for short).     

Long time behavior of stochastic {MHD} equations perturbed by multiplicative noises

In this paper, 2-dimensional (2D) magnetohydrodynamics (MHD) equations perturbed by multiplicative noises in both the velocity and the magnetic field is studied. We first considered the stability, or the upper semi-continuity, for equivalent random dynamical systems (RDS), and then applying the abstract result we established the existence and the upper semi-continuity of tempered random attractors for the stochastic MHD equations. This result shows that the asymptotic behavior of MHD equations is stable under stochastic perturbations.     

伪线性映射及其变分不等式解的特征

研究拓扑向量空间到其共轭空间的伪线性映射和其变分不等式问题,给出伪线性映射的几个等价形式,并对伪线性映射的变分不等式解集的特征进行了刻画.     

Random attractor for stochastic second-order non-autonomous stochastic lattice equations with dispersive term

In this paper, we consider the asymptotic behaviour of solutions to second-order non-autonomous stochastic lattice equations with dispersive term and additive white noises in the space of infinite sequences. We first transfer the stochastic lattice equations into random lattice equations, and prove the existence and uniqueness of solutions that generate a random dynamical system. Second, we prove the existence of a tempered random absorbing set and a random attractor for the system. Finally, we establish the upper semi-continuity of the random attractors as the coefficient of the white noise term tends to zero.     

A note on the article “Fuzzy totally continuous and totally semi-continuous functions”

A. Mukherjee has introduced the concepts of fuzzy totally continuous and fuzzy totally semi-continuous functions in 1999. Apart from the fact that fuzzy total continuity and fuzzy total semi-continuity were introduced in Mukherjee's paper without regard for whether they are topological notions (i.e. give rise to topological categories) or for whether there are deep, canonical examples of such concepts, we show in this note that there are mathematical errors in some of the claimed results of the paper as well.     

Survey on Vector Complementarity Problems

The paper aims at summarizing the main results on Vector Complementarity Problems (VCP), including the existence of a solution and the relations with Vector Variational Inequalities and Vector Optimization Problems. Particular attention will be given to a VCP with a variable domination structure, where the ordering cone depends on the unknown variable.     

Asymptotic behavior of a class of nonlinear evolution equations

In this paper, using a new method (or technology), we prove the existence and upper semi-continuity of the global attractors A_ω for a class of nonlinear evolution equations in D(A)×D(A), where the nonlinear term f satisfies a critical exponential growth condition.     

Hölder Continuity of Solutions to Parametric Weak Generalized Ky Fan Inequality

In this paper, by using a scalarization technique, we obtain sufficient conditions for Hölder continuity of the solution mapping for a parametric weak generalized Ky Fan Inequality in the case where the solution mapping is a general set-valued one. The result is different from the recent ones in the literature.