向量均衡问题的超有效性的对偶形式

在局部凸空间中引进了向量均衡问题的强超有效解、C-强超有效解、弱超有效解, C-弱超有效解、齐次超有效解、 C-齐次超有效解的概念,并在局部凸空间中用极理论为工具讨论了向量均衡问题的 C-弱超有效解, C-超有效解, C-齐次超有效解,以及C-强超有效解的对偶形式. 又在赋范线性空间中讨论了向量均衡问题的以上各种超有效解之间的等价性,并且在赋范线性空间具正规锥的条件下讨论了向量均衡问题的以上各种超有效解的对偶形式. 作为它的应用,给出了向量优化问题各种超有效解的对偶形式.     

具强制条件的向量均衡问题

研究锥伪单调、锥拟凸和上锥连续映射在某种强制性条件下的向量均衡问题解集的特征,建立强制性条件与向量均衡问题解集的关系,得到对偶向量均衡问题局部解集含于向量均衡问题解集的性质和向量均衡问题解集的非空性条件,给出在锥伪单调、锥拟凸和上锥连续映射条件下向量均衡问题解集的非空有界性与强制性条件的等价性．     

Three dimensional elastodynamics of 2D quasicrystals: The derivation of the time-dependent fundamental solution

The time-dependent differential equations of elasticity for 2D quasicrystals with general structure of anisotropy (dodecagonal, octagonal, decagonal, pentagonal, hexagonal, triclinic) are considered in the paper. These equations are written in the form of a vector partial differential equation of the second order with symmetric matrix coefficients. The fundamental solution (matrix) is defined for this vector partial differential equation. A new method of the numerical computation of values of the fundamental solution is suggested. This method consists of the following: the Fourier transform with respect to space variables is applied to vector equation for the fundamental solution. The obtained vector ordinary differential equation has matrix coefficients depending on Fourier parameters. Using the matrix computations a solution of the vector ordinary differential equation is numerically computed. Finally, applying the inverse Fourier transform numerically we find the values of the fundamental solution. Computational examples confirm the robustness of the suggested method for 2D quasicrystals with arbitrary type of anisotropy.     

Stability and Sensitivity Analysis of Solutions to Weak Vector Variational Inequalities

The paper mainly concerns the study of a parametric weak vector variational inequality. Robinson’s metric regularity and Lipschitzian stability of the solution mapping for the parametric weak vector variational inequality are firstly established. Then sensitivity analysis of the solution mapping for the parametric weak vector variational inequality is discussed. As applications, Lipschitzian continuity and differentiability of the solution mapping are also investigated for a parametric variational inequality.     

Solution of a regularized problem for a stationary magnetic field in a nonhomogeneous conducting medium by a finite element method

The use of a vector finite element method for solving a regularized stationary magnetic field problem, which is formulated in terms of a vector magnetic potential, is validated. A generalized solutionis approximated using first-order Nedelec vector elements of the second kind on tetrahedrons. The existence and uniqueness of the solution to a discrete regularized problem and its convergence to the generalized solution for the case of an inhomogeneous (in the electromagnetic properties) domain are justified. Some issues of the numerical solution to the discrete regularized problem are discussed. Approaches to optimize the algorithms are shown on a series of numerical experiments.     

Saddle points for vector valued functions: existence, necessary and sufficient theorems

Necessary and sufficient conditions for a point to be a weak saddle point of a vector valued function (i.e. to be a solution of the vector saddle point problem) are given. Also, an existence result for a vector saddle point to have a solution is given.     

Vector Variational Inequality and Vector Pseudolinear Optimization

The study of a vector variational inequality has been advanced because it has many applications in vector optimization problems and vector equilibrium flows. In this paper, we discuss relations between a solution of a vector variational inequality and a Pareto solution or a properly efficient solution of a vector optimization problem. We show that a vector variational inequality is a necessary and sufficient optimality condition for an efficient solution of the vector pseudolinear optimization problem.     

The Pascoletti–Serafini Scalarization Scheme and Linear Vector Optimization

The Pascoletti–Serafini scalarization scheme for general vector optimization problems is studied. It is specified to linear vector optimization to give minimal representation formulae for the weakly efficient solution set and the efficient solution set. Several facts on connectedness of the solution sets of Pascoletti–Serafini’s scalar auxiliary problems, both for linear vector optimization and for nonlinear vector optimization, are established.     

锥凸对称向量拟均衡问题解集的通有稳定性

在拓扑向量空间中,利用Ky Fan截口定理得到一个锥凸向量拟均衡问题弱Pareto解的存在性结果.作为该结果的应用,得到了一个对称向量拟均衡问题在支付映射为锥凸条件下弱Pareto解的存在性定理.该定理在较弱的条件下回答了Fu在文献[1]中提出的第二个问题,即在支付映射为锥凸且连续的条件下对称向量拟均衡问题的弱Pareto解是否存在.最后在赋范线性空间中研究了锥凸对称向量拟均衡问题弱Pareto解集的通有稳定性.     

弱向量变分不等式解集的连通性

研究拓扑向量空间到连续线性映射空间映射的弱向量变分不等式和与之相关 的纯量型变分不等式解集的关系, 引入弱和强一致连续概念,利用纯量型变分不等式 解集所表征的集值映射的特性给出弱向量变分不等式解集连通的一个充分条件。     

The least square prenucleolus and the least square nucleolus. Two values for TU games based on the excess vector

The nucleolus and the prenucleolus are solution concepts for TU games based on the excess vector that can be associated to any payoff vector. Here we explore some solution concepts resulting from a payoff vector selection based also on the excess vector but by means of an assessment of their relative fairness different from that given by the lexicographical order. We take the departure consisting of choosing the payoff vector which minimizes the variance of the resulting excesses of the coalitions. This procedure yields two interesting solution concepts, both a prenucleolus-like and a nucleolus-like notion, depending on which set is chosen to set up the minimizing problem: the set of efficient payoff vectors or the set of inputations. These solution concepts, which, paralleling the prenucleolus and the nucleolus, we call least square prenucleolus and least square nucleolus, are easy to calculate and exhibit nice properties. Different axiomatic characterizations of the former are established, some of them by means of consistency for a reasonable reduced game concept.     

Strong Vector Equilibrium Problems

In this paper, the existence of the solution for strong vector equilibrium problems is studied by using the separation theorem for convex sets. The arc-wise connectedness and the closedness of the strong solution set for vector equilibrium problems are discussed; and a necessary and sufficient condition for the strong solution is obtained.     

Generic Uniqueness of Solutions for a Class of Vector Ky Fan Inequalities

This paper is intended mainly to present some generic uniqueness results for a class of vector Ky Fan inequalities. By employing the methods of set-valued analysis, we prove that, in the sense of Baire category, most of the problems in a complete metric space, consisting of vector Ky Fan inequalities satisfying some conditions, have unique solution and that every vector Ky Fan inequality, possessing more than one solution, can be approached arbitrarily by a sequence of vector Ky Fan inequalities each of which has a unique solution. Our discussions are under two different settings. One setting is related to vector Ky Fan inequalities defined on a compact set; the other is related to vector Ky Fan inequalities defined on a noncompact set. The corollaries of our results generalized the corresponding results in the literature.     

具有控制结构的向量均衡问题与解集的性质

提出一种具有控制结构的向量均衡问题与向量映射的新的伪单调性概念,得到具有控制结构的向量均衡问题解的存在性及其解集的紧凸性．作为应用,得到具有控制结构的向量变分不等式与互补问题的解．     

On Efficient Solutions in Vector Optimization

A new characterization is obtained for the existence of an efficient solution of a vector optimization problem in terms of associated scalar optimization problems. The consequences for linear vector optimization problems are derived as a special case, Applications to convex vector optimization problems are also discussed.     

Hölder continuity and upper estimates of solutions to vector quasiequilibrium problems

In this paper, we establish the Hölder continuity of solution mappings to parametric vector quasiequilibrium problems in metric spaces under the case that solution mappings are set-valued. Our main assumptions are weaker than those in the literature, and the results extend and improve the recent ones. Furthermore, as an application of Hölder continuity, we derive upper bounds for the distance between an approximate solution and a solution set of a vector quasiequilibrium problem with fixed parameters.     

A Solution Concept for Fuzzy Multiobjective Programming Problems Based on Convex Cones

A solution concept for fuzzy multiobjective programming problems based on ordering cones (convex cones) is proposed in this paper. The notions of ordering cones and partial orderings on a vector space are essentially equivalent. Therefore, the optimality notions in a real vector space can be elicited naturally by invoking a concept similar to that of the Pareto-optimal solution in vector optimization problems. We introduce a corresponding multiobjective programming problem and a weighting problem of the original fuzzy multiobjective programming problem using linear functionals so that the optimal solution of its corresponding weighting problem is also the Pareto-optimal solution of the original fuzzy multiobjective programming problem.     

New optimality conditions for unconstrained vector equilibrium problem in terms of contingent derivatives in Banach spaces

This article presents necessary and sufficient optimality conditions for weakly efficient solution, Henig efficient solution, globally efficient solution and superefficient solution of vector equilibrium problem without constraints in terms of contingent derivatives in Banach spaces with stable functions. Using the steadiness and stability on a neighborhood of optimal point, necessary optimality conditions for efficient solutions are derived. Under suitable assumptions on generalized convexity, sufficient optimality conditions are established. Without assumptions on generalized convexity, a necessary and sufficient optimality condition for efficient solutions of unconstrained vector equilibrium problem is also given. Many examples to illustrate for the obtained results in the paper are derived as well.     

Vector optimization problems with nonconvex preferences

In this paper, some vector optimization problems are considered where pseudo-ordering relations are determined by nonconvex cones in Banach spaces. We give some characterizations of solution sets for vector complementarity problems and vector variational inequalities. When the nonconvex cone is the union of some convex cones, it is shown that the solution set of these problems is either an intersection or an union of the solution sets of all subproblems corresponding to each of these convex cones depending on whether these problems are defined by the nonconvex cone itself or its complement. Moreover, some relations of vector complementarity problems, vector variational inequalities, and minimal element problems are also given. While this paper was being revised in September 2006, Professor Alex Rubinov (the second author of the paper) left us due to the illness. This is a very sad news to us. We dedicate this paper to the memory of Professor Rubinov as a mathematician and truly friend.     

Existence and stability of solutions for a class of generalized vector equilibrium problems

In this paper, two existence theorems concerning the strong efficient solutions and the weakly efficient solutions of generalized vector equilibrium problems are derived by using the Fan-KKM Theorem and an existence theorem for the efficient solutions of generalized vector equilibrium problems is established by using the scalarization method. Moreover, the lower semicontinuity of the strong efficient solution mapping and the weakly efficient solution mapping to parametric generalized vector equilibrium problems are showed under suitable conditions with neither monotonicity nor any information of the solution mappings. Finally, some applications to the vector optimization problems and the Stackelberg equilibrium problem are also given.