向量拟平衡问题的本质解及解集的本质连通区

本文研究向量拟平衡问题,得到了向量拟平衡问题解的一个存在性结果,证明了在满足一定的连续性和凸性条件的问题构成的空间Y中,大多数(在Baire分类意义下)问题的解集是稳定的,并证明Y的某子集中,每个向量拟平衡问题的解集中至少存在一个本质连通区。作为应用,我们导出了多目标广义对策弱Pareto-Nash平衡点的存在性,证明了在满足一定的连续性和凸性条件的多目标广义对策构成的空间P中,大多数对策的弱Pareto-Nash平衡点是稳定的,并证明了P中的每个对策的弱Pareto-Nash平衡点集中至少有一个本质连通区。     

锥弱有效解的本质集和本质连通区

对于多目标规划问题,通过引入锥弱有效解集的概念,证明了任何连续有界泛函至少存在一个极小本质集和一个本质连通区.     

广义向量似变分不等式解集的通有稳定性及本质连通区的存在性

本文研究广义向量似变分不等式解集的稳定性．证明了在满足一定的连续性和凸 性条件的广义向量似变分不等式问题构成的空间Ｍ中，大多数（在Ｂａｉｒｅ分类意下）广 义向量似变分不等式问题的解集是稳定的，并证明了Ｍ中的每个广义向量似变分不等式 的解集至少存在一个本质连通区。     

集值映射向量优化问题的锥弱有效解的镇定性和稳定性

本文研究了集值映射向量优化问题的锥弱有效解的镇定性和稳定性,我们引进了集值映射向量优化问题的镇定性和稳定性的定义,并证明了集值映射问题优化问题的镇定性和稳定性的一些主要定理。     

一类集值映射向量优化问题的ε-有效解

集值映射向量优化问题是最优化理论中的一个重要方向.在集值映射为生成锥内部-锥一类凸(简记为ic-锥类凸)的假设条件下,利用择一定理,给出了集值映射向量优化问题ε-弱有效解和ε-有效解的最优性条件和ε-Lagrange乘子定理,是弱有效解和有效解相应结果的推广.     

关于无限对策Nash平衡点集的本质连通区

证明了任意纯策略集是紧度量空间和支付函数连续的n人无限非合作对策存在 Nash平衡点集的本质连通区．     

集值映射向量优化问题的锥弱有效解的镇定性和稳定性(英文)

本文研究了集值映射向量优化问题的锥弱有效解的镇定性和稳定性,我们引进了集值映射向量优化问题的镇定性和稳定性的定义,并证明了集值映射向量优化问题的镇定性和稳定性的一些主要定理.     

向量优化问题的几个收敛性结果

在文本中，获得了集合的弱有效元与真有效元的几个收敛性结果。然后，讨论了集值映射向量优化问题（VP）和它的近似问题（VP）n，在较强的假设条件下，获得了（VP）n的真有效解的几个收敛性结果。     

Hausdorff空间中向量优化问题弱有效解的存在性定理

Hausdorff空间中关于向量优化问题弱有效解的一个存在性定理被给出,另外,举出一个反例说明Flores-Bazan和Vera所作的一个猜想是不成立的.     

本质连通区的存在性和稳定性

本文首先给出了一个统一的本质连通区的存在性定理．应用这个定理，容易地导出了不动点集和Nash平衡点集本质连通区的存在性定理．此外，还首先给出了一个统一的本质连通区的稳定性定理．     

The Essential Component of the Solution Set for Vector Equilibrium Problems

In the paper, by using Ky Fan’s section theorem, we obtain an existence theorem for vector equilibrium problems. Motivated by the ideas of Kinoshita and McLennan, we introduce the concept of the essential component of the solution set for vector equilibrium problems, and we prove that there exists at least one essential component of the solution set for every vector equilibrium problem satisfying some conditions. Research was partially supported by the Natural Science Foundation of Guangdong Province, P. R. China.     

On Essential Stable Sets of Solutions in Set Optimization Problems

This paper introduces the essential stability for set optimization problems. Some kinds of essential stable sets of weakly minimal and minimal solutions are shown. The graph of minimal solution mappings is not necessarily closed, which is different from weakly minimal solution mappings. The existence of minimum essential sets of minimal solutions is proved.     

On Essential Components of the Solution Set of Generalized Games

In this paper, we prove that for any generalized game, there exists at least one essential connected component of the set of it's equilibrium points.     

Equilibrium Problems for Infinite Dimensional Vector Potentials with External Fields

We consider a minimal energy problem with an external field over noncompact classes of infinite dimensional vector measures $(\mu^i)_{i\in I}$ on a locally compact space. The components μ ⁱ are positive measures normalized by $\int g_i\,d\mu^i=a_i$ (where a _i and g _i are given) and supported by closed sets A _i with the sign +?1 or ??1 prescribed such that A _i ?∩?A _j ?=?? whenever ${\rm sign}\,A_i\ne{\rm sign}\,A_j$ , and the law of interaction between μ ⁱ , i?∈?I, is determined by the matrix $\bigl({\rm sign}\,A_i\,{\rm sign}\,A_j\bigr)_{i,j\in I}$ . For positive definite kernels satisfying Fuglede’s condition of consistency, sufficient conditions for the existence of equilibrium measures are established and properties of their uniqueness, vague compactness, and strong and vague continuity are studied. Examples illustrating the sharpness of the sufficient conditions are provided. We also obtain variational inequalities for the weighted equilibrium potentials, single out their characteristic properties, and analyze continuity of the equilibrium constants. The results hold, e.g., for classical kernels in $\mathbb R^n$ , $n\geqslant 2$ , which is important in applications.     

Essential Component and Essential Optimum Solution of Optimization Problems

In this note, the concept of essential component of optimum solutions is introduced. We obtain necessary and sufficient conditions for the existence of one essential component or one essential optimum solution in the optimum solution set of the optimization problems.     

A Finite Algorithm for Solving Infinite Dimensional Optimization Problems

We consider the general optimization problem (P) of selecting a continuous function x over a -compact Hausdorff space T to a metric space A, from a feasible region X of such functions, so as to minimize a functional c on X. We require that X consist of a closed equicontinuous family of functions lying in the product (over T) of compact subsets Y _t of A. (An important special case is the optimal control problem of finding a continuous time control function x that minimizes its associated discounted cost c(x) over the infinite horizon.) Relative to the uniform-on-compacta topology on the function space C(T,A) of continuous functions from T to A, the feasible region X is compact. Thus optimal solutions x ^* to (P) exist under the assumption that c is continuous. We wish to approximate such an x ^* by optimal solutions to a net {P _i }, iI, of approximating problems of the form min_{xX i} c i(x) for each iI, where (1) the net of sets {X _i } _I converges to X in the sense of Kuratowski and (2) the net {c _i } _I of functions converges to c uniformly on X. We show that for large i, any optimal solution x ^* _i to the approximating problem (P _i ) arbitrarily well approximates some optimal solution x ^* to (P). It follows that if (P) is well-posed, i.e., limsupX _i ^* is a singleton {x ^*}, then any net {x _i ^*} _I of (P _i )-optimal solutions converges in C(T,A) to x ^*. For this case, we construct a finite algorithm with the following property: given any prespecified error and any compact subset Q of T, our algorithm computes an i in I and an associated x _i ^* in X _i ^* which is within of x ^* on Q. We illustrate the theory and algorithm with a problem in continuous time production control over an infinite horizon.     

集值映射向量优化问题ε-超有效解集的连通性

本文在赋范向量空间中讨论集值映射向量优化问题的ε-超有效性，在锥-半连续和广义锥-次类凸的假设条件下，获得了ε-超有效点(解)集的连通性结果．     

Continuity of the Solution Set to Parametric Weak Vector Equilibrium Problems

In this paper, we obtain some stability results for parametric weak vector equilibrium problems in topological vector spaces. We provide sufficient conditions for the continuity of the solution set mapping in parametric weak monotone vector equilibrium problems. This research was partially supported by the National Natural Science Foundation of China (Grant 10561007) and the Natural Science Foundation of Jiangxi Province, China.     

Vector Equilibrium Problems. Existence Theorems and Convexity of Solution Set

The natural quasi-concavity of set-valued mappings in an ordered vector space is introduced. Existence theorems for vector equilibrium problems involving set-valued monotone mappings are obtained and the convexity of the solution set is shown.     

Infinite Dimensional Parametric Optimal Control Problems

In this paper we study parametric optimal control problems monitored by nonlinear evolution equations. The parameter appears in all the data, including the nonlinear operator. First we show that for every value of the parameter, the optimal control problem has a solution. Then we study how these solutions as well as the value of the problem respond to changes in the parameter. Finally, we work out in detail two examples of nonlinear parabolic optimal control systems.