向量均衡问题的弱有效解算法及其应用研究

Vector equilibrium problem is an important research content in the area of vector optimization. It includes optimization problem, variational inequality, complementary problem, saddle point problem and fixed point problem as special cases. And it is widely applied in many fields, such as Economics and Finance, Engineering and Technology, Traffic and Network..How to find a solution, i.e., algorithm, is an important research direction for vector equilibrium problem. At present, some researchers have investigated it preliminarily. However, the study in this field is just in the beginning stage and the corresponding results are not so many. Its theory is yet far from complete. On the other hand, large quantities of vector optimization problems arise in economics, manufacture and life. People need anxiously to find some feasible and effective algorithms to solve these problems, which can provide crucial theoretical basis for decision. Thus, how to propose appropriate algorithms for efficient solutions of vector equilibrium problem is an important project and needs urgently to be explored for people..The main purpose of this project is to study approximation algorithms for weakly efficient solutions of vector equilibrium problem and investigate its applications. These studies will enrich and develop the theory and algorithm for vector equilibrium problem itself, and provide important theoretical basis for its applications in diverse fields, such as traffic transportation, resource allocation, project management, etc. Moreover, the studies will promote further progress of some related problems, and so they have significant meaning to the development of the discipline of operation research.

向量均衡问题是向量优化领域中的重要研究内容。它包含了最优化、变分不等式、互补问题、鞍点问题以及不动点问题等重要问题为特例，在经济与金融、工程与技术、交通与网络等领域有广泛应用。.求解，即算法，是向量均衡问题的一个重要研究方向。目前，虽有学者对其进行了初步探讨，但由于这方面的研究才刚起步，结果较少，其理论还很不完善；另一方面，由经济、生产和生活中大量实际问题所形成的向量优化问题，又迫切需要人们找到一些可行且有效的方法来求解这些问题，为人们决策提供重要的理论依据。因此，如何建立向量均衡问题有效解的算法是一个亟待解决的重要课题。.本项目将着重研究向量均衡问题弱有效解的逼近算法，并讨论其应用。这些问题的研究不仅可以丰富和发展向量均衡自身的理论和算法，为向量均衡在交通运输、资源分配、工程管理等领域的应用提供重要理论依据，而且可以促进相关问题的进一步解决，对学科发展有重要意义。

向量均衡,以多目标决策和向量优化为应用背景,近年快速发展为运筹学的重要研究方向之一.目前,对其解的存在性研究已经获得比较丰富的成果,但对其算法研究却是刚起步,结果较少,理论还很不完善.另一方面,由经济、生产和生活中大量实际问题所形成的向量平衡问题,又迫切需要人们找到一些可行、有效和方便的计算方法来解决这些问题,为人们决策时提供重要的理论依据.因此,研究向量均衡问题的算法是一个很有意义的课题..本项目利用多非线性分析、凸分析、集值分析以及非光滑分析的相关理论与方法,研究了向量均衡问题的算法及其相关问题,获得了诸多创新结果：①建立了向量均衡问题解的存在性与解集的闭性和紧性新结论；②给出了向量均衡问题的迭代算法,并获得了其敛散性分析结果；③证明了模糊环境中动态变分不等式解的存在性定理,并给出了其解的迭代算法,还给出了算例,验证了算法的有效性；④得到了集优化问题的最优条件、解集的弧连通性以及解集的稳定性结果；⑤分析并证明了空间的若干拓扑性质.在国内外重要学术刊物上发表论文13篇,其中SCI收录论文10篇,CSCD收录论文2篇.培养或协助培养硕士研究生11人,其中已毕业6人,正在培养中5人.课题成员在资助期内还参加了全国性或国际性的重要学术会议16人次,及时交流和宣传了研究成果.本项目的研究完成了任务书中所提出的各项任务,达到了预期目标..本项目的研究结果不仅可以丰富和发展向量均衡、向量优化、向量变分不等式等相关问题的理论、方法和算法,而且可以为向量均衡在交通运输、资源分配、工程管理等领域的应用提供重要的理论依据,对学科的发展也有重要的意义.