均衡约束变分不等式问题的理论、算法及应用研究

In this project, by absorbing the existing theories and methods of one-level variational inequality and using the basic theories and techniques of some disciplines, such as functional analysis, convex analysis, nonsmooth analysis and computational mathematics, we investigate some theoretical nature and algorithms for variational inequalities with equilibrium constraints (abbreviated by VIEC). We explore some existence results for VIEC and characterize some properties of the set of solutions. We propose some of projection-type methods and proximal-type algorithms for solving VIEC. We introduce several gap functions for VIEC. Thus VIEC can be formulated into optimizatiom problems and solved it by using some effctive methods for optimization problems. As an application, some new problems arising in the logistics field after considering more constraints conditions are characterized as VIEC. The study of these problems can not only enrich and develop the theory and algorithms of variational inequalities, but also be applied to a large number of practical problems arising in many fields, such as transportation and logistics, economics and finance, communications engineering and technology. Therefore, it is of great importance to discipline and social economic development.

本项目以均衡约束变分不等式(双层问题)为研究对象。通过消化吸收单层变分不等式的理论和方法，综合应用泛函分析、凸分析、非光滑分析和计算数学等领域的知识展开研究。探索均衡约束变分不等式的可解性条件，刻画其解集性质；构造求解均衡约束变分不等式问题的投影类方法和邻近点类算法，研究均衡约束变分不等式的若干间隙函数，探索把该类问题转化为优化问题求解的途径；作为应用，把物流管理领域考虑更多约束因素后出现的一些新问题刻画为均衡约束变分不等式。上述问题的研究不仅可以丰富和发展变分不等式的理论、方法、技巧和算法，而且可以用于解决产生于交通与物流、经济与金融、通信工程与技术中的大量实际问题，对学科和社会经济发展都有重要意义。

均衡约束变分不等式源于自组网络的能量控制问题。本项目研究均衡约束、有界集约束、无界集约束的变分不等式解的存在性理论、投影型算法、扰动性分析以及以及在经济领域中的价格均衡问题的应用。项目研究取得了较好的成果。（1）解的存在性理论方面：在Hadamard流形上引入并研究H-半变分不等式，当约束集分别为有界和无界时，得到了解的存在性定理，并给出解存在的等价刻画；在欧氏空间中研究H-半变分不等式解的存在性定理并建立了Tikhonov正则化理论。（2）投影型算法方面：分别在欧氏空间、Hilbert空间、Banach空间中提出或发展了求解变分不等式、混合变分不等式等模型的新的投影型方法，分析收敛性，作出收敛率估计，给出数值算例。（3）解的扰动性理论方面：在欧氏空间、Banach空间中建立变分不等式、模糊变分不等式、广义似变分不等式等模型的扰动性理论，并研究相关理论在经济领域中均衡问题的潜在应用。该项目发展了变分不等式的理论和方法，对解决经济领域中的均衡问题具有较重要的意义和价值。