复逆变分不等式的若干问题及其应用研究

This proposal studies the following issues in complex inverse variational inequality: (1) Based on residual gap function, regularized gap function and D-gap function, error bounds for complex inverse variational inequality and the stability of these bounds are provided. (2) Well-posedness and Levitin-Polyak well-posedness of complex inverse variational inequality are obtained. (3) Regularization results for complex inverse variational inequality are explored. (4) Several effective iterative algorithms for solving complex inverse variational inequality are developed, and the convergence of these algorithms are discusssed. (5) As an application, the equivalence between evolutionary complex inverse variational inequality and time-dependent generalized Nash equilibrium control problem is established, the significance of time-dependent generalized Nash equilibrium control problem in real-life applications are revealed. The results of this project not only can enrich and develop complex inverse variational inequality theory, methods and techniques, but also can be used to solve many practical problems in communications, economics and finance, and traffic transportantion. It also has significance for subject and socioeconomic development.

本项目对复逆变分不等式中的以下几个问题进行研究：（1）基于残差间隙函数、正则化间隙函数及D-间隙函数研究复逆变分不等式的误差界，并讨论误差界的稳定性；（2）研究复逆变分不等式的Tykhonov适定性及L-P适定性分析；（3）研究复逆变分不等式的正则化结果；（4）研究求解复逆变分不等式的几类有效算法，且分析算法的收敛性；（5）作为应用，建立发展型复逆变分不等式与时间相依赖的广义Nash均衡控制问题的等价性，揭示时间相依赖的广义Nash均衡控制问题在现实问题中的重要意义。本项目的研究不仅可以丰富和发展复逆变分不等式的理论、方法与技巧，而且也可以用于解决产生于通信工程、经济与金融及交通运输等领域中的大量实际问题，对于学科与社会经济发展有深远的意义。

项目执行期间，我们以非线性分析、凸分析、变分不等式与微分方程等相关理论及方法，从以下几个方面展开研究：（1）研究了几类优化问题的严格可行性；（2）研究了一些扰动广义变分不等式的可解性；（3） 研究了几类微分方程行波解的分岔。相关研究工作已有10篇学术论文发表在国内外知名学术期刊上，其中9篇被SCI检索收录。本项目申请人还获得了中国博士后科学基金资助。本项目的研究成果不仅丰富和发展了变分不等式与微分方程理论、方法与技巧，而且可以用于研究工程技术、经济金融与优化控制等大量实际问题，具有重要的学术价值，对学科和经济发展都有意义。