动态逆混合变分不等式理论及算法研究

In this project, we investigate the theory with algorithms of dynamic inverse mixed variational inequalities, by the theory and techniques of variational inequalities, nonlinear analysis, equilibrium problem, optimization and convex analysis. We mainly study the following problems:. (1) Based on the theory of economic equilibrium and traffic network equilibrium, we consider the time-dependent equilibrium control problems, and characterize the optimal conditions as dynamic inverse mixed variational inequalities by the optimization theory and nonlinear analysis techniques;. (2) Based on the theory of variational inequalities and nonlinear analysis, we investigate the existence of solutions for dynamic inverse mixed variational inequalities;. (3) Using the properties of generalized f-projection operators, we construct algorithms for dynamic inverse mixed variational inequalities, and prove the convergence of iterative sequences generated by the algorithms.. The research of this project can develop and enrich the theory of inverse variational inequalities, and also can provide a new research way for traffic network equilibrium control problems, economic equilibrium control problems and normative flow control problems appeared in telecommunication networks.

本项目以凸分析、变分不等式和非线性分析的理论、方法和技巧为基础，结合均衡问题和最优化的相关理论，研究动态逆混合变分不等式的理论及算法。本项目主要研究下述问题：（1）借鉴经济均衡和交通网络均衡的理论，研究与时间相依赖的均衡控制问题，利用最优化理论和非线性分析方法将该问题的最优性条件转化为动态逆混合变分不等式；（2）借用变分不等式和非线性分析的理论、方法和技巧，研究动态逆混合变分不等式解的存在性；（3）利用广义f投影算子的性质，构造动态逆混合变分不等式解的迭代算法，分析算法的收敛性。本项目的研究不仅可以丰富和发展逆变分不等式的理论、方法和技巧，而且也可以为产生于交通网络均衡控制、经济均衡控制和远程通讯网络流量控制的大量问题提供一条新的研究途径。

近几十年，变分不等式理论及其推广形式在经济金融、交通运输、最优化、算子理论和工程科学有着广泛的应用。本项目以凸分析、变分不等式和非线性分析的理论、方法和技巧为基础，结合均衡问题和最优化的相关理论，深入研究了广义f投影算子的稳定性质、动态逆混合变分不等式的理论及算法，获得了一些有趣的结果。. 目前，本项目的部分研究成果已有2篇发表在国内外知名学术期刊上。