具有时滞效应的微分向量优化问题的理论、算法及应用研究

The theory, algorithm and applications of differential variational inequalities have received great attention from the scholars at home and abroad in recent years. In order to investigate the dynamic systems with vector constraint, we have introduced and studied differential vector variational inequalities. Since vector optimization problem is closely related to vector variational inequality, differential vector optimization problem is a new problem worth studying. In the research of dynamic system from the science and engineering, considering the delay effect has important practical significance. Therefore, we will make a thorough and systematic study of delay differential vector optimization problems in this project. Based on the conclusions of differential vector variational inequalities, by using some basic tools from nonlinear and convexity analysis, combining the theory, methods and skills of delay differential equation and vector optimization, we will obtain some results about the existence of the solution, and the stability analysis. Then we will consider the approximation algorithm and study the convergence analysis. In addition, we will apply the conclusions to solve some new differential game problems. The results of this project not only enrich and develop the theory of differential variational inequality and vector optimization problem, but also provide useful tools to study some problems in reality.

近年来，微分变分不等式的理论、算法及应用研究引起了国内外许多学者的关注。为了研究具有向量约束的动态系统，我们引入并研究了微分向量变分不等式问题。由于向量优化问题与向量变分不等式紧密相关，微分向量优化问题是值得研究的新问题。在研究大量产生于科学和工程的动态系统时，考虑时滞对系统的影响具有重要的现实意义。本项目以微分向量变分不等式的研究成果为基础，借用非线性分析和凸分析的一些研究工具，结合时滞微分方程和向量优化的理论、方法和技巧，对具有时滞效应的微分向量优化问题进行系统和深入的研究，获得该类问题解的存在性和稳定性等刻画结果，构造逼近解的算法、分析算法的收敛性，并将这些研究结果用于解决微分博弈中的一些新问题。本项目的研究不仅可以丰富和发展微分变分不等式和向量优化问题的相关理论和算法，而且可以为一些实际问题的研究提供有力的工具。

近年来，微分变分不等式的理论、算法及应用研究引起了国内外许多学者的关注。为了研究具有向量约束的动态系统，我们引入并研究了微分向量变分不等式问题。由于向量优化问题与向量变分不等式紧密相关，微分向量优化问题是值得研究的问题。在大量产生于科学和工程的动态系统时，考虑时滞对系统的影响具有重要的现实意义。本项目以微分向量变分不等式的研究成果为基础，借用非线性分析和凸分析的一些研究工具，结合时滞微分方程和向量优化的理论、方法和技巧，对具有时滞效应的微分向量优化问题进行深入系统的研究，获得该类问题解的存在性和稳定性等刻画结果，构造逼近解的算法、分析算法的收敛性。本项目的研究不仅可以丰富和发展微分变分不等式和向量优化问题的相关理论和算法，而且为一些实际问题的研究提供有力的工具。