Hadamard流形上纳什均衡问题的算法研究

纳什均衡理论是非合作博弈的基础，是20世纪最重大的科学进展之一，对经济学等社会科学影响巨大。流形是一种非线性的空间形式，是欧氏空间的推广。通过引入适当的黎曼度量，可以把欧氏空间中的一些非凸集和非凸函数转化为黎曼流形上的凸集和凸函数。. 本项目以Hadamard流形上纳什均衡问题为研究对象。主要研究四个问题：(1)引入Nikaido-Isoda函数，把纳什均衡问题转化为一般均衡问题，设计求解该均衡问题的算法；(2)研究具有优良性质的间隙函数，设计求解间隙函数最小点的下降算法；(3)研究求解Hadamard流形上纳什均衡点的投影型方法，证明全局收敛性，分析收敛率，给出应用的例子；(4)研究求解Hadamard流形上纳什均衡点的邻近点算法，证明全局收敛性，分析有限收敛性质，给出应用的例子。上述问题的研究可丰富和发展纳什均衡理论，具有重要意义。

The Nash equilibrium theory, which is regarded as one of the most significant scientific progress in the 20th Century, is the foundation of non-cooperative game theory. It is of the great influence in the social sciences such as economics. A manifold, which is a generalization of Euclidean space, is one of the nonlinear spaces. By introducing appropriate Riemannian metric, some nonconvex subsets and functions in Euclidean spaces can be seen as convex concepts in the sense of Riemannian metric.. In this project, Nash equilibrium problems on Hadamard manifolds will be studied in the following four aspects: (1) The concept of Nikaido-Isoda function for Nash equilibrium problem on Hadamard manifolds is introduced. Nash equilibrium problem is transformed into equilibrium problem on Hadamard manifolds by the Nikaido-Isoda function. And then some algorithms for solving the equilibrium problem are investigated; (2) Some nice gap functions for Nash equilibrium problems on Hadamard manifolds are investigated carefully and used for constructing some descent algorithms; (3) Some projection-type methods for solving Nash equilibria on Hadamard manifolds are introduced, where global convergence and the rate of convergence are analyzed, and some application examples are provided; (4) Some algorithms based on the proximal methods for solving Nash equilibria on Hadamard manifolds are introduced, where global convergence and finite convergence are analyzed, and some application examples are provided. The study of these problems can enrich and develop the Nash equilibrium theory. Therefore, it is of the great influence.