本项目拟利用矩阵半张量积方法，以势博弈为突破点，对不同网络环境下系统的分布式优化问题进行代数建模，分析和设计博弈学习算法，从而给出一种系统有效的、基于数学公式解的研究方法。首先，对于时不变网络，通过设计博弈参与者的效用函数，使优化问题的全局目标函数表示为势博弈的势函数，从而将优化问题的寻优求解转化为势博弈的均衡求解，给出可建模的代数验证条件。其次，对于时变网络，引入状态变量刻画动态变化的网络结构，建立基于状态的势博弈与优化问题之间的映射关系，给出可建模的代数验证条件。另外，对于不能建模为势博弈的系统，通过寻找最广泛意义下满足等式约束的一类势博弈，建立一般网络系统的博弈模型。最后，基于得到的博弈模型，设计基于代数状态空间表示的学习算法，得到博弈演化动态的代数形式，分析博弈均衡的收敛性和全局最优性。以上研究是对已有结果的扩展，对博弈优化理论的发展具有重要意义。

Using the semi-tensor product method and potential game as the breakthrough point, this project aims to study the game-based models and learning algorithms of distributed optimization problems in different networked environments, an effective method based on mathematical formulas is thus correspondingly provided. First, for time-invariant networks, the global objective function of an optimization problem can be described as a potential function of a potential game by designing the utility function for each player, and then an algebraic verification condition is given to assure that the optimization problem is converted into solving equilibrium of the potential game. Second, for time-varying networks, by introducing state variables to describe dynamical networked structures, an algebraic verification condition is obtained to make sure that the system can be formulated as a state-based potential game. Moreover, for a system that can not be modeled as a potential game, its game-based model is established by searching for a kind of the most general potential game satisfying the equality constraint. Finally, based on the obtained models, some game-based learning algorithms under algebraic state space representation are designed, then the evolutionary dynamics is converted into an algebraic form, and the convergence and global optimality of equilibrium are discussed. The above research is an extension of the existing results, which is of great significance to the development of game optimization theory.