## 1 Introduction  
In recent years, the generalized Nash equilibrium (GNE) [1, 2], has been increasingly applied to solve practical problems. Examples include electricity market scheduling [3], peer-to-peer electricity market analysis [4] and energy trading in combined heat and power market [5]. In a noncooperative game, there are both shared and unshared constraints. These constraints cause players' private costs and feasible decision sets to interact with each other. In such a situation, it is necessary to establish rules for the players. The players make decisions by exchanging information according to the rules. A Nash equilibrium can be found as long as the rules are appropriate. Algorithms for solving Nash equilibrium problems can be classified as centralized optimization algorithms and distributed optimization algorithms based on the way they solve optimization problems. In a centralized optimization algorithm, the entire problem is solved by a single entity with access to all the necessary information. In contrast, distributed optimization algorithms solve the problem by decomposing it into smaller sub-problems that are cooperatively solved by multiple players. This paper mainly studies distributed optimization algorithms. The information setting of distributed optimization algorithms is also divided into two types. The first type requires each player to publicly disclose their decision information. Another type only requires players to disclose information to their neighbors. In this paper, these rules are referred to as \"global information setting\" and \"partial information setting\", respectively. Since the 1950s, research on GNEP has gradually become a hot topic [6, 7, 8, 9]. For the solving of Nash equilibrium problems, many distributed algorithms have been studied in recent years, such as an algorithm based on the forward-backward operator partitioning method [10], a triangular preconditioned proto-dual algorithm [11] and an algorithm fully adapted to monotone variational inequalities [12]. In [13], [PERSON] proposes a distributed primal-dual algorithm for the computation of generalized Nash equilibria (GNE) in noncooperative games in networked systems. The algorithm handles games with shared affine coupling constraints via operator splitting methods. In contrast, the Algorithm 1 proposed in this paper has a higher convergence speed. [14] reformulates the GNE problem as a variational inequality problem and solves it through the doubly augmented operator splitting method (DAOS). This method can handle nonsmooth and nonconvex objective functions and constraints. In contrast, the Algorithm 1 proposed in this paper solves the optimization problem by using unconstrained optimization and linear mapping, which has higher computational efficiency. Moreover, some studies have combined the ideas of machine learning to solve optimization problems. By combining extremum-seeking control (ESC) with learning algorithms, [PERSON], [PERSON] and [PERSON] proposed a GNE-seeking algorithm in [15]. This algorithm can be executed in a partial information setting and does not require knowledge of the explicit expression of the cost function. In most of the current research, only algorithms under the local information setting are studied. This paper discusses two forms of algorithms under two information settings. It also shows intuitively through experiments that the algorithm under the local information setting has higher efficiency. ```
Initialize: For each player \(i\in\mathcal{N}\), let \(\mathbf{x}_{i,0}^{[i



* Two new variants of the Douglas-Rachford algorithm are proposed in this paper, denoted as Algorithms 1 and 2, which can solve Nash equilibrium problems. * Two pairs of new splitting operators are introduced for Algorithms 1 and 2, respectively, which can ensure the convergence and efficiency of the algorithms. * Algorithm 2 is proved in this paper, and it has a linear convergence rate under the global information setting, while Algorithm 1 has a faster convergence rate under the local information setting.