Alternating Inertial and Overrelaxed Algorithms for Distributed Generalized Nash Equilibrium Seeking in Multi-Player Games

This paper investigates the distributed computation issue of generalized Nash equilibrium (GNE) in a multi-player game with shared coupling constraints. Two kinds of relatively fast distributed algorithms are constructed with alternating inertia and overrelaxation in the partialdecision information setting. We prove their convergence to GNE with fixed step-sizes by resorting to the operator splitting technique under the assumptions of Lipschitz continuity of the extended pseudo-gradient mappings. Finally, one numerical simulation is given to illustrate the efficiency and performance of the algorithm.


Introduction
Game theory is the study of mathematical models for describing competition and cooperation interaction among intelligent rational decision-makers [1]. In the past few years, networked games have received increasing attention due to their wide applications in different areas such as competitive economy [2], power allocation in interference channel models [3,4], environmental pollution control [5], cloud computing [6], wireless communication [7][8][9], and adversarial classification [10,11].
The Nash equilibrium (NE) is a set of strategies where each player's choice is its best response to the choices of the other players of the game [12]. An NE in games with shared coupling constraints is referred to as generalized Nash equilibrium (GNE) [13]. In order to compute the GNE, a great number of algorithms have been proposed [14][15][16][17][18], most of which depend on full-decision information, i.e., each player is assumed to have full access to all of the other players' actions. However, such an assumption could be impractical in large-scale distributed networks [19,20]. To overcome this shortcoming, fully distributed algorithms under the partial-decision information setting have recently become a research topic that attracts recurring interest.
Under the partial-decision information setting, each player can communicate only with its neighbors (instead of all its opponents) via a certain communication graph. In this case, the player has no direct access to some necessary decision information involving its cost function. In order to make up for the missing information, the player estimates other players' actions and exchanges its estimates with neighbors. Such an estimate would tend to be the real actions of players by designing an appropriate consensus protocol [21]. So far, some efforts have been devoted to the GNE seeking problem with partial-decision information. For example, an adaption of a fictitious play algorithm for large-scale games is introduced in [22], and information exchange techniques for aggregative games are studied in [23]. An operator-theoretic approach has been introduced to analyze GNE problems [16,21], under which the problem is cast as finding a zero of a sum of monotone √ x x denotes the norm induced by the inner product ·, · . Φ 0 stands for a symmetric positive definite matrix. Similarly, the Φ-induced product is x, y Φ = Φx, y , and the Φ-induced norm is x Φ = Φx, x . ⊗ is the Kronecker product, and diag(A 1 , · · · , A n ) denotes the block diagonal matrix with A 1 , · · · , A N on its diagonal. Suppose A ∈ R m×n , then denotes the element of A i in the j-th row and k-th column.

Operator Theory
The following concepts are reviewed from [30]. Let A : R m → 2 R m be a set-valued operator. Denote Id as the identity operator, i.e., Define the resolvent of operator A as R A = (Id + A) −1 . An operator A is called monotone if ∀(x, u), ∀(y, v) ∈ graA, we have x − y, u − v ≥ 0. Moreover, it is maximally monotone if graA is not strictly contained in the graph of any other monotone operator, i.e., for every We can easily derive that if A is averaged then it is nonexpansive, and A is firmly nonexpansive if and only if it The normal cone operator of the set Ω is defined as Let the projection of x onto Ω be P Ω (x) = arg min y∈Ω x − y , and P Ω (x) = R N Ω (x) = (Id + N Ω ) −1 (x).

Graph Theory
Let the graph G = (N , E ) describe the information exchanged among agents, where N := {1, · · · , N} is the set of players and E ⊂ N × N is the edge set. If player i can obtain information from player j, then (i, j) ∈ E and j belong to player i's neighbor set , and the weighted Laplacian of graph G is L := Deg − W. If G is connected and undirected, then 0 is an eigenvalue of L, and the eigenvalues of L are 0 < s 2 (L) ≤ · · · ≤ s N (L) in ascending order.

Game Formulation
In this section, we build a mathematical setup about the problem considered. Consider a set of players N = {1, · · · , N}, where every player i ∈ N controls its local decision variable x i ∈ Ω i ⊆ R n i and Ω i is the private decision set of player i. Denote n := ∑ N i=1 n i and Ω := Ω 1 × · · · × Ω N ∈ R n , then the stacked vector of all the players' decisions x := col(x i ) i∈N ∈ R n is called the decision profile. We also write x = (x i , x −i ), where x −i := col(x j ) j∈N /{i} = col(x 1 , · · · , x i−1 , x i+1 , · · · , x N ) denotes all of the decisions except player i's.
The local objective function of each player i ∈ N is denoted by J i (x i , x −i ), and the affine coupling constrained set is defined as Here, A i and b i are the local data only accessible to player i. Define the feasible set of player i as which implies that the feasible set of each player depends on the action of the other players. Every player aims to optimize its objective function, and the game can be represented by the inter-dependent optimization problems ∀i ∈ N : min In order to deal with the coupling constraints and solve the problems, we define the Lagrange function of each player i ∈ N : where λ i ∈ R m + is a dual variable. According to optimization theory, if x * i is an optimal solution to (3), then there exists λ * i ∈ R n i such that the following KKT conditions are satisfied: By using the normal cone operator, the KKT conditions (6) are equivalent to Note that by the definition of a normal cone, one has N R m We consider the GNE with the same Lagrangian multipliers for every player, i.e., where F is the pseudo-gradient mapping of the game with the following form: is continuously differentiable and convex in x i , and Ω i is nonempty, compact and convex for each player i, then K is nonempty and satisfies Slater's constraint qualification.
Assumption 2. F is µ-monotone and θ 0 -Lipschitz continuous, i.e., for any point x and It follows from ( [15], [Theorem 4.8]) that x * solves V I(F, K) (8) if and only if there exists a λ * ∈ R m such that the KKT conditions are satisfied: where . Assumption 1 guarantees the existence of the v-GNE for game (3) by ( [31], [Corollary 2.2.5]). The goal of this paper is to design distributed algorithms for seeking the v-GNE under a partial-decision information setting, where both the computational cost and convergence rate are taken into consideration.

Alternating Distributed v-GNE Algorithms
In this section, we propose two kinds of distributed algorithms for seeking the v-GNE of game (3) with partial-decision information, where each player controls its own actions and exchanges information with its neighbors via the communication graph. Remark 1. Some GNE seeking algorithms with inertia and overrelaxation have been proposed [28,29]. Although the fast convergence of these algorithms has been validated numerically, more computation resources are inevitably required at each iteration. Note that the computation resources could be limited and expensive in many situations. Inspired by the above discussion, in this section we design distributed GNE seeking algorithms with alternated inertia and alternated overrelaxation, where both fast convergence rate and low computation consumption are taken into consideration.
Suppose that player i ∈ N controls its local decision x i ∈ R n i and λ i ∈ R m + (i.e., the estimation of λ * in (10)). In order to make up for the lack of non-neighbors' information, we introduce an auxiliary variable x i for each player i that provides the estimation of the other players' decisions. To be specific, represents player i's estimation vector except its own decisions. In addition, an auxiliary variable z i ∈ R m + is introduced for each player i ∈ N . We assume that each player exchanges its local variable {x i , λ i , z i } with its neighbor through the communication graph G.

Alternating Inertial Distributed v-GNE Seeking Algorithm
In this subsection, we propose an alternating inertial distributed algorithm for seeking the v-GNE, where the inertia is adopted intermittently (see Algorithm 1). Here, respectively. ρ is the inertial parameter, c is the coupling parameter, and τ i , ν i , σ i are the fixed step-sizes of player i in the update step. P Ω i is the projection operator on to the set Ω i . Let x := col(x i ) i∈N , z := col(z i ) i∈N and λ := col(λ i ) i∈N . Letx := col(x i ) i∈N with Let := col(x, z, λ) ∈ Ω, where Ω := R Nn × R Nm × R Nm + , and we define operators A, B and matrix Φ as follows: where R := diag((R i ) i∈N ) with R i := 0 n i ×n <i I n i 0 n i ×n >i , n <i := ∑ j<i n j and n >i := ∑ j>i n j .

Lemma 1.
Suppose Φ 0 and Φ −1 A is maximally monotone, then any limit point¯ = col(x,z,λ) of Algorithm 1 is a zero of A + B and a fixed point of T 2 • T 1 .
Proof. By the continuity of the right hand of (15),¯ = T(¯ ). Since Φ is positive definite, In order to show the convergence of the algorithm, the following assumptions are introduced. (11) is θ-Lipschitz continuous, i.e., there exists θ > 0 such that for any x and x ,
Let k → ∞, we have˜ = T˜ , which implies that˜ is a fixed point of T and thus 2 3 -restricted averaged, and then one obtains which implies that the odd subsequence { 2k+1 } also converges to˜ , and thus { k } converges to˜ . Note that Φ 0 and Φ −1 A is maximally monotone.˜ is a fixed point of T, and hence is a zero of A + B by Lemma 1. It follows from ( [21], [Theorem 1]) that given any˜ := col(x * , z * , λ * ) ∈ zer(A + B), then x * = 1 N ⊗ x * , and x * solves V I(F, K) (8), that is, x * is a v-GNE of game (3).

Alternating Overrelaxed Distributed v-GNE Seeking Algorithm
In this subsection, an alternating overrelaxed distributed algorithm is constructed for seeking the v-GNE, presented in Algorithm 2, and also that η is an overrelaxed parameter.
Here the partial-decision information setting is considered.

Algorithm 2 Distributed alternating overrelaxed v-GNE seeking.
Initialization: Similar to (15), we suppose that Φ 0 and Φ −1 A is maximally monotone, then Algorithm 2 is equivalent to where k = col(x k , z k , λ k ) and T is given in (15). Next, we prove the convergence of Algorithm 2 to a v-GNE.
Proof. Similar to Theorem 1, we first show the convergence of { 2k }, and then prove the convergence of { k }. Note that T = T 2 • T 1 is α-restricted averaged with α = 2 3 when δ > 1 β . Let * be any fixed point of T.
First, we consider the subsequence { 2k }, and according to (23) and (17), one has where the first equality holds due to αx which implies that { k+2 − * 2 Φ } is monotonically decreasing and bounded, and is thus convergent. Furthermore, Note that if { 2k } is bounded, there exists a convergent subsequence { 2n k } →˜ for some limit˜ . 2 3 -restricted averaged, then one obtains which implies that the sequence { 2k+1 } converges to the same limit of { 2k }, and thus { k } converges to˜ .

Numerical Simulation
In this section, we consider a classic Nash-Cournot game over a network as [21], where there are N firms and each firm i ∈ {1, · · · , N} produces commodities to participate in the competition over m markets (see Figure 1). Each market (denoted by M 1 , · · · , M m ) has limited capacity. Here, the partial-decision information setting is considered where each firm has limited access to its neighboring firms' information over the communication graph as in Figure 2.   We assume that firm i participates in n i markets by producing x i ∈ R n i amount of commodities and its production is limited by the set Ω i ∈ R n i . The local matrix A i ∈ R m×n i for firm i represents which markets it participates in. Specifically, for the j-th column of A i , its k-th element is 1 if and only if firm i delivers [x i ] j amount of production to market M k ; all other elements are 0. Each market M k has a maximal capacity of r k > 0, that is, Ax ≤ r, where A = [A 1 , · · · , A N ], x = col(x i ) i∈N ∈ R n , n = ∑ N i=1 n i and r = col(r k ) k=1,··· ,m ∈ R m . Suppose that each firm i has the production cost c i (x i ) : Ω i → R, and the price function P : R n → R m maps the total supply of each market to the market's price vector. The local objective function of firm i is where H i is a diagonal matrix with the elements randomly drawn from (1,8) and h i is randomly drawn from (1,2). The price function is taken as the linear function P =P − DAx withP = col(P k ) k=1,··· ,m ∈ R m and D = diag(d k ) k=1,··· ,m ∈ R m×m , whereP k and d k are randomly drawn from (10,20) and (1,3), respectively. Set the step-sizes as c = 100, τ i = 0.003, ν i = 0.02, and σ i = 0.003. First, Figure 3 shows that the convergence to the v-GNE can be guaranteed under Algorithms 1 and 2, and the trajectories of the local decision x i,k of firms 1, 6, 10, 11 are displayed in Figure 4. It can be seen in Figure 5 that the estimates on the firms 1 and 3 asymptotically tend to their real actions by using the proposed algorithms.
Then, it can be seen from Figure 6, that both of the proposed Algorithms 1 and 2 converge to the GNE x * with a faster convergence as compared with ( [21], [Alg. 1]), where Algorithm 1 has the fastest convergent rate. From Figure 7 we can see that the proposed Algorithm 1 also has a faster convergence rate than ( [25]

Remark 2.
It is worthwhile to note that the introduction of the inertia and overrelaxation steps has the potential of accelerating the convergence rate. As such, in this paper, the inertial and overrelaxed distributed algorithms are developed based on the pseudo-gradient method for seeking generalized Nash equilibrium in multi-player games. The similar inertia idea has been considered in the proximal-point algorithm (see ( [25], [Alg. 3])). However, the proximal-point algorithm generally needs to solve the optimization problem at each step k, which may be time-consuming and possibly costs a great amount of computation resources in many situations. As such, pseudo-gradient algorithms with inertia and overrelaxation were constructed in this paper, which successfully guarantees the convergence to v-GNE with a fast convergence rate. Moreover, we note that the introduction of the inertia and overrelaxation steps increases the computation burden, and thus two alternating inertial and overrelaxed algorithms are established in Algorithms 1 and 2 to balance the convergence rate and computation burden. In order to better display the effectiveness of our algorithms, we have added the comparison with ( [25], [Alg. 3]) in the simulation part (see Figure 7). From Figure 7, it can be seen that Algorithm 1 in this paper outperforms the ( [25], [Alg. 3]) in terms of the convergence rate.

Conclusions
This paper has studied the GNE computation issue in multi-player games with shared coupling constraints under the partial-decision information setting. Two distributed algorithms with alternating inertia and alternating overrelaxation have been developed, respectively, with fixed step-sizes. Both algorithms have guaranteed the convergence to the GNE under a mild assumption, which have the potential of improving the convergence rate and saving computation cost. Finally, one simulation example has been provided to show the effectiveness of the proposed algorithms. Further research topics can be focused on stochastic NE seeking problems subject to time-varying topologies with and without event-triggered communication protocols.