Aggregative game theory is a mathematical framework to model the interdependent optimal decision making problems for a set of noncooperative agents, whenever the decision of each agent is affected by some aggregate effect of all the agents. This feature emerges in several application areas, such as demand side management in the smart grid , e.g. for electric vehicles [3, 4] and thermostatically controlled loads [5, 6], demand response in competitive markets  and network congestion control .
Existence and uniqueness of Nash equilibria in (aggregative) noncooperative games is well established in the literature of operation research , [10, §12], and automatic control [11, 12]. For the computation of a game equilibrium, several algorithms are available, both distributed protocols [13, 14] and semi-decentralized schemes [15, 16, 17, 18]. Among these, an elegant approach is to characterize the desired equilibrium solutions as the zeros of an operator, possibly monotone, e.g. the concatenation of interdependent Karush–Kuhn–Tucker operators, and in turn formulate an equivalent fixed-point problem, which is solved via appropriate fixed-point iterations. Overall, the available methods can ensure global convergence to an equilibrium if the coupling among the cost functions of the agents is “well behaved”, e.g. if the problem data are convex and the so-called pseudo-gradient game mapping is (strictly, strongly) monotone .
A popular class of algorithms for Nash equilibrium seeking is that of projected-gradient algorithms [20, §12], [14, 16, 19]. Whenever the pseudo-gradient game mapping is strongly monotone, projected-gradient algorithms can ensure fast convergence to a Nash equilibrium, possibly via distributed computation and information exchange. It follows that projected-gradient methods have the potential to be fast, simple and scalable with respect to the population size. At the same time, in the context of Nash equilibrium seeking, the convergence analyses for the available projected-gradient methods are quite diverse in nature.
In this paper, we aim at a unifying convergence analysis for projected-gradient algorithms that are adopted for the computation of generalized Nash equilibria in aggregative games. Specifically, we adopt a general perspective based on monotone operator theory  to show that projected-gradient algorithms with sequential updates belong to the class of preconditioned forward-backward splitting methods, introduced in  for multi-agent network games.
The main technical contribution of the paper is to conceive a design procedure for the preconditioned forward-backward splitting method. The proposed design is based not only on the splitting of the monotone operator whose zeros are the game equilibria, but also on the choice of the so-called preconditioning matrix, which induces the quadratic norm adopted to show global convergence of the resulting algorithm. Since the convergence characterization of the forward-backward splitting method is well established, the advantage of the proposed design is that global convergence follows provided that some mild monotonicity assumptions on the problem data are satisfied.
Remarkably, we discover that two recent projected-gradient algorithms for Nash equilibrium seeking in aggregative games,  and , can be equivalently written as preconditioned forward-backward splitting methods with symmetric preconditioning matrix, despite their algorithmic formulation is “asymmetric”.
denotes the set of real numbers, and the set of extended real numbers. (
) denotes a matrix/vector with all elements equal to(); to improve clarity, we may add the dimension of these matrices/vectors as subscript. denotes the Kronecker product between matrices and ;
denotes the maximum singular value of;
denotes the set of eigenvalues of. Given vectors , .
Operator theoretic definitions
denotes the identity operator. The mapping denotes the indicator function for the set , i.e., if , otherwise. For a closed set , the mapping denotes the projection onto , i.e., . The set-valued mapping denotes the normal cone operator for the the set , i.e., if , otherwise. For a function , ; denotes its subdifferential set-valued mapping, defined as ; A set-valued mapping is -Lipschitz continuous, with , if for all , , ; is (strictly) monotone if for all , , ; is -strongly monotone, with , if for all , , ; is -averaged, with , if , for all ; is -cocoercive, with , if is -averaged. With , we denote the resolvent operator of , which is -averaged if and only if is monotone; and denote the set of fixed points and of zeros, respectively.
Ii Generalized aggregative games
Ii-a Mathematical formulation
We consider a set of noncooperative agents, where each agent shall choose its decision variable (i.e., strategy) from the local decision set with the aim of minimizing its local cost function , which depends on both the local variable (first argument) and on the decision variables of the other agents, (second argument).
We focus on the class of aggregative games, where the cost function of each agent depends on the local decision variable and on the value of the aggregation function , with . In particular, we consider average aggregative games, where the aggregation function is the average function, i.e.,
Thus, for each , there is a function such that the local cost function can be written as
Furthermore, we consider generalized games, where the coupling among the agents arises not only via the cost functions, but also via their feasible decision sets. In our setup, the coupling constraints are described by an affine function, , where and . Thus, the collective feasible set, , reads as
while the feasible decision set of each agent is characterized by the set-valued mapping , defined as where and . The set represents the local decision set for agent , while the matrix defines how agent is involved in the coupling constraints. For instance, the shared constraints in (3) may contain a sparsity pattern that can be defined via a graph, where each agent has a set of “neighbors” with whom to share some constraints.
Remark 1 (Affine coupling constraint)
Affine coupling constraints as considered in this paper are very common in the literature of noncooperative games, see , , , . Moreover, we recall that the more general case with separable convex coupling constraints can be reformulated as game via affine coupling constraints [24, Remark 2].
Next, let us postulate standard convexity and compactness assumptions for the constraint sets, convexity and differentiability assumptions for the local cost functions.
Standing Assumption 1 (Convex differentiable functions)
For each and the function is convex and continuously differentiable.
Standing Assumption 2 (Compact convex constraints)
For each , the set is nonempty, compact and convex. The set satisfies the Slater’s constraint qualification.
In summary, the aim of each agent , given the decision variables of the other agents, , is to choose a strategy, , that solves its local optimization problem according to the game setup previously described, i.e.,
From the game-theoretic perspective, we consider the problem to compute a Nash equilibrium, as formalized next.
Definition 1 (Generalized Nash equilibrium)
The collective strategy is a generalized Nash equilibrium (GNE) of the game in (4) if and for all
In other words, a set of strategies is a Nash equilibrium if no agent can improve its objective function by unilaterally changing its strategy to another feasible one.
Ii-B Variational equilibria and pseudo-gradient of the game
Within all the possible Nash equilibria, we focus on an important subclass of equilibria, with some relevant structural properties, such as “larger social stability” and “economic fairness” [9, Theorem 4.8], that corresponds to the solution set of an appropriate variational inequality. Let us first formalize the notion of variational inequality problem.
Definition 2 (Generalized variational inequality)
Consider a closed convex set , a set-valued mapping , and a single-valued mapping . The generalized variational inequality problem GVI, is the problem to find and such that
If for all , then GVI reduces to the variational inequality problem VI.
A fundamental mapping in a noncooperative game is the so-called pseudo-gradient, , defined as
Namely, the mapping is obtained by stacking together the subdifferentials of the agents’ objective functions with respect to their local decision variables.
Under Assumptions 12, it follows by [10, Proposition 12.4] that any solution of GVI is a Nash equilibrium of the game in (4). The inverse implication is not true in general, and actually in passing from the Nash equilibrium problem to the GVI problem most solutions are lost [10, §12.2.2]; indeed, a game may have a Nash equilibrium while the corresponding GVI has no solution. Note that, since the cost functions are differentiable (by Assumption 2), then GVI reduces to VI, which is commonly addressed in the context of game theory via projected gradient algorithms [14, 16, 19], [20, §12].
Under the postulated standing assumptions, it is shown in [10, Proposition 12.11] that a sufficient condition for the existence (and uniqueness) of a variational GNE of the game in (4) is the (strict) monotonicity of the pseudo-gradient in (5). Thus, let us assume strongly monotonicity of .
Standing Assumption 3 (Strong monotonicity)
The pseudo-gradient in (5) is -strongly monotone and -Lipschitz continuous, for some constants .
Iii Generalized Nash equilibrium as zero of the sum of two monotone operators
In this section, we exploit operator theory to recast the Nash equilibrium seeking problem into a monotone inclusion, namely, the problem of finding a zero of a set-valued monotone operator. As first step, we characterize a GNE of the game in terms of KKT conditions of the coupled optimization problems in (4). For each agent , let us introduce the Lagrangian function , defined as
where is the Lagrangian multiplier associated with the coupling constraints. It follows from [10, §12.2.3] that the set of strategies is a GNE of the game in (4) if and only if the following coupled KKT conditions are satisfied:
The constraint qualification in Assumption 2 is needed at this stage to ensures boundedness of the dual variables ’s.
In a similar fashion, we characterize a variational GNE in term of KKT conditions by exploiting the Lagrangian duality scheme for the corresponding VI problem, see [25, §3.2]. Specifically, is a solution of VI if and only if Then, the associated KKT optimality conditions read as
To cast (7) in compact form, we introduce the set-valued mapping , defined as
Essentially, the role of the mapping is that its zeros correspond to the variational generalized Nash equilibria of the game in (4), as formalized in the next statement.
Proposition 1 ([18, Th. 1])
To conclude this section, we note that the mapping can be written as the sum of two operators, defined as
The formulation is called splitting of , and will be exploited in different ways later on. We show next that the mappings and are both monotone, which paves the way for splitting algorithms.
First, consider in (10). The first term is maximally monotone, since normal cones of closed convex sets are maximally monotone and the concatenation preserves maximality [21, Prop. 20.23]; the second term
is linear and skew symmetric, i.e.,, thus maximally monotone [21, Ex. 20.30]. Then, the maximal monotonicity of follows from [21, Cor. 24.4], since . To prove that is -cocoercive, we note that for all , it holds that
The first and second inequalities follow from the -strong monotonicity and -Lipschitz continuity, respectively, of the mapping , postulated in Assumption 3.
Iv Preconditioned Forward-Backward splitting
In light of Lemma 1, the forward-backward (FB) splitting [21, §25.3] guarantees convergence to a zero of . In this section, we discuss a design procedure for FB algorithms, which is particularly useful when the resolvent of the operator cannot be computed explicitly. Moreover, we show that two existing algorithms for GNE seeking in aggregative games belong to this class of algorithms.
Iv-a Preconditioned Forward-Backward: Design Procedure
For any matrix , the following equivalence holds:
where and .
Consider a vector , then
where the first equivalence holds since .
In numerical analysis, represents a forward step with size and direction defined by , while represents a backward step. Directly from the iteration in (12), we have that
The choice of the preconditioning matrix in (13) plays a key role in the algorithm design. Next, we provide some general guidelines to design .
Design guidelines for the preconditioning matrix :
in (13) explicitly computable (necessary);
(convenient convergence analysis);
iterations in (12) sequential (convenient implementation).
Without loss of generality, we denote , then the inclusion in (13) reads in expanded form as
Consider a symmetric matrix , designed accordingly to the guidelines above, i.e.,
where and the coefficients (step sizes) and are chosen such that has positive eigenvalues (guideline 1), as formalized in the next lemma.
The matrix in (15) is positive definite if
where and .
The conditions in (16) directly follow by applying the the Schur’s complement on in (15). Now, we present the preconditioned Forward-Backward (pFB) algorithm associated with , which is the Banach–Picard iteration in (12) for .Algorithm 1: Preconditioned Forward Backward (pFB)
The iterations of Algorithm 1 are sequential (guideline 4), namely, the multiplier update, , exploits the most recent value of the agents’ strategies, .
In the next statement, we show the convergence of Algorithm 1 to a variational generalized Nash equilibrium, under suitable choices of the step sizes.
Theorem 1 (Global convergence of pFB)
The sequence defined by Algorithm 1, with step sizes , for all , and , with , globally converges to some , with as in (8).
See Section V.
Iv-B The Asymmetric Projection Algorithm in [16, Alg. 1] is a preconditioned Forward-Backward splitting
We note that Algorithm 1 (with equal step sizes) corresponds to the “asymmetric” projected algorithm (APA) proposed in [16, Alg. 1]. Therein, the algorithm design and its convergence analysis rely on a variational inequality formulation of the Nash equilibrium problem. Specifically, the authors define the convex set and the monotone mapping
and characterize the GNE as solutions of VI. Then, to solve VI in a semi-decentralized fashion, the authors propose an asymmetric implementation of the projection algorithm for variational inequalities [20, 12.5.1], in which each iteration is computed as
If the parameter in is chosen such that , then the unique solution in (17) is , where is the projection operator characterized by the asymmetric matrix in (18). Thus, the iteration in (17) equivalently reads as
which is nothing but a pFB associated with the splitting and the preconditioning matrix .
With (19), we showed that pFB algorithms based on different splittings and preconditioning matrices can lead to the same algorithm. From an operator-theoretic perspective, the convergence analysis is more convenient for the pFB with symmetric matrix (guideline 3), since the properties that , have with the standard inner product are preserved for , with inner product , as shown in Section V.
Iv-C The distributed algorithm for aggregative games on graphs in [14, §3] is a preconditioned Forward-Backward
In this section, we show that the synchronous distributed algorithm for NE seeking in network aggregative games proposed in [14, §3] can be written as a pFB splitting.
), but build an estimate of it by communicating over an undirected network with their neighboring agents.
Specifically, let be the set of underlying undirected edges between agents; let denote the set of neighbors of agent , with the convention that ; let be the degree matrix, where ; let be the adjacency matrix, such that if , otherwise. let be the Laplacian matrix and let us define the matrix , with , such that, given a vector , with , then
Let be the extension of in (5) to the augmented space of actions and estimates, defined as
Algorithm 2: Koshal–Nedić–Shanbhag algorithm
In the following statement, we show that Algorithm 2 is a pFB splitting with symmetric preconditioning matrix.
V Convergence Analysis
First, we show that the properties the mappings , have with the standard inner product are preserved for , with the inner product .
The mappings , , , satisfy the following properties in the -induced norm:
is -cocoercive and is -averaged, where
is maximally monotone and is -averaged.
(i): We need to show that for all the following condition holds:
We first provide an upper bound for the right hand side of (24). Let us denote for , then
where is symmetric and positive definite if the step sizes , are chosen as in Lemma 3. Moreover, it holds that , where is the smallest eigenvalue of . Now, we exploit the -cocoercivity of and the upper bound in (25) to define the cocoercivity constant in (24).
Thus, the mapping is cocoercive with constant w.r.t. the -induced norm. Since is -cocoercive, it follows from [21, Prop. 4.33] that is -averaged. (ii): Since is maximally monotone by Lemma (1) and is positive definite, if the step sizes are chosen as in Lemma 3, then the maximal monotonicity of follows from [26, Lemma 3.7].
Next, we show that the FB operator is averaged if the step sizes are chosen small enough.
where the second inequality in (28) holds for step sizes chosen as in (26). Moreover, if the step sizes are equal, i.e., for all , then holds for as in (27). We can now prove the convergence of Algorithm 1.
Proof of Theorem 1: The iterations in Alg. 1 are obtained explicitly by substituting into (14) and solving for , . Thus, Alg. 1 is the Banach–Picard iteration of the mapping , which is -averaged, with , by Lemma 5, if the step sizes satisfy (26). The convergence of the sequence generated by the Banach–Picard iteration of to follows by [21, Prop. 15.5].
By monotone operator theory, projected-gradient methods for generalized Nash equilibrium seeking in aggregative games are preconditioned forward-backward splitting methods, whose convergence has been established for problems with strongly monotone pseudo-gradient mapping.
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