A damped forward–backward algorithm for stochastic generalized Nash equilibrium seeking

A Distributed Forward–Backward Algorithm for Stochastic Generalized Nash Equilibrium Seeking

“…In a noncooperative game, convexity of the cost functions J i in the local variables x i (Assumption 2) does not imply monotonicity, nor cocoercivity (unless we restrict to the socalled jointly convex case [6]). Then, in light of Assumption 6, the two operators Ā and B in (10) have the following properties (in the Φ-induced norm).…”

“…Lemma 2: Let Assumptions 1-6 hold and let Φ 0. The operators Ā and B in (10) have the following properties:…”

“…where (Id+Φ −1 B) −1 represents the backward step and (Id−Φ −1 Â) is the forward one. We remark that  is a sampled and computable, approximation of the operator Ā in (10). By expanding (14) and solving for x k , z k and λ k , we obtain the distributed FB steps in Algorithm 1 with  as in (12), B as in (10) and…”

“…On the other hand, when considering a game-theoretic setup, it is desirable to consider distributed algorithms, in the sense that each agent should only know its local cost function and its local constraints. For this reason, preconditioning has been recently introduced in [1], see [10] for a preliminary extension of this method to the stochastic case.…”

“…A preliminary study related to this work is presented in [10]. In that paper, we consider a SGNEP and build a preconditioned FB algorithm with damping.…”