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

Abstract

We consider the stochastic generalized Nash equilibrium problem (SGNEP) with expected-value cost functions. Inspired by Yi and Pavel (Automatica, 2019), we propose a distributed GNE seeking algorithm based on the preconditioned forwardbackward operator splitting for SGNEP, where, at each iteration, the expected value of the pseudo-gradient is approximated via a number of random samples. Our main contribution is to show almost sure convergence of the proposed algorithm if the pseudogradient mapping is restricted (monotone and) cocoercive.

Online distributed tracking of generalized Nash equilibrium on physical networks

“…In reference [9], a Nesterov-based algorithm was proposed to seek the GNE of an energy sharing game among prosumers, where the proposed algorithm showed better convergence performance than two classical distributed methods: the alternating direction method of multipliers (ADMM) and the gradient descent method. References [10,11] separately studied the distributed forwardbackward algorithm for GNE seeking. The former studied a stochastic GNG, while the latter formulated an asynchronous algorithm paradigm.…”

Convergence of sequences: a survey

“…Unfortunately, Féjer monotonicity is hard to obtain, therefore the concept is typically relaxed to a quasi-Féjer property, where a vanishing error must be considered. Such an error term in the distance inequality is common in many equilibrium problems [11,7,35,20,22,36,1,37,23], especially in the stochastic case where the concept of quasi-Féjer monotone sequence was first introduced [38,39]. However, these properties are not necessarily enough to ensure convergence, hence, (quasi) Féjer monotonicity is often used in combination with convergence results on sequences of real numbers.…”

“…However, these properties are not necessarily enough to ensure convergence, hence, (quasi) Féjer monotonicity is often used in combination with convergence results on sequences of real numbers. These technical results have been used in many theoretical and computational applications that range from stochastic Nash equilibrium seeking [11,12,14] to machine learning [17,19,20].…”

“…The aim of this paper is to collect these results towards a complete overview, thus to be able to find the one that most suits the application at hand. In fact, many convergence results find their use in theoretical applications, such as Lyapunov stability analysis [1,2,3,4], variational analysis [5,6,7,8,9,10] and game equilibrium seeking [11,12,13,14], in automatic control, such as model predictive control [15] and network control problems [16], as well as in other engineering areas, e.g., training and learning in generative adversarial networks [17,18,19], vehicle flow control in traffic networks [20] and in modeling the prosumer behavior in smart power grids [11,14,21,22].…”

Distributed Nash Equilibrium Seeking for Generalized Convex Games with Shared Constraints

“…However, to the best of our knowledge, the partial decision information algorithm for constrained N -coalition games hasn't been proposed in the literature yet. Note that when the number of agents in each coalition is 1, the distributed GNE seeking problem for constrained N -coalition games can be reduced to a distributed GNE seeking problem for generalized noncooperative games (e.g., [13]- [15]). Thus, the studied problem is more general.…”