Semi-decentralized generalized Nash equilibrium seeking in monotone aggregative games

Abstract

We address the generalized Nash equilibrium seeking problem for a population of agents playing aggregative games with affine coupling constraints. We focus on semi-decentralized communication architectures, where there is a central coordinator able to gather and broadcast signals of aggregative nature to the agents. By exploiting the framework of monotone operator theory and operator splitting, we first critically review the most relevant available algorithms and then design two novel schemes: (i) a single-layer, fixed-step algorithm with convergence guarantee for general (non-cocoercive, non-strictly) monotone aggregative games and (ii) a single-layer proximal-type algorithm for a class of monotone aggregative games with linearly coupled cost functions. We also design novel accelerated variants of two algorithms via (alternating) inertial steps. Finally, we show via numerical simulations that the proposed algorithms outperform those in the literature in terms of convergence speed.

Operationally-Safe Peer-to-Peer Energy Trading in Distribution Grids: A Game-Theoretic Market-Clearing Mechanism

“…[13]- [17]. Among these methods, semi-decentralized ones [16] have been shown to be particularly efficient in terms of convergence speed. Here, we tailor Algorithm 6 in [16] for our P2P market game in (1).…”

“…Among these methods, semi-decentralized ones [16] have been shown to be particularly efficient in terms of convergence speed. Here, we tailor Algorithm 6 in [16] for our P2P market game in (1). Before presenting the algorithm, let us introduce, for each prosumer i ∈ N , the dual variable µ tr (i,j) ∈ R H , for all j ∈ N i , which are associated with the trading reciprocity (8b).…”

“…(ii) Algorithm 1 is semi-decentralized, i.e., the prosumers rely on a reliable central coordinator (i.e., the DNO) that gathers local variables in aggregative form and then broadcasts signals, such as dual variables, to all prosumers, see Figure 2. Such communication architecture is particularly efficient to design fast and scalable equilibrium seeking algorithms in games [16]; (iii) The local primal update of each prosumer (lines: [8][9][10][11] involves the solution of a quadratic programming problem 4 , for which very efficient solvers are available, e.g. [32]; (iv) The primal update of the DNO (lines: [26][27][28][29]…”

“…The derivation and convergence analysis of Algorithm 1 relies (for the most part) on the customized preconditioned proximal-point (cPPP) algorithm for generalized aggregative games proposed in [16, Algorithm 6]. The objective of this appendix is to show that the proposed market-clearing game (1), with cost functions and constraints sets defined in ( 17)- (19), satisfies all the technical conditions in [16,Theorem 2], among which is the existence of a variational GNE, i.e., item (i) of Proposition 1. Therefore, we invoke [16,Theorem 2] to prove convergence of Algorithm 1, i.e., item (ii) of Proposition 1.…”

“…The objective of this appendix is to show that the proposed market-clearing game (1), with cost functions and constraints sets defined in ( 17)- (19), satisfies all the technical conditions in [16,Theorem 2], among which is the existence of a variational GNE, i.e., item (i) of Proposition 1. Therefore, we invoke [16,Theorem 2] to prove convergence of Algorithm 1, i.e., item (ii) of Proposition 1. For a complete convergence analysis of the cPPP algorithm for aggregative games we refer to [16,Appendix C].…”