Projected-gradient algorithms for generalized equilibrium seeking in Aggregative Games are preconditioned Forward-Backward methods

TL;DR

This paper reveals that projected-gradient algorithms for finding equilibria in aggregative games are actually preconditioned forward-backward splitting methods, providing a unified operator-theoretic framework for their design.

Contribution

It demonstrates that existing projected-gradient methods are special cases of preconditioned forward-backward algorithms, offering a new unified perspective for designing equilibrium seeking algorithms.

Findings

Projected-gradient methods are preconditioned forward-backward methods.

Unified operator-theoretic framework for equilibrium algorithms.

Connections between recent methods and operator splitting techniques.

Abstract

We show that projected-gradient methods for the distributed computation of generalized Nash equilibria in aggregative games are preconditioned forward-backward splitting methods applied to the KKT operator of the game. Specifically, we adopt the preconditioned forward-backward design, recently conceived by Yi and Pavel in the manuscript "A distributed primal-dual algorithm for computation of generalized Nash equilibria via operator splitting methods" for generalized Nash equilibrium seeking in aggregative games. Consequently, we notice that two projected-gradient methods recently proposed in the literature are preconditioned forward-backward methods. More generally, we provide a unifying operator-theoretic ground to design projected-gradient methods for generalized equilibrium seeking in aggregative games.