Learning Generalized Nash Equilibria in Non-Monotone Games with Quadratic Costs

TL;DR

This paper introduces a distributed gradient method and a zero-order scheme for computing generalized Nash equilibria in non-monotone quadratic games, achieving linear and sublinear convergence respectively.

Contribution

It reformulates GNE problems into a convex program satisfying the PL condition, enabling distributed algorithms without strong monotonicity assumptions.

Findings

Distributed gradient method converges linearly to a GNE.

Zero-order method converges at rate O(1/t) using only local evaluations.

Applicable to non-monotone games with quadratic costs and linear constraints.

Abstract

We study generalized Nash equilibrium (GNE) problems in games with quadratic costs and individual linear equality constraints. Departing from approaches that require strong monotonicity and/or shared constraints, we reformulate the KKT conditions of the (generally non-monotone) games into a tractable convex program whose objective satisfies the Polyak-Lojasiewicz (PL) condition. This PL geometry enables a distributed gradient method over a fixed communication graph with global geometric (linear) convergence to a GNE. When gradient information is unavailable or costly, we further develop a zero-order fully distributed scheme in which each player uses only local cost evaluations and their own constraint residuals. With an appropriate step size policy, the proposed zero-order method converges to a GNE, provided one exists, at rate O(1/t).