Towards A Goldfarb-Idnani Variant for Strongly Monotone Linear-Quadratic Games

TL;DR

This paper proposes a variant of the Goldfarb-Idnani method for efficiently computing variational Nash equilibria in strongly monotone quadratic games with shared constraints, showing promising numerical performance.

Contribution

It introduces a simple Goldfarb-Idnani variant that maintains key properties for strongly monotone games, expanding its applicability beyond symmetric cases.

Findings

Method is potentially competitive with state-of-the-art algorithms.

Numerical results demonstrate effectiveness in game-theoretic linear model predictive control.

Convergence to an equilibrium is not always guaranteed in the new variant.

Abstract

We analyze a simple variant of the Goldfarb-Idnani (GI) dual active-set method for computing variational generalized Nash equilibria of strongly monotone N-player games with convex quadratic costs and shared affine inequality and equality constraints. We show that several properties of the GI algorithm are maintained in spite of having a possibly non-symmetric pseudogradient matrix in the joint KKT system of the game, although convergence to an existing equilibrium is not guaranteed as in the original algorithm. Our numerical results show that the method is potentially competitive with alternative state-of-the-art algorithms, including for computing solutions of game-theoretic linear model predictive control laws.