Convex Generalized Nash Equilibrium Problems and Polynomial Optimization

TL;DR

This paper introduces a polynomial optimization approach using Moment-SOS relaxations to compute or detect the nonexistence of convex GNEs in polynomial GNEPs, with proven convergence under certain conditions.

Contribution

It develops a novel method combining rational Lagrange multipliers and semidefinite relaxations for solving convex polynomial GNEPs, ensuring correctness and efficiency.

Findings

Method successfully computes GNEs in numerical tests.

The approach can detect nonexistence of GNEs.

Numerical experiments demonstrate efficiency and reliability.

Abstract

This paper studies convex Generalized Nash Equilibrium Problems (GNEPs) that are given by polynomials. We use rational and parametric expressions for Lagrange multipliers to formulate efficient polynomial optimization for computing Generalized Nash Equilibria (GNEs). The Moment-SOS hierarchy of semidefinite relaxations are used to solve the polynomial optimization. Under some general assumptions, we prove the method can find a GNE if there exists one, or detect nonexistence of GNEs. Numerical experiments are presented to show the efficiency of the method.