On the polyhedral homotopy method for solving generalized Nash equilibrium problems of polynomials

TL;DR

This paper introduces a numerical method combining polyhedral homotopy continuation and Moment-SOS relaxations to find all solutions or determine the nonexistence of generalized Nash equilibria in polynomial GNEPs.

Contribution

It presents a novel approach that guarantees finding all GNEs or confirming their absence in polynomial GNEPs, under generic conditions.

Findings

Method successfully finds all GNEs when they exist.

Method can detect nonexistence of GNEs.

Numerical experiments demonstrate efficiency.

Abstract

The generalized Nash equilibrium problem (GNEP) is a kind of game to find strategies for a group of players such that each player's objective function is optimized. Solutions for GNEPs are called generalized Nash equilibria (GNEs). In this paper, we propose a numerical method for finding GNEs of GNEPs of polynomials based on the polyhedral homotopy continuation and the Moment-SOS hierarchy of semidefinite relaxations. We show that our method can find all GNEs if they exist, or detect the nonexistence of GNEs, under some genericity assumptions. Some numerical experiments are made to demonstrate the efficiency of our method.