Generalized Nash equilibrium problems with quasi-linear constraints

TL;DR

This paper introduces a novel method for solving generalized Nash equilibrium problems with polynomial objectives and linear constraints, using Carathéodory's theorem and Moment-SOS relaxations for efficient computation.

Contribution

It provides a new representation of KKT sets via partial Lagrange multipliers and transforms the problem into polynomial optimization problems solvable by Moment-SOS relaxations.

Findings

Efficient computation of all GNEs demonstrated.

Method can detect nonexistence of GNEs.

Numerical experiments confirm computational efficiency.

Abstract

We study generalized Nash equilibrium problems (GNEPs) such that objectives are polynomial functions, and each player's constraints are linear in their own strategy. For such GNEPs, the KKT sets can be represented as unions of simpler sets by Carath\'{e}odory's theorem. We give a convenient representation for KKT sets using partial Lagrange multiplier expressions. This produces a set of branch polynomial optimization problems, which can be efficiently solved by Moment-SOS relaxations. By doing this, we can compute all generalized Nash equilibria or detect their nonexistence. Numerical experiments are also provided to demonstrate the computational efficiency.