Relaxation Methods for Generalized Nash Equilibrium Problems with Inexact Line Search

The generalized Nash equilibrium problem (GNEP) is an extension of the standard Nash game where, in addition to the cost functions, also the strategy spaces of each player depend on the strategies chosen by all other players. This problem is rather difficult to solve and there are only a few methods available in the literature. One of the most popular ones is the so-called relaxation method, which is known to be globally convergent under a set of assumptions. Some of these assumptions, however, are rather strong or somewhat difficult to understand. Here, we present a modified relaxation method for the solution of a certain class of GNEPs. The convergence analysis uses completely different arguments based on a certain descent property and avoids some of the technical conditions for the original relaxation method. Moreover, numerical experiments indicate that the modified relaxation method performs quite well on a number of different examples taken from the literature.