Chapter 4 GNEPs with No Special Structure

Abstract

As we saw in the previous chapter, there are some sound proposals for the analysis and solution of standard Nash problems and of jointly convex problems. However, when it comes to the analysis and solution of a gnep in its full generality, the situation is much bleaker. A gnep cannot be reduced to a VI, but it can be reduced to a quasi-variational inequality, as first noted in [14]. A quasi-variational inequality is a problem formally similar to a VI where, however, the feasible set K is not fixed but depends on x. Since the theory of quasi-variational inequalities is not nearly as developed as that of VIs, this reduction is not very useful, especially from the computational point of view. So we do not pursue this issue further. In this chapter, instead, after discussing very briefly some existence results, we consider in some detail two recent algorithmic developments that seem promising.