Generalized Nash Equilibrium Problems in Banach Spaces: Theory, Nikaido--Isoda-Based Path-Following Methods, and Applications
Michael Hintermüller, Thomas M. Surowiec, A. Kämmler · SIAM Journal on Optimization · 2015
Building upon the results in [M. Hintermüller and T. Surowiec, Pac. J. Optim., 9 (2013), pp. 251--273], a class of noncooperative Nash equilibrium problems is presented, in which the feasible set of each player is perturbed by the decisions of their competitors via a convex constraint. In addition, for every vector of decisions, a common “state” variable is given by the solution of an affine linear equation. The resulting problem is therefore a generalized Nash equilibrium problem (GNEP). The existence of an equilibrium for this problem is demonstrated, and first-order optimality conditions are derived under a constraint qualification. An approximation scheme is proposed, which involves the solution of a parameter-dependent sequence of standard Nash equilibrium problems. An associated path-following strategy based on the Nikaido--Isoda function is then proposed. Function-space-based numerics for parabolic GNEPs and a spot-market model are developed.