Abstract:

Recently a new solution concept for generalized Nash equilibrium problems was pub-
lished by the author. This concept selects a reasonable equilibrium out of the typically
infinitely many. The idea is to model the process of finding a compromise by solving
parametrized generalized Nash equilibrium problems. In this paper we propose an
algorithmic realization of the concept. The model produces a solution path, which is
under suitable assumptions unique. The algorithm is a homotopy method that tries to
follow this path. We use s emismooth Newton steps as corrector steps in our algorithm
in order to approximately solve the generalized Nash equilibrium problems for each
given parameter. If we have a unique solution path, we need three additional theo-
retical assumptions: a stationarity result for the merit function, a coercivity condition
for the constraints, and an extended Mangasarian–Fromowitz constraint qualification.
Then we can prove convergence of our semismooth tracing algorithm to the unique
equilibrium to be selected. We also present convincing numerical results on a test
library of problems from literature. The algorithm also performs well on a number of
problems that do not satisfy all the theoretical assumptions.

Reference:

• Dreves, A.: An algorithm for equilibrium selection in generalized Nash equilibrium problems. Computational Optimization and Applications. (2019), DOI 10.1007/s10589-019-00086-w.