On error bounds and Newton-type methods for generalized Nash equilibrium problems
- 345 Downloads
Error bounds (estimates for the distance to the solution set of a given problem) are key to analyzing convergence rates of computational methods for solving the problem in question, or sometimes even to justifying convergence itself. That said, for the generalized Nash equilibrium problems (GNEP), the theory of error bounds had not been developed in depth comparable to the fields of optimization and variational problems. In this paper, we provide a systematic approach which should be useful for verifying error bounds for both specific instances of GNEPs and for classes of GNEPs. These error bounds for GNEPs are based on more general results for constraints that involve complementarity relations and cover those (few) GNEP error bounds that existed previously, and go beyond. In addition, they readily imply a Lipschitzian stability result for solutions of GNEPs, a subject where again very little had been known. As a specific application of error bounds, we discuss Newtonian methods for solving GNEPs. While we do not propose any significantly new methods in this respect, some new insights into applicability to GNEPs of various approaches and into their convergence properties are presented.
KeywordsGeneralized Nash equilibrium problem Error bound Upper Lipschitz stability Newton-type methods
The authors thank Andreas Fischer for pointing out an inconsistency in the original version of Sect. 4.
- 7.Dreves, A., Facchinei, F., Fischer, A., Herrich, M.: A new error bound result for generalized Nash equilibrium problems and its algorithmic application. Technical report MATH-NM-1-2013, Technische Universität Dresden, Institute für Mathematik, January 2013. Available at Optimization Online Google Scholar
- 15.Facchinei, F., Pang, J.-S.: Finite-Dimensional Variational Inequalities and Complementarity Problems. Springer, New York (2003) Google Scholar
- 37.Solodov, M.V.: Constraint qualifications. In: Cochran, J.J., et al. (eds.) Wiley Encyclopedia of Operations Research and Management Science. Wiley, New York (2010) Google Scholar