On generalized Nash equilibrium problems in infinite-dimensional spaces using Nikaido–Isoda type functionals

Michael Ulbrich, Julia Fritz

Research output: Contribution to journalArticlepeer-review

Abstract

We present an analysis of generalized Nash equilibrium problems in infinite-dimensional spaces with possibly non-convex objective functions of the players. Such settings arise, for instance, in games that involve nonlinear partial differential equation constraints. Due to non-convexity, we work with equilibrium concepts that build on first order optimality conditions, especially Quasi-Nash Equilibria (QNE), i.e. first-order optimality conditions for (Generalized) Nash Equilibria, and Variational Equilibria (VE), i.e. first-order optimality conditions for Normalized Nash Equilibria. We prove existence of these types of equilibria and study characterizations of them via regularized (and localized) Nikaido-Isoda merit functions. We also develop continuity and (continuous) differentiability results for these merit functions under quite weak assumptions, using a generalization of Danskin's theorem. They provide a theoretical foundation for, e.g. using globalized descent methods for computing QNE or VE.

Original languageEnglish
JournalOptimization Methods and Software
DOIs
StateAccepted/In press - 2024

Keywords

  • (regularized) Nikaido–Isoda function
  • 46N10
  • 49J27
  • 49J50
  • 90C26
  • 90C48
  • 91A10
  • Danskin's theorem
  • Generalized Nash equilibrium problems
  • existence of equilibria
  • infinite-dimensional spaces
  • merit functions

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