@phdthesis{Harms2014,
  author    = {Harms, Nadja},
  title     = {Primal and Dual Gap Functions for Generalized Nash Equilibrium Problems and Quasi-Variational Inequalities},
  url       = {http://nbn-resolving.de/urn:nbn:de:bvb:20-opus-106027},
  school      = {Universit{\"a}t W{\"u}rzburg},
  year      = {2014},
  abstract  = {In this thesis we study smoothness properties of primal and dual gap functions for generalized Nash equilibrium problems (GNEPs) and finite-dimensional quasi-variational inequalities (QVIs). These gap functions are optimal value functions of primal and dual reformulations of a corresponding GNEP or QVI as a constrained or unconstrained optimization problem. Depending on the problem type, the primal reformulation uses regularized Nikaido-Isoda or regularized gap function approaches. For player convex GNEPs and QVIs of the so-called generalized `moving set' type the respective primal gap functions are continuously differentiable. In general, however, these primal gap functions are nonsmooth for both problems. Hence, we investigate their continuity and differentiability properties under suitable assumptions. Here, our main result states that, apart from special cases, all locally minimal points of the primal reformulations are points of differentiability of the corresponding primal gap function. Furthermore, we develop dual gap functions for a class of GNEPs and QVIs and ensuing unconstrained optimization reformulations of these problems based on an idea by Dietrich (``A smooth dual gap function solution to a class of quasivariational inequalities'', Journal of Mathematical Analysis and Applications 235, 1999, pp. 380--393). For this purpose we rewrite the primal gap functions as a difference of two strongly convex functions and employ the Toland-Singer duality theory. The resulting dual gap functions are continuously differentiable and, under suitable assumptions, have piecewise smooth gradients. Our theoretical analysis is complemented by numerical experiments. The solution methods employed make use of the first-order information established by the aforementioned theoretical investigations.},
  subject      = {Nash-Gleichgewicht},
  language  = {en}
}