Package: GNE 0.99-7

Introduction | GNEP as a nonsmooth equation | Notation and definitions | A classic example | Usage example | Localization of the GNEs | Benchmark of the complementarity functions and the computation methods | Initial point $z_0 = (4, -4, 1, 1)$ | Initial point $z_0 = (-4, 4, 1, 1)$ | Initial point $z_0 = (3, 0, 1, 1)$ | Initial point $z_0 = (0, 3, 1, 1)$ | Initial point $z_0 = (-1, -1, 1, 1)$ | Initial point $z_0 = (0, 0, 1, 1)$ | Conclusions | Special case of shared constraints with common multipliers | Constrained-equation reformulation of the KKT system | GNEP as a fixed point equation or a minimization problem | NI reformulation | QVI reformulation | The jointly convex case | In this subsection, we present reformulations for a subclass of GNEP called jointly convex case.Firstly, the jointly convex setting requires that the constraint function is common to all players $g^1=\dots =g^N= g$.Then, we assume, there exists a closed convex subset $X \subset \R^n$ such that for all player $i$,$$ | NIF formulation for the jointly convex case | QVI formulation for the jointly convex case | List of examples | Tables for the nonsmooth reformulation | Appendix for the nonsmooth reformulation | \subsection{Semismooth reformulation -- Shared constraint case\label{app:ceq:jointcase}}The Jacobian is given by$$\Jac \widetilde H(x, \tilde \lambda, \tilde w) =\left(\begin{matrix}\Jac_x \bar L (x, \tilde \lambda) & \Jac_{\tilde \lambda} \bar L (x, \tilde \lambda) & 0 \\Jac_x \tilde g(x) & 0 & I \0 & \diag[\tilde w] & \diag[\tilde \lambda]\end{matrix}\right),$$where$$\Jac_{\tilde \lambda} \bar L (x, \tilde \lambda)