Optimality Conditions for Generalized Ky Fan Quasi‐Inequalities with Applications

In this paper, the image space analysis is employed to study a generalized Ky Fan quasi‐inequality with cone constraints. By virtue of a nonlinear scalarization function and a positive linear operator, a nonlinear (regular) weak separation function and a linear regular weak separation function are introduced. Nonlinear and, in particular, linear separations for the generalized Ky Fan quasi‐inequality with cone constraints are characterized. Some necessary and sufficient optimality conditions, especially a saddle‐point sufficient optimality condition for the generalized Ky Fan quasi‐inequality with cone constraints, are obtained. As applications, some sufficient conditions for (weak) vector equilibrium flows of vector traffic equilibrium problems with capacity arc constraints, are derived.