Saddle points and gap functions for weak generalized Ky Fan inequalities

In this paper, we employ the image space analysis method to investigate a weak generalized Ky Fan inequality with cone constraints. Some regular weak separation functions are introduced, and generalized Lagrangian functions are constructed by using these regular weak separation functions. Under suitable convexity assumptions and Slater condition, the existence of solution for the weak generalized Ky Fan inequality with cone constraints is equivalent to a saddle point of the generalized Lagrangian functions. Moreover, we also use the regular weak separation functions to construct gap functions for the weak generalized Ky Fan inequality with cone constraints, and obtain its error bound.