Vector quasi-equilibrium problems: separation, saddle points and error bounds for the solution set

In this paper, we employ the image space analysis (for short, ISA) to investigate vector quasi-equilibrium problems (for short, VQEPs) with a variable ordering relation, the constrained condition of which also consists of a variable ordering relation. The quasi relatively weak VQEP (for short, qr-weak VQEP) are defined by introducing the notion of the quasi relative interior. Linear separation for VQEP (res., qr-weak VQEP) is characterized by utilizing the quasi interior of a regularization of the image and the saddle points of generalized Lagrangian functions. Lagrangian type optimality conditions for VQEP (res., qr-weak VQEP) are then presented. Gap functions for VQEP (res., qr-weak VQEP) are also provided and moreover, it is shown that an error bound holds for the solution set of VQEP (res., qr-weak VQEP) with respect to the gap function under strong monotonicity. Copyright Springer Science+Business Media New York 2014