Nonlinear separation methods and applications for vector equilibrium problems using improvement sets

In this paper, the image space analysis is applied to investigate a vector equilibrium problem using improvement sets and with matrix inequality constraints. First, a nonlinear scalar regular weak separation function is constructed by using the oriented distance function and the norm function. Then, a global saddle-point condition for a generalized Lagrange function is investigated. It is shown that the existence of a saddle point is equivalent to a nonlinear separation of two suitable subsets in the image space. Furthermore, a gap function and an error bound are obtained in terms of the nonlinear scalar regular weak separation function under suitable assumptions. As applications, an optimality condition, a gap function and an error bound for a strategic game with vector payoffs are also given.