Nonlinear Characterizations Of Several Approximate Solutions Of Set-Valued Equilibrium Problems With Ordering C One Has Empty Topological Interior

Scalarization is an important aspect of vector equilibrium problems,which can be used to study the optimality conditions of vector equilibrium problems.Because the classical nonlinear separation theorem depends on the nonemptiness of the topological interior of ordering cones,it is more valuable to study the vector equilibrium problem with empty topological interior while the relative algebraic interior,relative topological interior or quasi relative interior is not empty.In this paper,in a real linear space,approximate weakly efficient solutions of set-valued equilibrium problems are introduced in terms of ordering cones' relative algebraic interior,relative topological interior and quasi relative interior,respectively.Some properties are discussed for Minkowski nonlinear function.Optimality conditions for approximate weakly efficient solutions of unconstraint set-valued equilibrium problems are established in terms of ordering cones' relative algebraic interior,relative topological interior and quasi relative interior,separately.In addition,the necessary and sufficient conditions for approximate weakly efficient solutions are established to set-valued equilibrium problems with constraints in terms of ordering cones' relative algebraic interior.In this paper,in a real linear space,approximate Henig efficient solutions of set-valued equilibrium problems are first introduced via ordering cones' relative algebraic interior.Then,by using Minkowski nonlinear function,optimality conditions for unconstraint and constraint approximate Henig efficient solutions are established,respectively.