Optimality And Scalarization Of Approximate Solutions For Vector Equilibrium Problems

Vector equilibrium is a class of generalized mathematical models including vector variational inequalities,vector optimization,fixed point theory and complementarity problems,it is widely used in the fields of applied mathematics and operations management.The optimality condition is one of the important contents of optimization theory,and the scalarization is a powerful tool to realize the transformation of vector and numerical equivalence,it is of great significance to study the optimality conditions and scalarization of approximate solutions to vector equilibrium problems.Details of this article include:1.The optimality conditions of approximate quasi globally proper efficient solutions are studied.Firstly,the necessary condition of approximate quasi globally proper efficient solution is investigated by utilizing the separation theorem with respect to the quasi relative interior of convex set.Secondly,based on the Clarke subdifferential,the concept of approximate quasi pseudoconvex function is introduced.Under the assumption of introduced convexity,the sufficient condition is also presented.Finally,by using Tammer's function and constructing the nonlinear functional with mild conditions,the scalarization theorems of approximate quasi globally proper efficient solution are proposed.2.The optimality conditions in term of convexificators of approximate quasi weakly efficient solution are studied.Firstly,the necessary condition of approximate quasi weakly efficient solution is received by using the properties of convexificators.Secondly,the notion of approximate pseudoconvex function in the form of convexificators is introduced,and under the introduced generalized convexity assumption,a sufficient condition of approximate quasi weakly efficient solution is established.Finally,taking advantage of the special lemma with Tammer's function,the scalar characterizations of approximate quasi weakly efficient solution is described.3.The optimality conditions and duality theorems of approximate quasi weakly efficient solutions for constrained problems are studied.Firstly,the necessary condition for approximate quasi weakly efficient solution in approximate subdifferential form are established.Secondly,under the condition of quasi-pseudo-type-I function,the sufficient condition of quasi weakly efficient solution is obtained.Finally,the generalized approximate Mond-Weir dual model of constrained problem is introduced,and the weak,strong and inverse duality theorems of approximate quasi weakly efficient solution between the model and the original problem are established.