Optimality Of Approximate Solutions For A Class Of Nonsmooth Vector Equilibrium Problems

The vector equilibrium problems are of great extensions for vector optimization,vector variational inequality and complementarity problems.It has become one of the most important topics in the study of applicable mathematics because of its fields of applications such as mathematical programming and management science.It is well known that the exact solutions of the mathematical model usually don’t exist.However,the existences of approximate solutions are often weak.Therefore,it is of great significance to study optimality of approximate solutions for vector equilibrium problems.It is worth noting that most mathematical models are usually non-smooth,that is,the objective and constraint functions are nondifferentiable.Then,it has theoretical value and practical significance to study nonsmooth problems.The purpose of this thesis are to study the optimality of approximate solutions for a class of nonsmooth vector equilibrium problem,details of this article include:1.The optimality and scalarization for approximate solutions to a constrained vector equilibrium problem is devoted.The optimality conditions are given in terms of Michel-Penot subdifferentials,and the scalarization theorems are proposed via a strongly monotone cone convex function.We firstly establish a necessary condition for approximate quasi weakly efficient solution.Furthermore,a new definition for approximately pseudo-convex functions is introduced,and by utilizing it,a sufficient condition for approximate quasi weakly(Benson proper)efficiency is examined.Finally,by using the properties of Bishop-Phelps cone,we present the scalarization theorems for approximate quasi weakly(Benson proper)efficient solutions.2.The optimality condition and scalarization theorem for approximate quasi Henig efficient solutions to a constrained vector equilibrium problem is derived.Firstly,a newly generalized convexity is defined and its properties are discussed.Secondly,based on the generalized convexity and convex separation theorems,the optimality condition of approximate quasi Henig efficient solution is established.Meanwhile,we obtain the scalarization theorem of approximate quasi Henig efficient solution by a class linear function.3.The optimality and duality for robust approximate solutions to a Non-smooth Semiinfinite Multiobjective Optimization Problem(NSMOP)is investigated.Under suitable constraint qualifications,necessary conditions for robust approximate quasi weakly efficient solutions in terms of Clarke subdifferentials are established.Then,a newly generalized convexity is introduced,and it is applied to examine a sufficient condition for robust approximate quasi weakly efficient solutions.Finally,a Mond-Weir type approximate dual is introduced,and its relationship with(NSMOP)is also discussed.