Cone Characterization Of Solutions In Vector Optimization And Approximate Solutions Of Vector Variational Inequalities

Vector optimization theory and method play an important role in solving the opti-mal decision problem.It is widely used in many fields such as mathematical economy,communication engineering,traffic design and internet.Therefore,vector optimization theory and method have important guiding significance for many application fields.Hav-ing been studied in depth,its theory and methods can promote the development of social science and technology.Researching vector optimization problems requires a large num-ber of mathematical tools and methods,involving convex analysis,optimization basis,variational inequality and many other disciplines.In this paper,we mainly study some sufficient conditions and necessary conditions for Pareto proper effective point,Pareto effective point and strong effective point of vector optimization problem.The topological properties of the co-radiant set are studied.We have studied some properties and nonlin-ear scalarizations of approximate solutions to the vector variational inequality problem.The main research contents of this paper are as follows:In the first part,the concept of strong effective points is given by general cone order in the locally convex Hausdorff space.We use different cones that are contingent,tangent,normal cones to study the necessary and sufficient conditions for the Pareto proper effective point and strong effective point under the convex condition.Under the non-convex conditions,the sufficient conditions for the Pareto effective point and the necessary conditions for strong effective points are studied.In the second part,the co-radiant set that is an important tool to research the approximate solution of the vector optimization problem proposed by Gutierrez et al.Its topological property is studied in the topological linear space.Under the generalized convexity condition,we obtained the topological properties of the approximate convex co-radiant set.Under convex conditions,the topological linear properties of convex co-radiant sets are obtained and we give some examples illustrate the main results.In the third part,in the Banach space,we study some properties of a class of approx-imate solutions for(weak)vector variational inequalities problems proposed by co-radiant sets,and obtain the relationship between the(C,?)-approximate solutions of the vector variational inequalities problems and the(C,?)-approximate solutions of the weak vector variational inequalities problems.We also obtain the relationship of(C,?)-approximate solution sets of vector variational inequality problems.We also use the nonlinear s-calarization function which are Gerstewitz function and Hiriart-Urruty function to scalar the vector variational inequality problems.We use the scalar problems(Pi)and the s-calar problems(P3)to obtain the necessary conditions for(C,?)-approximate solution of the(weak)vector variational inequality problems.We use the approximate solution of the scalar problems(P2)to equivalently characterize(C,?)-approximate solutions of the(weak)vector variational inequality problems.In addition,the relationship between the approximate solution of the generalized vector variational inequality problem and the ap-proximate solution of the vector optimization problem is also studied in finite dimension Euclidean space.