Exceptional Family Of Elements For The Vector Optimization Problems On Hadamard Manifolds

Vector optimization theory has wide applications in economic management,transporta-tion,engineering technology,and many other fields.In recent years,the existence and boundedness of optimization problems(including vector optimization problems)have been conducted in-depth research by many scholars,and achieved fruitful results.In this thesis,we focus on existencess and boundedness of weakly efficient solutions for vector optimization problem and vector variational inequality by the method of using the exceptional family of elements on Hadamard manifolds.The thesis consists of four chapters,it is organized as follows:In Chapter 1,we introduce the backgrounds and developments of the vector optimization problem,vector variational inequality and the exceptional family of elements.Moreover,we recall some conceptions and lemmas which are used by this thesis.In Chapter 2,the exceptional family of elements was used to study the geodesic convex optimization problems on Hadamard manifolds.Firstly,we introduce a conception of the exceptional family of elements and some lemmas for the geodesic convex optimization prob-lems on Hadamard manifolds.Secondly,we prove the equivalence between the existence of solution to geodesic convex optimization problem and the nonexistence exceptional family elements.Finally,we introduce some coercivity conditions to guarantee the existence of a solution for geodesic convex optimization problem on Hadamard manifolds,study the re-lations between these coercivity conditions and the the nonexistence exceptional family of elements.In Chapter 3,we focus on existencess and boundedness of weakly efficient solutions for vector optimization problem on Hadamard manifolds.Firstly,we proved the vector optimiza-tion problem exist weakly efficient solutions and its equivalent scalar optimization problem has a solution on Hadamard manifolds.Secondly,we defined a conception of the exceptional family of elements for the vector optimization problems on Hadamard manifolds,through the vector optimization problem is transformed into a class of scalar optimization problem,and then prove that the existence of weakly efficient solutions for the vector optimization problem if and only if there not exist the exceptional family of elements,and gives a number of conditions that do not exist the exceptional family of elements.Finally,we also study the sufficient and necessary conditions for the vector optimization problems of weakly efficient solution set is not empty and bounded.In Chapter 4,we focus on existencess of weakly efficient solutions for vector set-valued mappings variational inequalities on Hadamard manifolds.Firstly,we proved the vector vari-ational inequalities existence of weakly efficient solutions and its equivalent scalar variational inequalities has a solution on Hadamard manifolds,and we defined a conception of the ex-ceptional family of elements for the vector variational inequalities on Hadamard manifolds.according to concern between variational inequality problems with the exceptional family of elements,we can proved that the nonexistence of exceptional family elements implies the existence of weakly efficient solutions for Kuratowski upper semicontinuous vector set-valued mappings variational inequalities on Hadamard manifolds.Secondly,we gives a number of conditions that do not exist the exceptional family of elements for vector variational in-equalities,thus we obtained some sufficient conditions for the existence of weakly efficient solutions for the vector variational inequalities on Hadamard manifolds.Finally,we got the vector optimization problems existence of weakly efficient solutions equivalent to that type of the existence of solutions for vector variational inequalities.The main feature of this paper is the first article take advantage of the exceptional family of elements method of vector optimization problems and vector variational inequality problems of weakly efficient solution on Hadamard manifolds,and obtain nonexistence of weakly efficient solutions implies the existence of exceptional family elements.