Study On Some Characterizations Of Solutions In Set-valued Optimization

Vector optimization is an important branch of mathematical programming,setvalued optimization is an important part of vector optimization,which is widely applied in many areas of mathematical economics,financial management,survival theory,engineering science,military decision making,and so on.Study about the topic involves many disciplines,such as: set-valued analysis,variational analysis,convey analysis,nonlinear analysis,nonsmooth analysis and partial ordering theory,etc.In this thesis,we mainly study the characterizations of solutions of set-valued optimization problems in four aspects: the characterizations of nonlinear scalarization without any convexity conditions;the characterizations of Linear scalarization,Lagrange multiplier theorem,duality theorem and saddle point theorem under generalized convexity;the connectedness of solution sets in the sense of linear scalarization;the differential properties of sets maps–sensitivity.This thesis includes five parts as follows:In the first part,firstly,we establish nonlinear separation theorem by virtue of Gerstewit nonlinear scalarization functions in real linear spaces without any topological structure.Then,we give nonlinear scalarization characterizations of weak efficient solutions and Benson proper efficient solutions by the nonconvex separation theorem without any convexity assumption.In the second part,firstly,we propose a new class of generalized cone convex setvalued maps called relatively solid generalized cone subconvexlike in real linear spaces and prove the relationships between it and previous generalized cone convexity.Then,under the assumption of relatively solid generalized cone subconvexlike,the characterizations of Benson proper efficiency of set-valued optimization problem are established by means of linear scalarization,Lagrange multipliers theorems,saddle points theorems and duality theorems.In the third part,firstly,we propose the concept of nearly E-subconvexlike setvalued maps,under the assumption of E-subconvexlike in real linear spaces,we establish linear scalarization theorems,Lagrange multiplier theorems of E-global proper efficient solutions and Optimality conditions of E-weak efficient solutions.Then,we study the nonlinear scalarization characterizations of E-global proper efficient solutions by Gerstewit nonlinear scalarization functions and the nonconvex separation theorem.In the forth part,firstly,under the the assumption of E-subconvexlike of the binary function in real locally convex Hausdorff topological vector spaces,we establish the linear scalarization of weak efficient solutions,Benson proper efficient solutions,Heing proper efficient solutions of set-valued vector equilibrium problems under improvement sets.Then,by means of the linear scalarization results,we obtain the connectedness of weak efficient solutions sets,Benson proper efficient solutions sets,Heing proper efficient solutions sets of set-valued vector equilibrium problems.In the last part,we study the sensitivity of proper perturbation maps of set-valued optimization problems by employing second-order composed contingent derivative in Banach spaces.Firstly,we give a feasible set map and defined a proper perturbation map(sets map)based on Benson proper efficient solutions,discuss relationships between second-order composed contingent derivative of the feasible set map and that of its profile map.Then,by using a standard separation theorem for convex sets,we discuss the relationships between the second-order composed contingent derivative of proper perturbation maps and the set of Benson proper minimal points of the second-order composed contingent derivative of the feasible set maps.