New Second-order Derivatives Of Set-valued Mappings And Its Application In Vector Optimization

The optimality conditions and sensitivity analysis and stability analysis of set-valued optimization problems are important parts of set-valued optimization theory researching.In this thesis,we mainly study the second-order derivatives type optimality conditions of unconstrained set-valued optimization problems and constrained set-valued optimization problems,second-order sensitivity analysis and stability analysis of parameterized vector optimization problems.And the specific content is as follows:1.By using the radial cone and weak efficiency,we first introduce the secondorder weakly composed radial epiderivatives of set-valued maps,which optimizes the epiderivatives of set-valued maps in some references,and also establish some important properties and existence theory of the epiderivatives.Then,by virtue of the second-order weakly composed radial epiderivatives of set-valued maps and its properties,we establish a second-order optimality necessary condition of Benson proper efficient solutions for constrained set-valued optimization problems under the assumption of generalized cone-convexity,and also obtain a second-order optimality sufficient condition and a second-order optimality necessary condition of Benson proper efficient solutions for constrained set-valued optimization problems without any assumption of convexity,and some results we obtained improve the corresponding ones in some literatures.2.We introduce the generalized second-order composed radial epiderivatives of set-valued mappings by virtue of the radial cone and efficiency,which optimizes the epiderivatives of set-valued maps in some references,and discuss existence theory and some basic properties of the epiderivatives.Then,under the assumptions of generalized cone-convexity or without of convexity,we investigate the second-order sufficient optimality conditions and the second-order necessary optimality conditions of weakly efficient solutions for unconstrained set-valued optimization problems and Henig proper efficient solutions for constrained set-valued optimization problems by using the generalized second-order composed radial epiderivatives of set-valued mappings,and some results we obtained improve and imply the corresponding ones in some literatures.At the same time,we give specific examples to verify some main results we obtained.3.We introduce the second-order composed radial contingent derivatives of set-valued maps based on the radial cone and contingent cone,and investigate some significant properties of the derivatives,such as chain rules.Then,by virtue of the second-order composed radial contingent derivatives of set-valued maps and some of its properties,we establish the second-order sensitivity results of the second-order derivatives of the perturbation map for parametric vector optimization problems.In addition,under suitable assumptions,we investigate the upper semicontinuity and lower semicontinuity of the second-order derivatives of the perturbation map for parametric vector optimization problems.