Kernelled And Adjoint Kernelled Quasidifferential:Research And Applications

Quasidifferential calculus has been widely used in nonsmooth analysis and optimization.In quasidifferential calculus,the problem of quasidifferential of quasidifferentiable function be-ing not unique is essential.In order to solve this problem,this disseration focuses on finding a subclass of quasidifferentiable function class,which has good algebraic properties.The size of the subclass is as large as possible and there is an automatic and effective method of select-ing the representation of the quasidifferential of the quasidifferentiable function in the subclass.In this disseration,firstly,a method of selecting the representation of the quasidifferential is constructed,and by which a subclass of quasidifferentiable function class is defined.Then,through exploring the algebric properties of the subclass,the method of selecting the quasid-ifferential representation element can be improved,and a new subclass of quasidifferentiable function class is defined.The process repeats untill the expected function class is founded.The optimality theory of the optimization problem of the function in the expected function class is established.The main results of this disseration can be summarized as follows:1.In Chapter 3,in n-dimensional Euclidean space,a new important property of Demyanov difference of convex compact sets is given.Some properties of the tangent and normal cones of convex compact set at its extreme points are given.Several properties of the support function of convex compact set are proposed.Some sufficient conditions and necessary and sufficient con-ditions for Demyanov difference and Minkowski difference of convex compact sets being equal are given.The kernelled quasidifferential of quasidifferentiable function and K-differentiable function are redefined,and some relevant operational properties are given.The existence of the kernelled quasidifferential is explored,some sufficient conditions are proposed.Under some condition,a formula of the kernelled quasidifferential is presented.The structure of K-differentiable function class is analysed.2.In Chapter 4,the notions of adjoint kernelled quasidifferential of quasidifferentiable function and K*-differentiable function are proposed.Some operational properties of ad-joint kernelled quasidifferential and K*-differentiable function are given.The existence of the adjoint kernelled quasidifferential is explored,some sufficient conditions are given.Un-der some condition,a formula of the adjoint kernelled quasidifferential is presented.The structure of K*-differentiable function class is analysed.With the help of exploring the re-lationship between K-differentiable function and K*-differentiable function,the notion of K°-differentiable function is proposed.Some relevant operational properties and the struc-ture of K°-differentiable function class are given.The expectations of the functions from a class of random K°-differen-tiable functions are proved to be K°-differentiable functions.3.In Chapter 5,based on the operational properties of K°-differentiable function and the theory of kernelled quasidifferential and adjoint kernelled quasidifferential,the optimality con-ditions for K°-differentiable optimization problems are considered.The first-order optimality conditions for unconstrained K°-differentiable optimization problems are given.Optimality conditions for inequality constrained K°-differentiable optimization,including geometry type optimality condition and Fritz John necessary of optimality condition,are proposed.Under regular conditions,optimality conditions for K-differentiable optimization with equality and inequality constrained,including geometry type optimality condition and Fritz John necessary of optimality condition,are given.Measureing the risk in VaR and CVaR,a model on cor-porate investment and financial decisions is constructed,which is a random K°-differentiable optimization problem.