Set-Valued Optimization And Bilevel Set-Valued Optimization:Theory And Algorithm

In this dissertation,we mainly consider set-valued optimization problems and bilevel set-valued optimization problems.With the help of concepts of second-order and higher-order derivatives and subdifferential,optimality conditions,sensitivity analysis and approaches of transforming the bilevel optimization problem into a single level optimization problem are studied.For some special set-valued optimization problems and bilevel set-valued optimization problems,we further apply characterization techniques to design some effective solution methods.The main contents of this dissertation can be summarized as follows:Firstly,concepts of new second-order composed derivatives and higher-order inner derivatives of set-valued mappings are introduced,and some important properties are discussed,for instance,convexity,subadditivity,Lipschitz property and chain rule.Moreover,we apply the separation theorem of convex sets,directional compactness and domination property to establish KKT optimality conditions,in which the derivatives of the objective and the constraint functions are considered in separated ways,and some results focused on sensitivity analysis.Next,a class of nonsmooth semivectorial bilevel optimization problems with a convex lower level problem violating the Slater constraint qualification are considered.The perturbed KKT transformation and the complementary approximate KKT transformation are proposed to transform the bilevel optimization problem into a single-level optimization problem,and then the approximate solution of the original bilevel optimization problem is obtained.Moreover,the numerical results show that our proposed approaches are effective.Finally,the lower-level optimal value transformation and the KKT transformation are presented to transform the semi-set-valued bilevel optimization problem into a single-level optimization problem based on two different characterization techniques.And then necessary optimality conditions for the semi-set-valued bilevel optimization problem are established.Moreover,the numerical results show that our proposed approaches are effective.